On Utility of Wealth Maximization
On Utility of Wealth Maximization Bahman Angoshtari August 18, 2009
A DISSERTATION SUBMITTED FOR THE DEGREE OF Master of Science in Applied Mathematics (Financial Engineering) UNDER SUPERVISION OF Prof. A. Bagchi and Prof. F. Jamshidian
Abstract This project is concerned with the continuous-time portfolio choice problem, also known as Merton's problem, when the opportunity set is stochastic (e.g. when the interest rate and/or volatility is stochastic). There are two main approaches for solving continuous-time portfolio problems: the classical stochastic control approach and the so called martingale approach. The main contribution of this project is to develop a new approach, called the
direct approach.
Unlike the stochastic control approach, it is not based on the Markov state assumption and can be extended to the general semimartingale market (though we have not tried to do so). Its advantage over the martingale approach is that the direct approach, as its name suggests, is dealing with the primal problem
directly.
So, unlike the martingale approach, the completeness
or incompleteness of the market has not so much aect on it. Furthermore we are able to obtain the general form of the optimal portfolio policy directly, and not through a dual problem.
Contents 1 Introduction
2
2 Preliminaries
6
2.1
The market model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.2
Consumption pairs, wealth processes and self- nancing strategies . . . . . . . . .
8
2.3
Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3 Stochastic Control Theory
11
4 Martingale or Duality Approach
14
4.1
Proof of the Martingale Approach . . . . . . . . . . . . . . . . . . . . . . . . . .
5 The Direct Approach 5.1
15
22
Proof of the Direct approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Examples
24
32
6.1
Logarithmic Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
6.2
Power Utility
35
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1
Original Merton Setting
6.2.2
Complete Gaussian Market
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36 40
7 Conclusions and Recommendations
47
A Stochastic Exponential and Logarithms
49
B Other Results
51
Bibliography
54
1
Chapter 1 Introduction∗ A fundamental concern for investors is the problem of portfolio optimization when they trade between a risk-free asset and a large number of risky assets. One of the most important decisions many people face is the choice of a portfolio of assets for retirement savings. These assets may be held as a supplement to de ned-bene t public or private pension plans; or they may be accumulated in a de ned contribution pension plan, as the major source of retirement income. In either case, a dizzying array of assets is available. Institutional investors also face complex decisions. Some institutions invest on behalf of their clients, like pension funds and insurance companies. Others, such as foundations and university endowments, are similar to individuals in that they seek to nance a long-term stream of discretionary spending.
The investment
options for these institutions have also expanded enormously since the days when a portfolio of government bonds was a norm. Modern nance theory is often thought to have started with the mean-variance analysis of Markowitz in the early 1950s (see [63]). This made the portfolio choice theory the original subject of modern nance. He introduced the mean-variance as a criterion for portfolio selection. It is the criterion mostly used by practitioners in spite of its unrealistic hypothesis such as normal returns, single period, etc. The continuous-time portfolio problem has its origin in the pioneering work of Merton [64, 65]. It is concerned with nding the optimal investment strategy of an investor. In other words, how much of which security she should hold at every time instant between now and a time horizon
T,
to maximize her expected utility from intermediate consumption and accumulated
wealth at the end of the time horizon. In the classical Merton problem the investor can allocate her money into a risk-less savings account and
d
dierent risky stocks. Using the methods of
stochastic optimal control, Merton derived a nonlinear partial dierential equation (Bellman equation) for the value function of the optimization problem. He also produced the closed-form solution of this equation for the special cases of power, logarithmic, and exponential utility functions. A drawback of his approach, however, was the assumption of a constant investment opportunity set (i.e.
constant or deterministic factors).
This made the model unrealistic,
speci cally for long-term investors. There are two main approaches for solving continuous-time portfolio problems: the classical stochastic control approach and the so called martingale approach. The stochastic control approach, which has been used since the seminal work of Merton, is based on the requirement of Markov state processes and involves solving a highly nonlinear PDE, the so called ∗
This literature study is partly excerpted from Zariphopoulou [87].
2
Hamilton-Jacobi-Bellman (HJB) equation.
The strength of this approach is in its access to
the well-developed theory of PDEs and the corresponding numerical techniques.
Using this
approach, the cases of special utilities (namely, the exponential, power and logarithmic) have been extensively analyzed. In these cases, convenient scaling properties reduce the associated HJB equation to a more tractable quasilinear one. However, when the utility function is general, very little is known about the maximal expected utility as well as the form and properties of the optimal policies once the log-normality assumption is relaxed and correlation between the stock and the factor is introduced. This is despite the Markovian nature of the problem at hand, the advances in the theories of fully nonlinear PDEs and stochastic control, and the computational tools that exist today. Speci cally, general results on the validity of the Dynamic Programming Principle, regularity of the value function, existence and veri cation of optimal feedback controls, representation of the value function and numerical approximations are still lacking. A more recent approach to the problem of expected utility maximization, which permits us to avoid the assumption of Markov asset prices, is based on duality characterizations of portfolios provided by the set of martingale measures.
For the case of a complete nancial
market, where the set of martingale measures is a singleton, this
martingale methodology
was
developed by Pliska [72], Cox and Huang [24, 25] and Karatzas, Lehoczky and Shreve [46]. Considerably more di cult is the case of incomplete nancial models.
It was studied in a
continuous-time diusion model by He and Pearson [43] and by Karatzas, Lehoczky, Shreve and Xu [47]. The central idea here is to solve a dual variational problem and then to nd the solution of the original problem by convex duality, similarly to the case of a complete model. This powerful approach is applicable to general market models and yields elegant results for the value function and optimal consumption and terminal wealth. However, the optimal portfolio must be then characterized via martingale representation results for the optimal wealth process, so little can be said about the structure and properties of the optimal investments. Another weakness of this approach is the lack of appropriate numerical methods. The lack of rigorous results for the value function when the utility function is general limits our understanding of the optimal policies. Informally speaking, the optimal portfolio consists of two components.
The rst is the so-called myopic portfolio and has the same functional
form as the one in the classical Merton problem.
The second component, usually referred
to as the excess hedging demand, is generated by the stochastic factor.
Conceptually, very
little is understood about this term. In addition, the sum of the two components may become zero which implies that it is optimal for a risk averse investor not to invest in a risky asset with positive risk premium. A satisfactory explanation for this counter-intuitive phenomenon, related to the so-called market participation puzzle (see [5, 18, 44]), is also lacking. Besides these di culties, there are other issues that limit the development of an optimal
static
investment theory in complex market environments.
One of them is the
choice of the
utility function at the speci c investment horizon.
Direct consequences of this choice are,
from one hand, the lack of exibility to revise the risk preferences at other times and, from the other, the inability to assess the performance of investment strategies beyond the prespeci ed horizon. Addressing these limitations has been the subject of a number of studies and various approaches have been proposed. With regards to the horizon length, the most popular alternative has been the formulation of the investment problem in
(0, +∞)
and incorporating
either intermediate consumption or optimizing the investors long-term optimal behavior (see, among others, [48, 81]).
Investment models with random horizon have also been examined
3
([22]). The revision of risk preferences has been partially addressed by recursive utilities (see, for example, [32, 76, 77]). Recently, Musiela and Zariphopoulou developed a forward performance criterion which addresses both issues of the horizon length and revision of risk preferences (see [68, 87]). Let us now focus on the main subject of this project, the use of stochastic factor models in continuous-time portfolio choice problems, and brie y discuss the existing body of work. Stochastic factors have been used in portfolio choice to model asset predictability, stochastic volatility, and interest rates. The predictability of stock returns was rst discussed in [34, 35, 38]; see also [9, 10, 14, 15]. More complex models were analyzed in [1, 8]. The role of stochastic volatility in investment decisions was studied in [3, 21, 38, 39, 42, 70, 78], and others. Models that combine predictability and stochastic volatility were analyzed, among others, in [50, 54, 61, 86, 71]. In a dierent modeling direction, stochastic factors have been incorporated in asset allocation models with stochastic interest rates (see, for example, [11, 12, 16, 23, 26, 28, 30, 75, 79, 83]). From the technical point of view, the analysis is not much dierent. However, various technically interesting questions arise (see, for example, [52, 54, 73]). Classical textbooks on the subject include [17, 31, 48], among others. More speci cally, Korn and Kraft ([52]), Zariphopoulou [86], Pham [71], and Fleming and Hernández-Hernández [36] used stochastic control approach to handle the optimal consumption and asset allocation problems with stochastic opportunity set. However they took some restrictive assumptions on stochastic factors as well as the market price of risk, which excludes models such as Heston's stochastic volatility model. Kramkov and Schachermayer ([55, 56]) provided minimal conditions for the validity of martingale approach on a general semimartingale market. Korn and Kraft ([53]) provided some counter examples to highlight the fact that uncritical application of the two main approaches of solving continuous-time portfolio problems can lead to wrong conclusions if only the necessary and not the su cient conditions of the main results are checked. Chacko and Viceira [21] considered recursive utility (including power utility) over intermediate consumption.
They assumed a stochastic volatility model where the reciprocal
of volatility follows a mean-reverting square-root process which is instantly correlated with stock returns.
They derived analytic expressions for the optimal consumption and portfolio
policies which where exact for the case of power utility and approximate for the case of recursive utility. Castaneda and Hernandez ([20]) used a combination of martingale approach and stochastic control theory to nd explicit solutions for power and logarithmic utility functions. Kraft ([54]) proved a veri cation result which covers Heston's stochastic volatility model for the power utility. Ekeland and Ta in ([33]) studied the problem of optimal portfolio choice in a bond market described in the general HJM framework. They proved the existence of an optimal portfolio in two cases: when the driving Wiener process is nite-dimensional and when the Wiener process is in nite dimensional but the market price of risk is deterministic. Ringer and Tehranchi ([73]) considered the same problem, but with a Markovian HeathJarrowMorton model of the interest rate term structure driven by an in nite-dimensional Wiener process. They gave su cient conditions for the existence and uniqueness of an optimal trading strategy. Karatzas and Kardaras ([45]) introduced the concept of whose wealth appears
better
numeraire portfolio, a trading strategy
when compared to the wealth generated by any other strategy, in
the sense that the ratio of the two processes is a supermartingale. They derived necessary and su cient conditions for the numeraire portfolio to exist. Liu ([61]) solved dynamic portfolio choice problems, up to the solution of an ordinary dierential equation (ODE), when the asset returns are quadratic and the agent has a power utility function.
4
He also considered three
special cases of his model: a pure bond portfolio problem where the bond returns is described by quadratic term structure model (which includes Gaussian and CIR models), a pure stock portfolio problem when the volatility follows Heston model, and a mixed bond-stock portfolio problem with quadratic term structure and Heston stochastic volatility. The main contribution of this project is developing a new approach to the continuous-time optimal consumption and terminal wealth problem with stochastic opportunity set. It will be called the
direct approach.
It is based on probabilistic arguments, and it can potentially be
extended to the general semimartingale market, though we have not tried to do so.
Hence,
unlike the stochastic control approach, it is not based on the Markov state assumption and it can be used to obtain general results. Its advantage over the martingale approach is that the direct approach, as its name suggests, is dealing with the primal problem
directly.
So, unlike
the martingale approach, the completeness or incompleteness of the market has not so much aect on it. Furthermore we are able to obtain the general form of the optimal portfolio policy directly, and not through the dual problem. The rest of the report is organized as follows. In chapter 2 we de ned the market speci cations, derived some elementary results, and formalized the portfolio choice problem. Chapter 3 is a literature review about stochastic control approach to portfolio choice. In chapter 4 we present, axiomatically, the martingale approach for incomplete markets. Chapter 5, on generalities of the direct approach, contains the main results of the project. In chapter 6, we considered three special cases: the case of logarithmic utility under general speci cation of the market, the case of power utility under Black-Scholes type market (the original Merton setting), and the case of power utility with Gaussian term structure. Finally in chapter 7 we have summarized the main results and suggested some topics for further research. For the sake of completeness, we have included, in the appendices, some results on stochastic exponentials and logarithms, change of measure, and Gaussian term structure models.
5
Chapter 2 Preliminaries 2.1 The market model We start by de ning the market. This de nition will be used throughout chapter 4, chapter 5 and section 6.1.
But in chapter 3 and section 6.2, we will take some special cases of the
de nition which will be de ned separately. Note that the market is allowed to be
stochastic and adapted price processes are without jump.
the coe cients are all
(but
not necessarily Markov
incomplete,
processes), and the
De nition 2.1. (General market model) Consider a stochastic basis Ω, F, (Ft )t∈[0,T ] , P Ft
is the ltration generated by
k
independent Brownian motions
The market consists of a bank account process
B = (Bt ),
(e.g. bonds, stock, etc.) represented by the price process take
n ≤ k.
W =
, where T .
(1) (k) Wt , . . . , Wt
n other zero-dividend assets T (1) (n) A = At , . . . , At . We always and
Assume that the price processes are positive continuous semi-martingales (Itô
processes) with the following dynamics:
´. ´. µ ds + 0 Σs dW s t , At = A 0E 0 s ´t Bt = e 0 rs ds .
(2.1)
r, the short rate, is an adapted and integrable process. µ, the drift term in real measure, n × 1 adapted and integrable vector process. Finally, for all t ∈ [0, T ], Σt is an n × k > adapted and almost surely of rank n for all t ∈ [0, T ], and Σt Σt is integrable.
Here
is an
Remark
2.2
.
Note that if
n = k,
then the market is complete. In this case
Σ−1 t
exists for all
t ∈ [0, T ]. In the next theorem we parametrize all equivalent martingale measures (EMMs) and state price densities (SPDs) of the market.
Theorem 2.3. (Parametrization of EMMs and SPDs) Consider the market of de nition 2.1. T (k) For any EMM Q, there exist a unique predictable process λ = λ(1) , . . . , λ satisfying the t t system Σλ = µ − r1n×1 ,
6
(2.2)
such that Q can be written as (λ)
dQ dP
= ´ZT , . = E − 0 λ> s dW s .
Z (λ)
(2.3)
Furthermore the corresponding SPD is given by πt = e
−
´t 0
rs ds
ˆ . > λs dW s , E − 0
and the process
f W
(2.4)
t
de ned by
ˆ
t
λs ds,
ft , Wt + W
(2.5)
0
is a Q-Brownian motion. Proof.
Any measure
Q,
equivalent to
P,
Z (λ) for some predictable process
Zt , Et
can be written as:
(λ)
dQ dP
=´ZT . = E − 0 λ> s dW s ,
> (1) (k) λ = λt , . . . , λ t .
(2.6)
To see this we may start by de ning the
dQ
. Then by the representation property of W , the stochastic logarithm dP ´. > of Zt can be represented by Log (Z) = − λ dW s for some predictable process λ. 0 s Z (λ) ). Theorem B.2 in the Equation (2.4) follows from the de nition of SPD (i.e. π , B h i martingale
appendix implies that follows that
f W
f W
is in fact a
de ned by equation (2.5) is a
Q-martingale.
Since
f W
= t,
it
Q-Brownian motion (this result can also be obtained by the Girsanov
theorem).
e , A , and note that Q is an EMM if the process A B e can be found as follow: A
Consider the discounted price processes
e A
is a
Q-martingale.
The
e = A
Q-dynamics
1 A B
= e
−
´.
0 ru du
ˆ E
of
ˆ
.
.
µu du +
Σu dW u ˆ . ˆ . = E (µu − ru 1n×1 ) du + Σu dW u 0 ˆ0 . ˆ . f u − λu du = E (µu − ru 1n×1 ) du + Σ u dW 0 0 ˆ . ˆ . f = E (µu − ru 1n×1 − Σu λ) du + Σ u dW u . 0
0
0 It follows that
e A
is a
Q-martingale
0 only if equation (2.2) holds.
7
(2.7)
n < k (incomplete market case), then we may use equation (2.2) to express n λ in terms of (k − n) other elements of λ, and in this way we nd a parametrization in terms of (k − n) market price of risk processes. On the other hand, if n = k (the
Note that if elements of of EMMs
complete market), then equation(2.2) gives us the unique market price of risk process as:
λt = Σ−1 t (µt − rt 1n×1 ) , Σ−1 t bt .
(2.8)
2.2 Consumption pairs, wealth processes and self- nancing strategies Next, we rede ne the well-established concept of self- nancing strategies. We will use equation(2.10) (u,c) of remark 2.5 frequently throughout the chapters 4, 5, and 6. Also note that we used W
W
for the wealth process, while
De nition 2.4.
w
Take
is used for the Brownian motion.
process for which the stochastic integrals
c = (ct )t∈[0,T ]
well-de ned. Also let process
W (u,c) = W
(u,c)
(u,c) Wt
> (1) (n) ut , . . . , u t be an adapted t∈[0,T ] ´. > ´. are u (diag (A))−1 dA, and 0 (1 − u> 1n×1 ) dB B 0
as the initial wealth.
be a non-negative predictable process. Note that for the
, de ned by
t∈[0,T ]
ˆ
ˆ
.
−1
>
u (diag (A))
, wE
(u,c)
Wt
≥ 0
dA + 0
We also de ne
u,
SF (w)
2.5
.
We call
dB (1 − u 1n×1 ) − B
W (u,c)
(u, c), wealth w .
c
given initial
From equation (2.1) we have
ˆ
.
cs ds ,
(2.9)
0
the wealth process corresponding to a
the proportional consumption rate process
as the set of all pairs
consumption rate process
Remark
t ∈ [0, T ].
for all
self- nancing strategy
.
>
0 we have
u =
Let
where
u
c,
and initial wealth
w.
is a self- nancing strategy for the
(diag (A))−1 dA = µu du + Σu dW u .
So equation
(2.9) can be rewritten as:
W
(u,c)
ˆ
ˆ
.
u> s
ˆ
.
(1 −
.
−
cs ds ˆ . ˆ . > > = wE us (µs − rs 1n×1 ) + rs − cs ds + us Σs dW s .
= wE
(µs ds + Σs dW s ) +
u> s 1n×1 )rs ds
0
0
0
(2.10)
0
0 Note that we may re-write equation(2.9) as
(u,c) n n (i) P P (i) dAt (i) dWt t ut (i) + (1 − ut ) dB − ct dt, (u,c) , Bt Wt At i=1 i=1 (u,c) W0 = w, Now we may interpret
(i)
ut
as the percentage of wealth invested in the
as the percentage investment in the bank account, and of percentage of wealth.
8
c
(2.11)
i-th
asset,
(1 −
n P
(i)
ut )
i=1 as the rate of consumption in terms
Also note that we may alternatively consider the total rate consumption (u,C) (u,C) a little misuse of notation, the wealth process W = Wt by: t∈[0,T ]
ˆ (u,C) Wt
and de ne, with
t
−1 Ws(u,C) u> dA , w+ s (diag (A)) 0 ˆ t ˆ t dBs (u,C) > Cs ds. − + Ws (1 − us 1n×1 ) Bs 0 0
(2.12)
(u,C)
≥ 0 for all t ∈ [0, T ], we call u a self- nancing strategy for the total consumption C . Obviously we would have W (u,C) ≡ W (u,c) , if we de ne
Then if rate
C
Wt
ct ,
Ct (u,C)
So from now on, we denote the wealth process by total consumption. Whether
c
.
(2.13)
Wt
W (u,c)
for both cases of proportional and
is a proportional or total consumption will be known from the
context.
consumption pair, which formalize the consumption behavior of an entity. We identify those consumption pairs which can be nanced by a self nancing strategy as aordable consumption pairs. We will also need the concept of a
De nition 2.6.
consumption pair
(C, Z) consisting of an adapted non´T negative total consumption rate process C = (Ct )t∈(0,T ] with Cs ds < ∞, and an FT -measurable 0 non-negative random variable describing terminal lump-sum consumption Z . We denote the set of all consumption pairs by C . A
is an ordered pair
De nition 2.7. For a consumption pair (C, Z), initial wealth w and a self- nancing strategy u (u,C) (for the total consumption C ), we say that (C, Z, u) is budget-feasible at w if WT ≥ Z a.s. A consumption pair (C, Z) is aordable with initial wealth w , if there exist a self- nancing strategy u
such that
(C, Z, u)
is budget-feasible at
pairs aordable with initial wealth
w.
We denote by
C(w)
the set of all consumption
w.
2.3 Problem Formulation The traditional criterion for optimal portfolio choice has been based on maximal expected utility (for the historical perspective see [88]). The tradition is either assuming an additively h´ i T time-separable utility function for intermediate consumption of the form E u(t, C )dt (in t 0
optimal consumption problem ), or utility of terminal wealth only E [U (WT )] (the portfolio choice problem with this criterion is also called an asset h´ i T allocation problem ), or the sum of these two components E 0 u(t, Ct )dt + U (WT ) . this case the problem is usually called the
On the other hand, it is worth to mention that it has long been recognized by economists that preferences may not be Intertemporally separable. In particular, the utility associated with the choice of consumption at a given date is likely to depend on past choices of consumption. According to Browning [13], this idea dates back to the 1890 book `Principles of Economics'
9
by Alfred Marshall.
For example high past consumption generates a desire for high current
consumption. Generalizations of standard time-separable preferences that have been suggested
recursive or stochastic dierential utility (see, for example, [32, 76, 77]), habit formation criterion (see [67]), and forward performance criterion (see [68, 87]). include
In this project we adapt the traditional criterion of expected utility of terminal wealth and time-separable utility of intermediate consumption. Speci cally we de ne
utility functionals
as
follows.
De nition 2.8.
J : C → R is de ned by: ˆ T u(t, Ct )dt + U (Z) , J(C, Z) , E
A total utility functional
(2.14)
0 with the following assumption on functions u(., .) and U (.): + - u : [0, T ] × R → R is continuous, and for each t ∈ [0, T ], and concave. + - U : R →
R
u(t, .) : R+ → R
is increasing
is increasing and concave.
u(., .) or U (.) is non-zero. Furthermore, either U is strictly concave or zero, t ∈ [0, T ], u(., t) is strictly concave or zero. refer to the function u (., .) as the consumption utility function, and the function U (.)
- At least one of or for all We as the
terminal utility functions.
We may now give the formal de nition of Merton's problem.
Problem 2.9.
(Merton's Problem) Consider a total utility functional
ˆ J(C, Z) , E
T
u(t, Ct )dt + U (Z) .
(2.15)
0 Then for the initial wealth
w,
we de ne Merton's problem as:
sup
(u,c)
J(cW (u,c) , WT
(u,c)∈SF (w)
10
) .
(2.16)
Chapter 3 Stochastic Control Theory∗ In this chapter we present the stochastic control approach without going into details. For the sake of simplicity, we only consider two assets throughout this chapter: a risky stock and a bank account. As already pointed out in the rst chapter, the underlying assumption of the stochastic control approach is assuming a Markov state processes.
More speci cally we should restrict
our de nition of market as follows. The stock price process
B = (Bt )t≥0
S = (St )t≥0
and the bank account
have the following dynamics:
dSt St dBt Bt The stochastic factor
Y = (Yt )t≥0
(1)
= µ (Yt ) dt + σ (Yt ) dWt , = r (Yt ) dt.
(3.1)
is assumed to satisfy:
p (1) (2) . dYt = b (Yt ) dt + d (Yt ) ρdWt + 1 − ρ2 dWt Here
(1) (2) W t = Wt , Wt
Next we de ne the
is a standard Brownian motion.
value function V (w, y, t; T ) as ˆ
V (w, y, t; T ) =
(3.2)
sup E (u,c)∈SF (w)
T
u(s, cs Ws(u,c) )ds
+
(u,c) (u,c) U (WT )|Wt
= w, Yt = y .
(3.3)
t
As solution of a stochastic optimization problem, the value function is expected to satisfy the Dynamic Programming Principle (DPP), namely for all
V (w, y, t; T ) =
sup E (u,c)∈SF (w)
h
V
s ∈ (t, T ),
Ws(u,c) , Ys , s; T
(u,c) |Wt
i
= w, Yt = y .
(3.4)
This is a fundamental result in optimal control and has been proved for a wide class of optimization problems. For a detailed discussion on the validity and strongest forms of the DPP in problems with controlled diusions, we refer the reader to [37]. Key issues are the measurability and continuity of the value function process as well as the compactness of the set of admissible controls. ∗
It is worth mentioning that a proof speci c to the problem at hand has not been
This chapter is mainly excerpted from Zariphopoulou [87].
11
produced to date. Recently, a weak version of the DPP was proposed by Bouchard and Touzi [7] where conditions related to measurable selection and boundness of controls are relaxed. Besides its technical challenges, the DPP exhibits two important properties of the value function process.
Speci cally,
V (w, y, s; T ), s ∈ [t, T ],
is a supermartingale for an arbitrary
investment strategy and becomes a martingale at an optimum (provided certain integrability conditions hold). One may, then, view
V (w, y, s; T ) as the intermediate (indirect) utility in the
relevant market environment. It is worth noticing, however, that the notions of utility and risk aversion for times
t ∈ [0, T )
are tightly connected to the investment opportunities the investor
has in the speci c market.
Observe that the DPP yields a backward in time algorithm for (u,c) the computation of the maximal utility, starting at expiration with U WT and using the martingale property to compute the solution for earlier times. For this reason, we refer to this formulation of the optimal portfolio choice problem as backward. Fundamental results in the theory of controlled diusions yield that if the value function is smooth enough then it satis es the HJB equation,
Vt + bVy + 21 d2 Vyy 1 2 2 2 + sup w u σ V + {w [(µ − r) u + r] − C} V + wuρσdV + u (t, C) = 0. ww w wy 2 (u,C)∈SF (w)
(3.5) Moreover, optimal policies may be constructed in a feedback form from the rst-order conditions in the HJB equation, provided that the candidate feed-back process is admissible and the wealth SDE has a strong solution when the candidate control is used. The latter usually requires further regularity on the value function. In the reverse direction, a smooth solution of the HJB equation that satis es the appropriate terminal and boundary (or growth) conditions may be identi ed with the value function, provided the solution is unique in the appropriate sense. These results are usually known as the
veri cation theorem
and we refer the reader to [37, 85] for a general
exposition on the subject. In maximal expected utility problems, it is rarely the case that the arguments in either direction of the veri cation theorem can be established. Indeed, it is very di cult to show a priori regularity of the value function, with the main di culties coming from the lack of global Lipschitz regularity of the coe cients of the controlled process with respect to the controls and from the non-compactness of the set of admissible policies.
It is, also, very di cult to
establish existence, uniqueness and regularity of the solutions to the HJB equation.
This is
caused primarily by the presence of the control policy in the volatility of the controlled wealth process which makes the classical assumptions of global Lipschitz conditions of the equation with regards to the non linearities fail.
Additional di culties come from state constraints
and the non-compactness of the admissible set. Regularity results for the value function (3.4) for general utility functions have not been obtained to date except for the special cases of homothetic preferences (see, for example, [36, 54, 66, 71, 86]).
The most general result in
this direction, and in a much more general market model, was recently obtained by Kramkov and Sîrbu [57] where it is shown that the value function is twice dierentiable in the spatial argument but without establishing its continuity. Some answers to the questions related to the characterization of the solutions to the HJB equation may be given if one relaxes the requirement to have classical solutions. An appropriate class of weak solutions turns out to be the so called viscosity solutions ([27, 60, 59], and [82]). The analysis and characterization of the value function in the viscosity sense has been carried out for the special cases of power and exponential utility (see, for example, [86]).
12
However,
proving that the value function is the unique viscosity solution of (3.5) has not been addressed. A key property of viscosity solutions is their robustness (see [59]). If the HJB has a unique viscosity solution (in the appropriate class), robustness can be used to establish convergence of numerical schemes for the value function and the optimal feedback laws.
Such numerical
studies have been carried out successfully for a number of applications. However, for the model at hand, no such studies are available. Numerical results using Monte Carlo techniques have been obtained in [29] for a model more general than the one herein. Important questions arise on the dependence, sensitivity and robustness of the optimal feedback portfolio in terms of the market parameters, the wealth, the level of the stochastic factor and the risk preferences. Such questions are central in nancial economics and have been studied, primarily in simpler models in which intermediate consumption is also incorporated (see, among others, [2, 51, 58, 69], and [74]). For diusion models with and without a stochastic factor, qualitative results can be found in [29, 50, 61, 84], and, recently, in [6] (see, also [62] for a general incomplete market discrete model). However, a qualitative study for general utility functions and/or arbitrary factor dynamics has not been carried out to date. Let's have a review on the existing results for the most frequently used utilities, namely, the exponential, power and logarithmic ones.
They have convenient homogeneity properties
which, in combination with the linearity of the wealth dynamics in the control policies, enable us to reduce the HJB equation to a quasilinear one. Under a
distortion transformation
(see, for
example, [86]) the latter can be linearized and solutions in closed form can be produced using the Feynman-Kac formula. The smoothness of the value function and, in turn, the veri cation of the optimal feedback policies follows easily. Multi-factor models for these preferences have been analyzed by various authors. The theory of BSDE has been successfully used to characterize and represent the solutions of the reduced HJB equation (see [49]). The regularity of its solutions has been studied using PDE arguments by [71] and [66], for power and exponential utilities, respectively. Finally, explicit solutions for a three factor model can be found in [61].
13
Chapter 4 Martingale or Duality Approach The Martingale method has been given increasing attention since it was conducted by Pliska [72], Cox and Huang [24, 25] and Karatzas, Lehoczky and Shreve [46].
Martingale method
allows us to solve the problems of utility maximization in a very elegant manner. However, the Martingale method is not omnipotent. When the market is incomplete, traditional Martingale method will be problematic.
He and Pearson [43], and Karatzas, Lehoczky, Shreve and Xu
[47] generalized the approach to incomplete markets. The central idea here is to solve a dual variational problem and then to nd the solution of the original problem by convex duality. In a general incomplete market, the dual problem is a stochastic variational problem where the control variables are the consumption process
c,
the terminal wealth
Z,
and market price
of risk processλ. Namely,
h´ i T J(C, Z) , E 0 u(t, Ct )dt + U (Z) subject to : h´ i T (λ) (λ) E 0 πt Ct dt + πT Z = w, n o −1 −1 λ = Σ ΣΣT b + Σ> ΣΣ> Σ − Ik×k a(C,Z,λ) , (C, Z) ∈ C, λ is an MPR.
sup
(4.1)
see remark 4.8 below. For general semimartingale markets and general utility functional, necessary and su cient conditions for validity of the martingale approach has been proposed, see [55, 56]. These conditions guarantee that the duality gap is zero (i.e. the optimal value of the primal and dual problem are the same), and that the optimal consumption pair
(c, Z) given by
the dual problem is actually aordable. However they only provide the existence of the optimal portfolio
u,
and to nd the portfolio policy one should solve a representation problem.
The situation is much more tractable once we assume that the market is complete. In this case the market price of risk is unique, hence the dual problem would be only in terms of the consumption pair
(c, Z).
But then we can solve this
static problem directly to nd the optimal
consumption and terminal wealth. Speci cally the optimal consumption pair is,
Ct∗ , I(ηπt , t) for t ∈ (0, T ) , Z ∗ , IF (ηπT ). where
IF (.)
and
I(., t) are the inverses of U 0 (x) and
14
∂ u(x, t), ∂x
(4.2)
π = (πt ) is the state price density
process, and the constant
η
is the unique solution of the following equation:
ˆ
T
E πT IF (ηπT ) +
πτ I(ηπτ , τ )dτ = w.
(4.3)
0 See remark 4.10 below. This is actually the original martingale approach. Yet again, to obtain the optimal portfolio we need to solve a representation problem. In the next subsection we present the Martingale approach in details.
4.1 Proof of the Martingale Approach The essence of the martingale approach in incomplete markets, also known as duality approach, is in remark 4.8 and theorem 4.9. In summary the proof goes like this:
•
First we identify a necessary and su cient condition under which an arbitrary consump-
(C, Z)
tion pair
is aordable. It will be done in two steps, lemma 4.6 and proposition
4.7.
•
Then we relate Merton's problem to the dual problem containing only the consumption pairs and not the optimal portfolio
u
(see equation 4.26).
•
Then in theorem 4.9, we identify conditions under which we can solve this dual problem.
•
Finally we consider the case of complete markets in remark4.10.
To express the martingale approach, we need the following de nitions and lemmas.
De nition 4.1. risk process
λ
Assume that we are given a consumption pair (C, Z) and a market-price of (λ) (C,Z,λ) (along with the corresponding SPD π ). Then we de ne the process W
as:
(λ) (C,Z,λ) πt W t
, Et
ˆ (λ) πT Z
T
πτ(λ) Cτ dτ
+
.
(4.4)
t
Lemma 4.2. (Inada condition) We say that a strictly concave, increasing and dierentiable function F : R+ → R satis es Inada conditions if inf F 0 (x) = 0 and supF 0 = +∞. If F : R+ → x x R satis es Inada conditions, then the function IF , which is the inverse of F 0 , is well de ned as a strictly decreasing continuous function on (0, ∞) which image is (0, ∞). De nition 4.3. (Condition A)
U (x) is dierentiable on (0, ∞), strictly concave and satis es Inada conditions. Either (for all t) u(x, t) is zero or (for all t) u(x, t) is dierentiable on (0, ∞), strictly concave and satis es Inada conditions. Also for all η > 0 and any SPD π we assume: ˆ T E πT IF (ηπT ) + πτ I(ηπτ , τ )dτ < ∞. (4.5) Either
U (x)
is zero or
0 Here
IF (.)
and
I(., t)
are the inverses of
U 0 (x)
and
15
∂ u(x, t). ∂x
Lemma 4.4. Consider problem 2.9. Assume that there exists a unique constant η (λ) > 0 such that: ˆ (λ) (λ) E πT IF (η (λ) πT ) +
condition A
holds, then for any MPR λ,
T
πτ(λ) I(η
(λ) πτ(λ) , τ )dτ
= w.
(4.6)
0
Here I(., t) and IF (.) are the inverses of the derivatives of u(., t) and U (.), respectively. Proof.
De ne
ˆ (λ) (λ) f (η) , E πT IF (ηπT ) +
T
πτ(λ) I(ηπτ(λ) , τ )dτ
.
(4.7)
0
Condition A along with lemma 4.2 imply that one or both of IF (.) and I (., t) (for all t ∈ (0, T )) are strictly decreasing continuous functions on
(0, ∞)
which images are
w>0 by η (λ).
inherits these two properties, we can conclude that for any which satis es
f (η) = w.
We denote this unique solution
(0, ∞).
f (.) η>0
Since
there exist a unique
We need to prove the numeraire invariance property of our de nition of budget feasibility (de nition 2.7).
Lemma 4.5. (numeraire invariance) Letπ = (πt ) be an arbitrary de ator and w be the initial wealth. A strategy (C, Z, u) is budget feasible at w if and only if : ´t (u,C) (u,C) > t ∈ [0, T ] π W = π w + πs Ws us (diag (πs As ))−1 d (πA)s t 0 t 0 d(πB) ´t ´t (u,C) > s + 0 πs Ws 1 − us 1n×1 πs Bs − 0 πs Cs ds ≥ 0 (u,C) πT WT ≥ πT Z a.s.
Proof.
The proof is a simple application of product rule.
By de nition
(C, Z, u)
(4.8)
is budget
feasible if and only if:
(u,C) ´ t (u,C) > W = w + Ws us (diag (As ))−1 dAs t ∈ [0, T ] t 0 dB ´ t ´ t (u,C) > + 0 Ws 1 − us 1n×1 B − 0 Cs ds ≥ 0 (u,C) WT ≥ Z a.s.
(4.9)
The second conditions in equation (4.8) are equivalent to the second equations above. To show that the rst condition in equation(4.8) is implied by the rst condition in equation(4.9), we apply the product rule to obtain:
d Note that:
(u,C) πt Wt
(u,C)
= πt dWt
(u,C)
+ Wt
dπt + d π, W (u,C) t
ˆ . −1 (u,C) (u,C) > d π, W = d π, Ws us (diag (As )) dAs t =
0 (u,C) > Wt ut (diag (At ))−1
16
d[π, A]t
(4.10)
(u,C) in equation(4.10), we will obtain: d π, W (u,C) and dWt dBt (u,C) > (u,C) −1 > = πt Wt ut (diag (At )) dAt + Wt 1 − ut 1n×1 − Ct dt Bt
By substituting for
d πW (u)
t
(u,C)
+Wt =
=
(u,C)
dπt + Wt
−1 u> d[π, A]t t (diag (At ))
−1 {πt dAt + diag (At ) 1n×1 dπt + d[π, A]t } u> t (diag (At )) {πt dBt + Bt dπt } (u,C) +Wt 1 − u> − πt Ct dt t 1n×1 Bt d (πB)t (u,C) (u,C) > 1 1 − u> Wt ut (diag (At ))−1 d (πA)t + Wt − πt C (4.11) n×1 t dt t Bt (u,C)
Wt
which in turn, will give us the rst condition in equation(4.8). Finally to show the converse, i.e. that the rst condition in equation(4.9) is also implied (π) by the rst condition in equation(4.8), we start by de ning the process as At = πt A t , (π) (π) (π) Bt = πt Bt ,Ct = πt Ct , and Z = πT Z . Then equation(4.8) is equivalent to assuming (π) (π) (π) (π) that (C , Z , u) is budget budget feasible at w with price processes given by At and Bt . 1 Now by considering the de ator and using the same argument we used form equation(4.10) πt to (4.11), we will reach the rst condition in equation (4.9). As mentioned above, the main idea of martingale approach is to identify conditions under which an arbitrary consumption pair
(C, Z) is aordable.
The following lemma is the rst step
towards this goal.
Lemma 4.6. Assume we are given a consumption pair (C, Z) and a market-price of risk process λ. Then (C, Z) is aordable with initial wealth w if and only if : (i) ˆ T
(λ)
(λ)
πt Ct dt + πt Z ≤ w.
E
(4.12)
0
(n) (ii)- There exist a predictable process u = u(1) t , . . . , ut
ΣTt ut − λt =
T
t∈[0,T ] 0 W , π (λ) W (C,Z,λ) t . (λ) (C,Z,λ) πt W t
satisfying: (4.13)
Furthermore if these conditions hold, then u is the self- nancing strategy for the consumption pair (C, Z). Proof.
(i). Assume (C, Z) to be aordable, i.e. there u such that (C, Z, u) is budget feasible at w. By using lemma4.5
First we prove the necessity of condition
exist a self- nancing strategy for de ator
π,
we will obtain:
´t (u,C) (u,C) > = w + 0 πs W s us (diag (πs As ))−1 d (πA)s t ∈ [0, T ] πt Wt d(πB) ´t ´t (u,C) > s + 0 πs W s 1 − us 1n×1 πs Bs − 0 πs Cs ds ≥ 0 (u,C) πT W T ≥ πT Z a.s.
17
De ne a process
N
by:
ˆ Nt ,
(u,C) πt W t
ˆ
t
+
πs Cs ds
(4.14)
ˆ
0 t
= w+
−1 Ws(u,C) u> s (diag (As ))
t
Ws(u,C) 1 − u> s 1n×1
d (πA)s + 0
0
π (λ) A
Note that by de nition,
π (λ) B
and
are (local) martingales, so
N
d (πB)s .(4.15) Bs
is a non-negative
local martingale, and hence a super martingale. The supper-martingale property give us the desired equation (4.12):
ˆ
T
E πT Z +
πt Ct dt ≤ E [NT ] ≤ N0 = w.
(4.16)
0
(ii)
To prove the necessity of condition we may proceed as follow. From remark (2.5), C (C,Z,λ) as the proportional W is the wealth process of a self- nancing strategy u with c , W (C,Z,λ) consumption and w as the initial wealth, if and only if:
(C,Z,λ) Wt
ˆ = wE
ˆ
.
0 Z (λ) , where B properties of stochastic integrals we may write:
=
u> u Σu dW u
.
(4.17)
0
Recall from remark (2.3) that
π (λ) W (C,Z,λ) =
.
> uu (µu − ru 12×1 ) + ru − cu du + π ,
Z (λ) = E −
´.
λ> dW s 0 s
.
So by using the
Z (λ) (C,Z,λ) W B ´ . E − 0 λ> s dW s ´.
B e 0 rs ds 0 ˆ . ˆ . > > × wE uu (µu − ru 12×1 ) + ru − cu du + uu Σu dW u 0
0
w [− ´ . λTs dW s ,´ . uTs Σs dW s ] 0 = e 0 B0 ˆ . ˆ . > > > ×E us (µs − rs 1n×1 ) − cs ds + us Σs − λs dW s 0 0 ˆ . ˆ . > w > > = E us (µs − rs 12×1 − Σs λs ) − cs ds + us Σs − λs dW s B0 0 0 ˆ . ˆ . w > > E − cs ds + us Σs − λs dW s . (4.18) = B0 0 0 Hence we obtain:
d π (λ) W (C,Z,λ)
t
(λ)
(C,Z,λ)
> −ct dt + u> t Σt − λt dW t (λ) (λ) (C,Z,λ) > Σ − λ = −πt Ct dt + πt Wt u> t t t dW t . = πt W t
On the other hand from equations (4.4) and the martingale representation property of
(4.19)
W,
we
have
d π (λ) W (C,Z,λ)
(λ)
+ πt Ct dt = t 18
(λ) (C,Z,λ) 0 π W , W t dW t .
(4.20)
From the last two results we can conclude
(λ)
(C,Z,λ)
πt Wt
which is equation(4.13). Finally to prove the converse, suppose that conditions
W
(u,c)
ˆ , wE
(λ) (C,Z,λ) 0 > = π W , W Σ − λ , u> t t t t
(i)
and
(ii)
ˆ
.
u> u
hold. Then by de ning
.
u> u Σu dW u
(µu − ru 12×1 ) + ru − cu du +
,
0
0
(ii)
(C,Z,λ) we may invert the proof of necessity of condition , to conclude that W ≡ W (u,c) . Now (C,Z,λ) we may use the de nition of W along with condition to prove that u is indeed a self- nancing strategy and
(C, Z, u)
(i)
is budget feasible.
In the following proposition we improve the result of the previous lemma by making the
u.
aordability condition free of any self- nancing strategy
Also we give an expression for the
corresponding self- nancing strategy, if a consumption pair is aordable.
Proposition 4.7. A consumption pair (C, Z) is aordable with initial wealth w if and only if for a speci c market price of risk λ satisfying −1 λ , Σ> ΣΣT (µ − r1n×1 ) o n −1 + Σ> ΣΣ> Σ − Ik×k
we have
ˆ
>
E
(λ) πt Ct dt
+
W , π (λ) W (C,Z,λ)
0 ! t
,
(λ) (C,Z,λ) π t Wt
(λ) πt Z
(4.21)
≤ w.
(4.22)
0
Furthermore, if this condition holds, the corresponding self- nancing strategy is given by: −1 u = ΣΣ>
Proof.
( µ − r1n×1 + Σ
0 !) W , π (λ) W (C,Z,λ) t (λ)
(C,Z,λ)
.
0
De ne
(C,Z,λ)
at
,
[W,π(λ) W (C,Z,λ) ]t (λ)
πt
(C,Z,λ)
Wt
and
(4.23)
πt W t
bt , µ−r1n×1 .
Then from condition
(4.6), and theorem (2.3) we have the following system for vectors
λ
and
(ii) of lemma
u:
( (C,Z,λ) ΣTt ut − λt = at Σt λ t = b t . By solving for
Note that rank
(4.24)
λ from the rst equation and substituting in the second equation we will obtain (C,Z,λ) Σt ΣTt ut − at = bt . (4.25) ΣΣT
=
rank (Σ)
= n,
so the
n×n
matrix
rearranging the last result we will obtain equation (4.23).
ΣΣT
is invertible.
By substituting for
u
Now by
in the rst
equation of the system above, we will obtain equation (4.21). Hence the necessary and su cient condition of lemma (4.6) can be restated as this proposition.
19
Remark
4.8
.
Note that from proposition (4.7), Merton's problem is equivalent to the following
variational problem:
h´
i
T 0
J(C, Z) , E u(t, Ct )dt + U (Z) subject to : h´ i T (λ) (λ) E 0 πt Ct dt + πT Z = w, o n −1 −1 Σ − Ik×k a(C,Z,λ) , λ = Σ ΣΣT b + ΣT ΣΣT (C, Z) ∈ C, λ is an MPR.
sup
(4.26)
This problem, generally known as the dual of problem , can be de ned even in a general semimartingale setting (see [55, 56] ). Now we are ready to prove the main result.
Theorem 4.9. (Martingale Approach) Consider problem 2.9 and assume that holds. De ne (λ) (λ) Ct , I(η (λ) πt , t) for t ∈ (0, T ) , (λ)
Z (λ) , IF (η (λ) πT ), 0
(C,Z,λ)
at
,
(λ)
πt
(C,Z,λ)
Wt
(4.27)
. Also de ne bt , µ − r1n×1 , and
where IF (.) and I(., t) are the inverses of U 0 (x) and [W ,π(λ) W (C,Z,λ) ]t
condition A
∂ u(x, t) ∂x
so that (λ)
(C,Z,λ)
dπt Wt
(λ)
(C,Z,λ)
= πt Wt
(C,Z,λ)
at
dW t .
(4.28)
If an MPR λ satis es >
λ=Σ
> −1
ΣΣ
o n (λ) (λ) > > −1 b + Σ ΣΣ Σ − Ik×k a(C ,Z ,λ) ,
(4.29)
then, under that speci c MPR λ, the optimal consumption is Ct(λ) and the optimal terminal wealth is Z (λ) . Furthermore the corresponding optimal self- nancing strategy is given by: > −1
u = ΣΣ
(C (λ) ,Z (λ) ,λ)
b + Σa
,
(4.30)
and we have W (u,C) = Z (λ) . Proof.
Consider the following problem: sup
J(C, Z) , E
subject to h´ T E 0
h´ T 0
(λ) πt Ct dt
(C, Z) ∈ C, λ is
a
+
i u(t, Ct )dt + U (Z)
(λ) πT Z
= w, given MPR.
Since there is not enough constraint to make the consumption pair
λ,
(4.31)
i
this problem is a relaxation of Merton's problem.
(C, Z)
aordable under
This means that the solution of this
problem gives an upper bound for the Merton's optimal solution. Now the main idea is that if
20
this solution of this relaxation problem satis es equation (4.21) as well, then it would be the solution for Merton's problem. First we try to nd the solution of equation (4.31). The Lagrangian is:
ˆ
T
L (C, Z, η) , E
u(t, Ct )dt + U (Z) ˆ T (λ) (λ) −η E πt Ct dt + πT Z − w . 0
(4.32)
0 To check the rst order conditions we rst nd the directional derivatives:
i h´ T (λ) 0 u (t, C + εx ) − ηπ x dt , t t t t 0 h i (λ) d L (C, Z + εy, η) = E U 0 (Z + εy) − ηπt y , dε h´ i T (λ) (λ) d L (C, Z, η) = E 0 πt Ct dt + πt Z − w, dη
d L (C dε
+ εx, Z, η) = E
(4.33)
x = (xt ) is an arbitrary predictable process, y is an arbitrary FT -measurable variable, and η is a positive number. The rst order conditions give us the solution, where
d | L (C, Z, η) dη η=0
= 0 =⇒ η = η (λ) , (λ) =⇒ Ct = Ct , =⇒ Z = Z (λ) .
d | L (C + εx, Z, η) = 0 dε ε=0 d | L (C, Z + εy, η) = 0 dε ε=0
random
(4.34)
Note that we only checked the necessary conditions of optimality, and not the su cient (second order) conditions. On the other hand equation (4.21), the su cient condition for aordability of
C (λ) , Z (λ)
,
translates into equation (4.29). So, as already mentioned, if λ also satis es this condition, then C (λ) , Z (λ) is also the optimal consumption pair in Merton's problem. Finally equation (4.30) follows directly from proposition (4.7).
Remark
4.10
.
Note that if the market is complete, i.e. n = k , then Σ would be invertible and λ = Σ−1 b. In this case equation (4.29) is automatically
the market price of risk would be
satis ed. This means that the relaxed problem in the proof of theorem (4.9) is equivalent to the Merton problem, and the solution is given by equations (4.27) and (4.30). Furthermore we may simplify equation (4.30) as follow:
u = ΣΣ>
−1
b + Σ>
21
−1
a(C
(λ) ,Z (λ) ,λ)
.
(4.35)
Chapter 5 The Direct Approach In this chapter we present an new approach for tackling the problem of portfolio choice.
It
has the strong aspects of both existing approaches: like the stochastic control theory, it deals with the problem in its original form, and like the martingale approach, it uses probabilistic argument which can be extended to general market settings. Recall that from equation (2.10) we have:
W
(u,c)
ˆ
ˆ
.
= wE
u> s
.
u> s Σs dW s
(µs − rs 1n×1 ) + rs − cs ds +
0
.
(5.1)
0
The main idea of the direct approach is to use the representation above to convert Merton's problem into
ˆ sup u∈P, c∈P +
Here
P
is the space of
n
T
E
(u,c) u(t, ct Wt )dt
+
(u,c) U (WT )
.
(5.2)
0
dimensional predictable vector processes, and
P + is
the space of non-
negative predictable processes. Then to nd a candidate for the optimal control, we may check the rst order optimality conditions:
(
Where the Lagrangian
L
d | L(u + v, c) = 0, d =0 d | L(u, c + y) = 0. d =0
(5.3)
is of the form,
ˆ
T
L (u, c) , E
(u,c) u(t, ct Wt )dt
+
(u,c) U (WT )
.
(5.4)
0
c ≥ 0, we should consider the casehof zero consumption separately. i (u,c) Lagrangian as L (u) , L (u, 0) = E U (WT ) , and the rst order
Also because of the constraint In this case we de ne the
d | L(u + v) = 0 (we also de ne W (u) , W (u,0) ). d =0 In the remainder of this chapter, we will show that the rst order conditions stated above
condition would be
can be converted into a set of equations that can be solved to obtain the optimal solutions. Namely, for the case of zero consumption we will have:
−1 ut = Σt Σ> t
( (µt − rt 1n×1 ) + Σt 22
0 ) W , V (u) t (u)
Vt
,
(5.5)
and for the general case we have:
(u,c) = Vt
(u,c) (u,c) t, ct Wt Wt , 0 [W ,V (u,c) ]t > −1 (µt − rt 1n×1 ) + Σt . ut = Σt Σt (u,c) ∂ u ∂x
(5.6)
Vt
Here the processes
V (u)
(u,c)
Vt
and
, Et
h´
T t
V (u,c) are
de ned as follows,
h i (u) (u) (u) Vt , Et U 0 WT WT , i (u,c) (u,c) (u,c) (u,c) ∂ 0 W . u s, c W c W ds + U W s s s s T T ∂x
(5.7)
For the main result see theorem 5.6, and equations 5.52 and 5.55. Note that the main hurdle [W,V u ]0t in implementing this approach is to nd explicit expressions for in the case of zero Vtu 0
consumption, and
V
(u,c)
and
[W,V (u,c) ]t
for the general case. (u,c) Vt We should mention that we have not looked at conditions which guarantee the optimality
of the candidates, i.e.
the counterparts of veri cation results in stochastic control approach
(see [37, 85]) or the necessary and su cient condition of martingale approach (see [55, 56]). As pointed out by Korn and Kraft [53], these results are crucially important and uncritical application of any method for continuous-time portfolio optimization can be misleading in the case of a stochastic opportunity set. We have proposed nding such conditions for validity of the direct approach as a future line of research (see chapter 7). Besides these technically challenging issues, there are a number of interesting questions on the economic properties of the optimal portfolios. From (5.5) and (5.6) one sees that the optimal portfolio consists of two terms, namely,
um = t
Σt Σ> t
−1
(µt − rt 1n×1 ) ,
uht =
Σt Σ> t
−1
Σt
[W , V ∗ ]0t . Vt∗
(5.8)
The rst component is known as the myopic investment strategy. It corresponds functionally to the investment policy followed by an investor in markets in which the investment opportunity set remains constant through time. The myopic portfolio is always positive for a nonzero market price of risk. The second term is called the excess hedging demand. It represents the additional investment caused by the presence of the stochastic factor. It does not have a constant sign, for the > −1 signs of the correlation coe cient Σt Σt Σt , and the volatility of the optimal V ∗ process are not de nite. The excess hedging demand term vanishes in the uncorrelated case; and when the volatility of the stochastic factor process is zero (the latter case can be reduced to the classical Merton one). Finally, the excess hedging demand term also vanishes for the case of logarithmic utility (see section 6.1 and [10]). It is worth to mention that despite the nomenclature
hedging demand,
a rigorous study
for the precise characterization and quanti cation of the risk that is not hedged has not been carried out. Indeed, in contrast to derivative valuation where the notion of imperfect hedge is well de ned, such a notion has not been established in the area of investments (see [80] for a special case).
23
As a nal observation, note that total allocation in the risky assets might become zero even if the risk premium is not zero. This phenomenon, related to the so called market participation puzzle, appears at rst counter intuitive, for classical economic ideas suggest that a risk averse investor should always retain nonzero holdings in an asset that oers positive risk premium. We refer the reader to, among others, [5, 18, 44]. In the next section we will present in detail the argument used to obtain the main results of the direct approach.
5.1 Proof of the Direct approach We start by identifying
(u+v,c) (u,c+y) d d | W and | W , which in some sense, are the building d =0 t d =0 t
blocks of our proof.
Lemma 5.1. Let u be a self- nancing strategy for the proportional consumption rate c. Then for all locally bounded predictable processes v and y we have ˆ t ˆ t d (u+v,c) (u,c) T T T v s µs − rs 1n×1 − Σs Σs us ds + |=0 Wt = Wt v s Σs dW s , (5.9) d 0 0 ˆ t d (u,c+y) (u,c) |=0 Wt = −Wt ys ds. (5.10) d 0
Proof.
De ning
Yt
Yt
as
ˆ t ˆ tn o > (uu + v u )T Σu dW u . , (uu + v u ) (µu − ru 1n×1 ) + ru − cu du +
(5.11)
0
0 From equation(2.10) we have
(u+v,c)
Wt
d (u+v,c) |=0 Wt d
1
= wE (Y )t = weYt − 2 [Y ]t . So we 1 (u,c) d = Wt |=0 Yt − [Y ]t . d 2
may conclude:
(5.12)
From equation(5.11), after interchanging the order of dierentiation and integration, we would obtain
ˆ
d |=0 Yt = d
ˆ
t
v Tu 0
t
v Tu Σu dW u .
(µu − ru 1n×1 ) du +
(5.13)
0
Also from equation(5.11) it follows that
ˆ
[Y ]t =
t T
(uu + v u ) Σu dW u ˆ
0 t
(uu + v u )T Σu d[W u ]ΣTu (uu + v u )
= ˆ0 t
(uu + v u )T Σu (Ik×k du) ΣTu (uu + v u )
= ˆ0 t
(uu + v u )T Σu ΣTu (uu + v u ) du.
= 0
24
(5.14)
By dierentiating from the last result we would have
d |=0 [Y ]t = d
ˆ
t
2v Tu Σu ΣTu uu du.
(5.15)
0
Equation(5.9) follows by substituting from equation (5.13) and (5.15) into equation (5.12). For obtaining equation (5.10) we may start by de ning
ˆ Yet ,
ˆ
t
uTu
t
uTu Σu dW u .
(µu − ru 1n×1 ) + ru − (cu + yu ) du +
0
(5.16)
0
Similar to equation (5.12) we have:
But
h i d 1 (u,c+y) (u,c) d |=0 Wt = Wt |=0 Yet − Ye . d d 2 t h i ´t d d e = − y du , and | Y | Ye = 0. Hence we obtain equation d =0 t d =0 0 u
(5.17)
(5.10).
t
In the following two lemmas, for the case of zero consumption, we nd an expression for d | L(u + v) in terms of the martingale V (u) of equation (5.51). d =0
Lemma 5.2. For the special case of zero consumption, de ne the Lagrangian as: h i (u) L (u) , E U (WT ) ,
(5.18)
where W (u) = W (u,0) . Then for all locally bounded predictable processes v , we have: h d (u) (u) 0 WT |=0 L(u + v) = E U WT d ˆ T ˆ T T v s µs − rs 1n×1 − Σs Σs us ds + × 0
Proof.
T
v Ts Σs dW s
(5.19)
0
By dierentiating equation (5.18), and interchanging the order of dierentiation with
expectation, we would have
d d (u+v) |=0 L(u + v) = E |=0 U (WT ) d d (u) d (u+v) 0 = E U (WT ) |=0 WT . d Now by substituting for
(5.20)
(u+v) d from equation (5.9) we will get the result. | W d =0 T
Lemma 5.3. For the special case of zero consumption, we de ne the martingaleV (u) as: (u)
Vt
h i (u) (u) , Et U 0 WT WT .
(5.21)
Then for all locally bounded predictable processes v , we have d |=0 L(u + v) = E d
ˆ
ˆ
T
Vs(u) v Ts
µs − rs 12×1 −
0
Σs ΣTs us
T
v Ts Σs d
ds + 0
W,V
(u)
s
.
(5.22)
25
Proof.
By considering the fact that
(u) (u) VTu = U 0 WT WT ,
equation (5.19) can be rewritten as
ˆ T ˆ T d T u T T v s Σs dW s . |=0 L(u + v) = E VT v s µs − rs 12×1 − Σs Σs us ds + d 0 0
(5.23)
By integration by parts we have:
VTu ˆ =
ˆ 0
T
ˆ
T
v Ts
µs − rs 12×1 −
Σs ΣTs us
ds + ˆ T µs − rs 12×1 − Σs Σs us ds +
T
v Ts Σs dW s
0 T
Vsu v Ts Σs dW s Vsu v Ts 0 0 ˆ t ˆ T ˆ t T T T v s Σs dW s dVtu v s µs − rs 12×1 − Σs Σs us ds + + 0 0 0 ˆ T v Ts Σs d [W , V u ]s . +
(5.24)
0 Note that, subject to some technical conditions, the second and third terms in the right-handside are zero-mean martingales. Now by taking expectation we will get the result. The following two lemmas are the counterparts of lemmas 5.2 and 5.3, for the case of non-zero consumption.
Lemma 5.4. For the general case with consumption, de ne the Lagrangian as: ˆ L (u, c) , E
T
(u,c) u(t, ct Wt )dt
+
(u,c) U (WT )
.
(5.25)
0
Then for all locally bounded predictable processes v , we would have: d |=0 L (u + v, c) d ˆ t ˆ T ˆ t ∂ (u,c) (u,c) T T T = E u t, ct Wt ct Wt v s µs − rs 12×1 − Σs Σs us ds + v s Σs dW s dt 0 ∂x 0 0 ˆ T ˆ T (u,c) (u,c) 0 T T T WT v s µs − rs 12×1 − Σs Σs us ds + v s Σs dW s , (5.26) +E U WT 0
0
and for all locally bounded predictable processes y, we have: d |=0 L (u, c + y) d ˆ T ˆ t ∂ (u,c) (u,c) = E u t, ct Wt Wt y t − ct ys ds dt 0 ∂x 0 ˆ T (u,c) (u,c) 0 −E U WT WT ys ds . 0
26
(5.27)
Proof.
By dierentiating equation (5.25), and interchanging the order of dierentiation with
expectation and then with integration, we would have
d |=0 L (u + v, c) = E d
ˆ
T
d d (u+v,c) (u+v,c) |=0 u(t, ct Wt )dt + |=0 U (WT ) d 0 d ˆ T d ∂ (u,c) (u+v,c) u t, ct Wt ct |=0 Wt dt = E d 0 ∂x (u,c) d (u+v,c) 0 +E U (WT ) |=0 WT . d
Now by substituting for
(5.28)
(u+v,c) (u+v,c) d d | W and | W from equation (5.9) we will get d =0 t d =0 T
equation (5.26). For obtaining equation (5.27), by dierentiating equation (5.25) we would have:
ˆ
T
d d (u,c+y) (u,c+y) |=0 u(t, (ct + yt ) Wt )dt + |=0 U (WT ) d 0 d ˆ T d ∂ (u,c) (u,c+y) = E u t, ct Wt ct |=0 Wt dt d 0 ∂x ˆ T ∂ (u,c) (u,c+y) u t, ct Wt yt Wt dt (5.29) +E 0 ∂x (u,c) d (u,c+y) 0 +E U (WT ) |=0 WT . (5.30) d
d |=0 L (u, c + y) = E d
Now substituting for
(u,c+y) (u,c+y) d d | W and | W from equation(5.10) would result in d =0 t d =0 T
equation (5.27).
Lemma 5.5. De ne the processV (u,c) by Vs(u,c)
ˆ , Es s
T
∂ (u,c) (u,c) (u) (u) 0 WT . u t, ct Wt ct Wt dt + U WT ∂x
(5.31)
Then for all locally bounded predictable processes v , we would have: d |=0 L(u + v, c) = E d
ˆ
ˆ
T
Vs(u,c) v Ts
µs − rs 12×1 −
Σs ΣTs us
v Ts Σs d
ds +
0
T
W,V
0
and for all locally bounded predictable processes y, we have: d |=0 L (u, c + y) = E d
Proof.
ˆ
T
∂ (u,c) (u,c) (u,c) u t, ct Wt Wt − Vt yt dt . ∂x
0
(u,c)
s
,
(5.32)
(5.33)
To derive equation (5.33), we note that by equation (5.27)
d |=0 L (u, c + y) = E d
ˆ 0
T
∂ (u,c) (u,c) u t, ct Wt Wt yt dt − I − II, ∂x
27
(5.34)
where
ˆ t ∂ (u,c) (u,c) u t, ct Wt ct Wt I , E ys ds dt 0 ∂x 0 ˆ T (u,c) (u,c) 0 ys ds . II , E U WT WT ˆ
T
(5.35)
0 For
I
we may write
ˆ
T
ˆ
t
∂ (u,c) (u,c) E I = u t, ct Wt ct Wt ys dsdt ∂x 0 0 ˆ Tˆ T ∂ (u,c) (u,c) = E u t, ct Wt ct Wt ys dtds ∂x 0 s ˆ Tˆ T ∂ (u,c) (u,c) E Es = u t, ct Wt ct Wt ys dtds ∂x 0 s ˆ T ˆ T ∂ (u,c) (u,c) ys Es = E ct Wt dt ds . u t, ct Wt s ∂x 0 h i (u,c) (u,c) (u,c) 0 For handling II , rst we de ne a martingale Gs , Es U W T WT . integration by parts to the de nition of II gives us ˆ T (u,c) ys ds II = E GT 0 ˆ T ˆ T ˆ t (u,c) (u,c) ys Gs ds + ys ds dGt = E 0 0 0 ˆ T (u,c) ys Gs ds , = E
(5.36)
Then applying
(5.37)
0 where we used the fact that, subject to some technical conditions, the second term in the second line is a martingale so its expectation vanishes. Now substituting for
I
and
II
in equation (5.34)
will result in equation (5.33). Similarly, to derive equation (5.32), we will start by equation (5.26)
d |=0 L (u + v, c) = I + II + III, d
(5.38)
where
ˆ
ˆ t ∂ (u,c) (u,c) T T I , E u t, ct Wt ct Wt v s µs − rs 12×1 − Σs Σs us ds dt (5.39) 0 0 ∂x ˆ T ˆ t ∂ (u,c) (u,c) T II , E u t, ct Wt ct Wt v s Σs dW s dt (5.40) 0 ∂x 0 h (u,c) (u,c) III , E U 0 WT WT ˆ T ˆ T T T T v s Σs dW s . (5.41) × v s µs − rs 12×1 − Σs Σs us ds + T
0
0 28
Expression
III
can be handled by following the same argument as in the proof of lemma 5.3,
which gives us
ˆ
ˆ
T
Ves(u,c) v Ts µs − rs 12×1 − Σs ΣTs us ds +
III = E 0
T
v Ts Σs d 0
where
i h (u,c) (u,c) (u,c) 0 e WT . Vs , Es U W T
For handling
I,
ˆ
T
h
W , Ve (u,c)
i
,
(5.42)
s
(5.43)
we follow a calculation similar to equation (5.36)
ˆ
t
∂ (u,c) (u,c) T T u t, ct Wt ct Wt v s µs − rs 12×1 − Σs Σs us dsdt E ∂x 0 0 ˆ Tˆ T ∂ (u,c) (u,c) T T E u t, ct Wt ct Wt v s µs − rs 12×1 − Σs Σs us dtds ∂x s 0 ˆ Tˆ T ∂ (u,c) (u,c) T T dtds E Es u t, ct Wt ct Wt v s µs − rs 12×1 − Σs Σs us ∂x 0 s ˆ T ˆ T ∂ (u,c) (u,c) T T Es E u t, ct Wt ct Wt dt v s µs − rs 12×1 − Σs Σs us ds (. 5.44) s ∂x 0
I = = = =
For handling
II ,
rst we de ne:
Vs(u,c,t)
, Es
∂ (u,c) (u,c) u t, ct Wt ct Wt ∂x
(5.45)
Now integration by parts would give us
ˆ t ∂ (u,c) (u,c) T v s Σs dW s u t, ct Wt ct Wt E ∂x 0 ˆ t (u,c,t) T = E Vt v s Σs dW s 0 ˆ t ˆ t ˆ s ˆ t (u,c,t) T T (u,c,t) T (u,c,t) = E Vs v s Σs dW s + v u Σu dW u dVs + v s Σs d W , V s 0 0 0 0 ˆ t v Ts Σs d W , V (u,c,t) s , = E (5.46)
0
where we assume that the rst two integrals in the third line are zero mean martingales. We
29
may now proceed as follow:
ˆ
II = = = =
ˆ t ∂ (u,c) (u,c) T E u t, ct Wt ct W t v s Σs dW s dt ∂x 0 0 ˆ T ˆ t T (u,c,t) v s Σs d W , V. dt E s 0 0 ˆ T ˆ t (u,c,t) 0 T dsdt v s Σs W , V. E s 0 0 ˆ T ˆ T T (u,c,t) 0 E v s Σs W , V. dtds s 0 s "ˆ 0 # ˆ T
T
T
v Ts Σs
= E ˆ
V.(u,c,t) dt
W, .
0 T
v Ts Σs d
= E
ˆ
ds s
T
V.
W,
(u,c,t)
dt .
.
0
(5.47)
s
Finally we put everything back together to obtain equation (5.32):
d |=0 L (u + v, c) = I + II + III d ˆ T ˆ Es = E 0
ˆ
T
+E 0
ˆ
T
+E 0
ˆ +E ˆ = E
T
T
∂ (u,c) (u,c) u t, ct Wt ct Wt dt s ∂x × v Ts µs − rs 12×1 − Σs ΣTs us ds ˆ T T (u,c,t) v s Σs d W , V. dt . s (u,c) > T e Vs v s µs − rs 12×1 − Σs Σs us ds h i > (u,c) v Σs d W , Ve s
0 T
ˆ
(5.48)
(5.49)
s
T
∂ (u,c) (u,c) (u,c) (u,c) 0 WT u t, ct Wt ct Wt dt + U WT Es 0 s ∂x ×v Ts µs − rs 12×1 − Σs ΣTs us ds ˆ T ˆ T h i T (u,c,t) (u,c) e +E v s Σs d W , V. dt + d W , V s 0 . s ˆ T = E Vs(u,c) v Ts µs − rs 12×1 − Σs ΣTs us ds 0 ˆ T T (u,c) +E v s Σs d W , V , (5.50) s 0
Where in the last step we used the fact that
(u,c)
Vs
Finally we are ready to prove the main result.
30
=
´T s
(u,c,t)
Vs
(u,c) dt + Ves .
Theorem 5.6. (the Direct Approach) Consider problem 2.9. For the case of zero consumption, de ne the martingale V (u) by h
(u) Vt
, Et U
0
(u) WT
(u) WT
i
.
(5.51)
Then a self- nancing strategy u (with no consumption) satis es the rst-order optimality conditions, i.e. dd |=0 L(u + v) = 0 for all bounded predictable process v , if and only if −1 Σt ΣTt
ut =
[W , V u ]0t (µt − rt 1n×1 ) + Σt . Vtu
(5.52)
For the general case (i.e. nonzero consumption) de ne the processV (u,c) by ˆ
(u,c) Vt
T
, Et t
∂ (u,c) (u,c) (u,c) (u,c) 0 WT . u s, cs Ws cs Ws ds + U WT ∂x
(5.53)
Then the pair (u, c) satis es the rst-order optimality conditions, i.e. (
d | L(u + v, c) = 0 d =0 d | L(u, c + y) = 0 d =0
(5.54)
for all bounded predictable process v and y, if and only if (u,c) = Vt
∂ u ∂x
(u,c) t, ct Wt
(u,c)
Wt
, 0 [W ,V (u,c) ]t T −1 (µt − rt 1n×1 ) + Σt . ut = Σt Σt (u,c)
(5.55)
Vt
Proof.
First consider the case of zero consumption. By lemma (5.3) the rst order condition
ˆ
becomes
T
E
v Ts
Vsu
µs − rs 12×1 −
Σs ΣTs us
+ Σs [W , V
u 0 ]s ds
= 0,
(5.56)
0
for all bounded
predictable
processes
v.
Since the coe cient in the braces is predictable, we
may conclude that it must vanish:
Vsu µs − rs 12×1 − Σs ΣTs us + Σs [W , V u ]0s = 0,
for all
0 < s < T.
(5.57)
This last result is equivalent to equation (5.52). For the general case (with nonzero consumption) we use lemma (5.5) to rewrite the rst order conditions as:
E
h´
for all bounded
T 0
E n
h´ n T 0
(u,c)
v Ts Vs
o i (u,c) − Vt yt dt = 0, 0 o i µs − rs 12×1 − Σs ΣTs us + Σs W , V (u,c) s ds = 0,
∂ u ∂x
(u,c)
t, ct Wt
predictable process v and y.
(u,c)
Wt
(5.58) (5.59)
Again since all terms in the braces are predictable,
they must vanish:
∂ u ∂x
(u,c)
Vs
(u,c)
(u,c)
(u,c)
− Vt = 0, for all 0 < t < T, 0 µs − rs 12×1 − Σs ΣTs us + Σs W , V (u,c) s = 0, for all 0 < s < T. t, ct Wt
Wt
This last system is equivalent to equation (5.55).
31
(5.60)
Chapter 6 Examples The most frequently used utilities are the exponential, power and logarithmic ones. Exponential case case has been extensively studied not only in optimal investment models but, also, in indierence pricing where valuation is done primarily under exponential preferences (see [19] for a concise collection of relevant references). However, a well known criticism of the exponential utility is that the optimal portfolio does not depend on the investors wealth.
While
this property might be desirable in asset equilibrium pricing, it appears to be problematic and counter intuitive for investment problems. Here we focus on logarithmic and power utilities.
6.1 Logarithmic Utility In this section, we consider the general market of de nition 2.1 and the logarithmic utility functional of the following form,
ˆ
T
log (Ct ) dt + log (Z) .
J(C, Z) , E
(6.1)
0 The logarithmic utility plays a special role in portfolio choice.
As it will be shown in this
section, the optimal portfolio is myopic, namely,
u = ΣΣT
−1
(µ − r1n×1 ) ,
This is a well-known fact, and the associated myopic portfolio, also know as the
portfolio,
(6.2)
growth optimal
has been extensively studied in the general semimartingale market settings (see, for
example, [4] and [45]). The associated optimal wealth is the so-called "numeraire portfolio". It has also been extensively studied, for it is the numeraire with regards to which all wealth processes are supermartingales under the historical measure (see, among others, [40] and [41]). Using the general results of chapter 4 and 5 would be an overkill for the case of logarithmic utility, since there is a much more simpler way.
Actually the idea of the direct approach (u,c) stem from this solution method. Note that from equation(2.10) we have Wt = wE (Y )t = Yt − 21 [Y ]t we , where,
Yt ,
´t 0
´t u> ) + rs − cs ds + 0 u> s (µs − rs 1n×1 s Σs dW s , ´t > > [Y ]t = 0 us Σs Σs us ds.
32
(6.3)
The optimum criterion can be expanded as follow,
ˆ
T
E
(u,c) log(ct Wt )dt
0
ˆ = E
ˆ
T
log (ct ) dt + 0
+
T
(u,c) log(WT )
(u,c) log(Wt )dt
(6.4)
+ log
(u,c) WT
.
(6.5)
0
Now we may with the second term above to obtain,
h i (u,c) E log(WT ) 1 = log(w) + E YT − [Y ]T 2 ˆ T 1 > > > = log(w) + E − us Σs Σs us + us (µs − rs 1n×1 ) + rs − cs ds . 2 0
(6.6)
The third term can also be written in the form,
ˆ
T
E 0
ˆ
T
ˆ
T
− T log(w) 1 [Y ]t dt 2
Yt dt − 0 0 ˆ T ˆ t 1 > > > = E − us Σs Σs us + us (µs − rs 1n×1 ) + rs − cs dsdt 2 0 0 ˆ T ˆ t > us Σs dW s dt +E 0 0 ˆ T ˆ T 1 > > > = E − us Σs Σs us + us (µs − rs 1n×1 ) + rs − cs dtds 2 0 s ˆ T ˆ t > + E us Σs dW s dt 0 0 ˆ T 1 > > > (T − s) − us Σs Σs us + us (µs − rs 1n×1 ) + rs − cs ds , = E 2 0 ´t > the last step we assume u Σ dW s to be a martingale. Substituting back 0 s s = E
where in
(u,c) log(Wt )dt
(6.7)
the last
two results in the optimal criterion gives us,
ˆ
T
E
(u,c) log(ct Wt )dt
+
(u,c) log(WT )
0
= (1 + T ) log (w) ˆ T 1 > > > +E (1 + T − s) − us Σs Σs us + us (µs − rs 1n×1 ) + rs ds 2 0 ˆ T +E {log(cs ) − (1 + T − s)cs } ds .
(6.8)
0 Now it su ces to maximize the quadratic integrand
1 > u 7→ − u> Σs Σ> s u + u (µs − rs 1n×1 ) , 2 33
(6.9)
and the integrand
c 7→ {log(c) − (1 + T − t)c}.
Finally elementary calculus will give us the
optimal portfolio as of equation (6.2) and the optimal consumption as,
1 . 1+T −t
ct =
(6.10)
Applying the direct approach is also straightforward. (u,c) and V :
(u,c) Vt
, Et
h´ T t
h
Vtu
0
(u) WT
(u) WT
First we nd the martingales
i
= 1, , Et U i (u) (u,c) (u,c) (u) ∂ 0 W = 1 + T − t. u s, c W c W ds + U W s s s s T T ∂x
(6.11)
0
V (u)
0 = W , V (u,c) t = 0. So for the case of zero consumption, by using t equation (5.52), the optimal portfolio would be: Obviously
W , V (u)
ut = Σt ΣTt
−1
(µt − rt 1n×1 ) ,
(6.12)
which is the same as what we found earlier by martingale approach. For the case of non-zero consumption, equation (5.55) will give use the same portfolio strategy as above, and the optimal consumption as:
(u,c)
Vt
=
∂ 1 (u,c) (u,c) u t, ct Wt Wt =⇒ ct = , ∂x 1+T −t
(6.13)
which is what we have already found in equation (6.10). Finally let's take the martingale approach. Besides proving that the optimal portfolio and consumption is of the form (6.2) and (6.10), we will also prove that the optimal market price of risk, and the optimal wealth process are given by:
λ = ΣT ΣΣT w (1 + T − t) ´0t = e (1 + T )
(C (λ) ,Z (λ) ,λ) Wt
−1
b,
(6.14)
−1
rs +(µs −rs 1n×1 )T (Σs ΣT s)
ˆ
.
(µ − r1n×1 )
×E
T
(µs −rs 1n×1 ) ds
−1 Σs ΣTs
Σs dW s
0 We start by nding an expression for I(y, t) = IF (y) = y1 and we obtain:
E
η (λ) ˆ
(λ) πT IF (η
(λ) (λ) πT )
of lemma 4.4. Note that
T
πτ(λ) I(η
+
(λ) πτ(λ) , τ )dτ
∂ u (t, x) ∂x
1+T . η (λ)
=
0 Then equation (4.6) will give us
(λ)
Ct
η (λ) =
.
(6.15)
t
= U 0 (x) =
1 , so x
(6.16)
1+T , and we have: w
1
(λ)
= I(η (λ) πt , t) =
η
1
(λ)
Z (λ) = IF (η (λ) πT ) =
(λ) (λ) πt (λ)
η (λ) πT
34
= =
w (λ)
(1 + T ) πt w
(λ)
(1 + T ) πT
.
,
(6.17)
(6.18)
The next step is to derive the process
(λ)
(C (λ) ,Z (λ) ,λ)
πt Wt
(λ)
(λ)
π (λ) W (C ,Z ,λ) . From equation (4.4) we may conclude: ˆ T (λ) (λ) (λ) (λ) πτ Cτ dτ = Et πT Z + t
w (1 + T − t) = . (1 + T ) But then
0 W , π (λ) W (C,Z,λ) t = 0,
(6.19) 0
so
a(C
(λ) ,Z (λ) ,λ)
,
[W ,π(λ) W (C,Z,λ) ]t (λ)
πt
(C,Z,λ)
Wt
=0
and equation (4.29)
λ as in equation (6.14). Now we may use theorem (4.9) to conclude that, only for (λ) (λ) is the optimal consumption pair. To nd equation (6.10), particular MPR λ, Ct , Z
would give us this
we observe that
(λ)
ct = Now by substituting for
Ct
(C (λ) ,Z (λ) ,λ) Wt
(C (λ) ,Z (λ) ,λ)
(λ)
w
=
πt W t
(1 +
(λ) (C (λ) ,Z (λ) ,λ) T ) πt Wt
.
(6.20)
in the denominator from equation (6.19) we will
obtain the result. Equation (6.2) directly follows from equation (4.30). Finally by substituting (λ) for πt from equation (2.4) into equation (6.19) we would have:
(C (λ) ,Z (λ) ,λ)
Wt
And substituting for
=
w (1 + T − t) (λ)
(1 + T ) πt w (1 + T − t) ´ = ´. − 0t ru du (1 + T ) e E − 0 λTu dW u ˆ . w (1 + T − t) ´0t (rs +λTs λs )ds T λs dW s . e E = (1 + T ) 0 t λ
(6.21)
from equation (6.14) will give us equation (6.15).
6.2 Power Utility In this section we take the power utility functional, de ned as:
ˆ
T
J(C, Z) , E 0
Ctγ Zγ dt + . γ γ
(6.22)
Then we consider the problem originally solved by Merton, and redrive the solution. After that we consider the case where the interest rate is stochastic, and follows a multi-factor Gaussian model. The optimal portfolio in the two cases dier in the following ways (compare theorem 6.3 and theorem 6.8),
•
In the Merton setting, the optimal portfolio only includes the myopic term, while the excess hedging demand term (containing the covariance between zero-coupon bonds and assets) enters in the optimal portfolio for the Gaussian term structure case.
•
In the Merton setting the optimal portfolio is deterministic whether we consider consumption or not, but for the Gaussian term structure case the optimal portfolio is deterministic when there is no consumption, and stochastic otherwise.
35
6.2.1
Original Merton Setting
Consider a market with deterministic coe cients, de ned as follow.
De nition 6.1. A market with deterministic coe cients is de ned as a special case of de nition 2.1 where
r, µ,
and
Σ
are deterministic functions of time, and
n = k
(i.e.
the market is
complete). In this case we write:
rt = r(t) µt = µ(t) Σt = Σ(t).
(6.23)
After deriving a useful result in lemma 6.2, we will identify Merton's solution in theorem 6.3.
Lemma 6.2. Consider the market de ned in de nition6.1 and let λ(t) and π = (πt ) be the unique market-price-of-risk and SPD identi ed in theorem 2.3, equations (2.8) and (2.4), respectively . Then for any constant ξ we have: πtξ = m(t) × Λt h i Es πtξ = m(t) × Λs
for all
t,
(6.24)
for all
s < t,
(6.25)
where the deterministic function m(t) and the martingale Λ = (Λt ) are de ned as: ´
t ξ−1 T λ (s)λ(s)}ds 2 m(t) , eξ 0 {−r(s)+ , ´. T Λ , E −ξ 0 λ (s) dW s .
Proof.
(6.26)
By using proposition A.4 (part iii) of appendix A, we obtain equation (6.24) as follow:
πtξ
´ ˆ . ξ T − 0t ru du = e E − λ (s) dW s 0
t
ˆ . ξ T −ξ 0 r(s)ds = e E − λ (s) dW s 0 t ˆ . ´t ´. T ξ(ξ−1) T [− −ξ 0 r(s)ds λ (s)dW ] s t 0 = e E −ξ λ (s) dW s e 2 0 t ˆ . ´t ξ−1 T λT (s) dW s . = eξ 0 {−r(s)+ 2 λ (s)λ(s)}ds E −ξ ´t
0
(6.27)
t
To obtain equation (6.25), we only need to take conditional expectation from the last result, and use the fact that
m(.)
is deterministic and
Λ
is a martingale:
h i Es πtξ = Es [m(t) × Λt ] = m(t)Es [Λt ] = m(t) × Λs .
36
(6.28)
Theorem 6.3. Consider the market de ned in de nition6.1 . Then: (i) Consider the Merton's problem with the total utility as: J(u) = E where
(u) WT
γ ,
γ
(6.29)
γ > 0 is a constant. Then the optimal portfolio is deterministic and is given by: u=
−1 1 ΣΣT (µ − r1n×1 ) 1−γ
(6.30)
(ii) Consider the Merton's problem with the total utility as: ˆ J(c, u) = E
T
(u,c)
ct Wt
γ
γ
0
dt +
(u,c)
WT
γ
γ
i.e. the case of power utility with consumption. Then the optimal portfolio is the same as the case of no consumption (i.e. equation 6.30). The optimal proportional consumption is also deterministic and is given by: ct = ´ T t
m(t)
m(s)ds + m(T )
,
(6.31)
where the function m(t) is de ned as: ´
t ξ−1 T m(t) , eξ 0 {−r(s)+ 2 λ (s)λ(s)}ds ,
with ξ = Proof.
γ γ−1
(6.32)
.
As pointed out in remark 4.10, since the market is complete, equation (4.29) holds
automatically. We only need to nd an expression for
(λ) (λ) a(C ,Z ,λ) ,
and then equations (4.27)
and (4.35) will give us the optimal solution. Also note that since the market is complete and −1 the unique MPR is λ = Σ (µ − r1n×1 ), we may drop the function argument (λ) in η (λ), (λ) π ,etc. (i) To apply the martingale approach we start with the function
IF (y) = y
1 γ−1 .
So
η
η
of lemma 4.4. We have
can be derived as follow:
E [πT IF (ηπT )]
=
h γ i 1 η γ−1 E πTγ−1 = w γ−1
w =⇒ η = h γ i E πTγ−1
.
(6.33)
(0,Z (λ) ,λ) de ned in theorem 4.9. Note that by taking The next goal is to nd an expression h ifor a ξ γ ξ , γ−1 in lemma 6.2 we have Et πT = m(T ) × Λt , from which we may also conclude: h i h i E πTξ = E0 πTξ = m(T ) × Λ0 = m(T ). 37
(6.34)
π (λ) W (0,Z
Now consider the martingale
(λ)
(0,Z (λ) ,λ)
πt Wt
(λ) ,λ)
of equation (4.4),
, Et [πT IF (ηπT )] h γ i 1 = η γ−1 Et πTγ−1 w = (m(T ) × Λt ) m(T ) = w × Λt ˆ . γ T λ (s) dW s . = wE − γ−1 0 t
(6.35)
This last result gives us
(0,Z (λ) ,λ)
a
,
W , π (λ) W (C,Z,λ)
0 t
(λ) (0,Z (λ) ,λ) πt Wt
γ λ (t) γ−1 γ = − Σ−1 b. γ−1
= −
(6.36)
Finally by using equation (4.35) we nd the optimal portfolio:
(ii) Here
u(x, t) = U (x) =
ΣΣT
−1
xγ , so γ
b + ΣT
−1
(λ)
(λ)
a(C ,Z ,λ) −1 −1 γ = ΣΣT b − ΣT Σ−1 b γ−1 −1 1 = ΣΣT b. 1−γ
u =
(6.37)
1
I(y, t) = IF (y) = y γ−1 .
ˆ
Hence equation 4.6 would become:
T
w = E
πt I(ηπt , t)dt + πT IF (ηπT )
0
ˆ
T
1 γ−1
= E
πt (ηπt ) dt + πT (ηπT ) ˆ T γ γ 1 γ−1 γ−1 γ−1 = η E πt dt + πT ,
1 γ−1
0
(6.38)
0 And we obtain
η
as:
γ−1
w η = h´ γ γ i T γ−1 γ−1 E 0 πt dt + πT To nd an expression for
(λ) (λ) a(C ,Z ,λ) ,
(6.39)
rst note that from equation 6.25 of lemma 6.2, we
38
have
Et πsξ = m(s) × Λt ˆ Et
t < s. So: ˆ T h i ξ ξ Et πsξ ds + Et πTξ = πs ds + πT t ˆ T = m(s) × Λt ds + m(T ) × Λt t ˆ T m(s)ds + m(T ) × Λt . =
for all
T
t
(6.40)
t Now by applying the product rule we obtain:
ˆ
T
πsξ ds
Et
+
πTξ
ˆ
T
m(s)ds + m(T ) ˆ T ˆ t Λτ d m(s)ds + m(T ) + 0 . τ ˆ t ˆ T m(s)ds + m(T ) dΛτ + 0 τ ˆ T ˆ t = m(s)ds + m(T ) − Λτ m (τ ) dτ 0 0 ˆ t ˆ T − m(s)ds + m(T ) Λτ ξλT (τ ) dW τ 0 τ ˆ t ˆ T Λτ m (τ ) dτ m(s)ds + m(T ) − = 0 0 ˆ t ˆ T ξ ξ Eτ πs ds + πT ξλT (τ ) dW τ . −
=
0
t
0
(6.41)
τ
In the last step we used equation (6.40). From this last result we may conclude:
ˆ
T
πTξ
πsξ ds
ˆ
T
+ E = m(s)ds + m(T ) 0 0 h h´ ii0 T ξ ξ W , E. . πs ds + πT h´ i t = −ξλ (τ ) T ξ ξ Et t πs ds + πT From the rst equation above we will nd the following explicit form for 1
η γ−1 = E Now consider the process
h´ T
π (λ) W (C
(C (λ) πt Wt
w
πsξ ds 0
+
(λ) ,Z (λ) ,λ)
)
(λ) ,Z (λ) ,λ
πTξ
i = ´T 0
w m(s)ds + m(T )
(6.42)
(6.43)
η:
.
(6.44)
of equation (4.4):
ˆ
T
, Et πt I(ηπt , t)dt + πT IF (ηπT ) t ˆ T 1 ξ ξ = η γ−1 Et πs ds + πT , t 39
(6.45)
where we took
ξ=
γ . By using equation (6.43) we may conclude γ−1
h i0 C (λ) ,Z (λ) ,λ) (λ) ( W,π W
(C (λ) ,Z (λ) ,λ) at ,
t
(C (λ) πt W t
(λ) ,Z (λ) ,λ
)
ii0 h h´ T W , Et t πsξ ds + πTξ i t = −ξλ (τ ) h´ = T ξ ξ Et t πs ds + πT But this is exactly what we found in equation (6.36) for the case of zero consumption. Hence the optimal portfolio would be also given by equation (6.30). To obtain the optimal consumption, we may proceed as follow:
πt Ct = πt I(ηπt , t) 1
= η γ−1 πtξ 1
= η γ−1 × m(t) × Λt ˆ T 1 m(t) = ´T η γ−1 m(s)ds + m(T ) Λt m(s)ds + m(T ) t t ˆ T 1 m(t) ξ ξ = ´T η γ−1 Et πs ds + πT m(s)ds + m(T ) t t m(t) (C (λ) ,Z (λ) ,λ) πt Wt . = ´T m(s)ds + m(T ) t Finally by using the relation
ct =
Ct
(C (λ) ,Z (λ) ,λ) W
(6.46)
, we will get the optimal proportional consump-
t
tion of equation (6.31).
6.2.2
Complete Gaussian Market
Consider the complete Gaussian market de ned bellow.
De nition 6.4.
By a
Gaussian Market
we are referring to the special case of de nition 2.1
where the volatility and the market price of risk process are deterministic, i.e.
λt = λ (t).
Furthermore the short-rate process
r
follows a
k -factor
Σt = Σ(t)
and
Gaussian short rate model.
That is, we have:
rt =
k X
(i)
rt = 11×k r t ,
(6.47)
i=1 where
r = r(1) , . . . , r(k) are independent Gaussian process (i) (i) ft(i) drt = ϑ(i) (t) + α(i) (t) rt dt + δ (i) (t)dW
with the following dynamics:
f or i = 1, . . . , k,
(6.48)
or equivalently,
f t. dr t = (θ (t) + diag (α (t)) r t ) dt + diag (δ (t)) dW Here
ϑ(i) (.), α(i) (.)
and
δ (i) (.)
are given deterministic functions.
40
(6.49)
We will also need to consider
De nition 6.5.
The
τ -maturity
τ -maturity
bonds explicitly.
bond price process
(τ )
Bt
B
(τ )
=
(τ ) Bt
is de ned as:
´τ − t rs ds , EQ e t
(6.50)
T σ (τ ) (t) to be the deterministic volatility of B (τ ) and ρ(A,τ ) = ρ(1,τ ) (t) , . . . , ρ(n,τ ) (t) to be the deterministic instantaneous correlation between the returns of assets A and the τ (τ ) maturity bond B . (τ ) (τ ) The following lemma will introduce a family of martingales F = Ft which play a Also assume
main role in nding the optimal solutions.
Lemma 6.6. For a maturity τ > 0 and an arbitrary constant ξ , de ne a family of martingales (τ ) F (τ ) = Ft as: E F
(τ )
t
−ξ
, Et e
´τ 0
rs ds
(6.51)
Then for the Brownian motion W we have:
Proof.
W , F (τ )
0
= ξσ (τ ) .
First we show that for some deterministic function
(6.52)
f (τ )
we have:
´τ ´τ −ξ 0 ru du Et e−ξ 0 ru du = f (τ )EQ e . t To see this, recall from equation (2.5) that
f u = dW u + λ (u) du. dW
(6.53)
Now from equation(B.2)
of appendix B, we have:
rt
ˆ t ˆ t −1 −1 fu Υ (u)θ(u)du + Υ (u)diag (δ (t)) dW = Υ(t) r 0 + 0 0 ˆ t ˆ t −1 −1 = Υ(t) r 0 + Υ (u)θ(u)du + Υ (u)diag (δ (t)) dW u 0 0 ˆ t Υ−1 (u)diag (δ (t)) λ(u)du +Υ(t) 0
, r˜ t + g(t).
(6.54)
r˜ , 11×k r˜ and g (t) , 11×k g(t). Note that rt = r˜t + g (t). Also, since r and P-dynamics of r˜ are identical, the same is true for r and r˜. So: ´τ ´τ ´τ Et e−ξ 0 ru du = e−ξ 0 g(u)du Et e−ξ 0 r˜u du ´τ ´ −ξ 0τ g(u)du Q −ξ 0 ru du = e Et e .
De ne of
And we may take
f (τ ) , e−ξ
´τ 0
g(u)du
.
41
the
Q-dynamics
(6.55)
´t
r du|Fs is Gaussian, and that its conditional variance s u is deterministic (to see this in a special case, refer to equation B.7 in appendix B). So we may Now it is a well known fact that
write:
E F
(τ )
t
, Et e
−ξ
´τ 0
ru du
´τ −ξ 0 ru du = f (τ )EQ e t ´τ ´t −ξ t ru du = f (τ )e−ξ 0 ru du EQ e t = f (τ )e−ξ = f (τ )e = f (τ )e = f (τ )e Since Vart
´τ t
ru du
−ξ
−ξ
−ξ
´t 0
´t 0
´t 0
´t 0
´τ Q ´τ ξ2 ru du −ξEt ( t ru du)+ 2 Vart ( t ru du)
e
ru du
e
ru du
e
ru du
e
ξ2 −ξ 2
ξ2 −ξ 2
ξ2 −ξ 2
´τ Vart ( t ru du)
e−Et (
´τ Vart ( t ru du)
´τ ξ − t ru du EQ e t
´τ Vart ( t ru du)
Bt
Q
(τ )
´τ t
ξ
ru du)+ 21 Vart (
´τ t
ru du)
ξ
.
(6.56)
is deterministic, we may conclude that:
F (τ ) = ξLog B (τ ) + BV. for some process
BV
(6.57)
of bounded variation. From this we may conclude
0 0 W , F (τ ) t = ξ W , Log B (τ ) t = ξσ (τ ) (t),
(6.58)
which is equation (6.52). The following lemma is a counterpart of lemma 6.2, and will play a crucial rule.
Lemma 6.7. Consider the market de ned in de nition 6.4 and let π = (πt ) be the SPD process. Then for any constant ξ we have: (t)
π ξ = m(t) × Λt , h it Es πtξ = m(t)Λ(t) for s < t, s
(6.59) (6.60)
where the deterministic function m(t) and the family of martingales Λ(t) are de ned as follow: ´t
ξ−1
T
m(t) , eξ 0 (−ξσ (s)+ 2 λ(s)) λ(s)ds , T ´. Λ(t) , E ξ 0 σ (t) (s) − λ (s) dW s .
Proof.
(t)
(6.61)
With a calculation similar to the way we obtain equation(6.27), we have:
πtξ
´ ˆ . ξ T − 0t rs ds = e E − λ (s) dW s 0 tˆ . ´ ´ ξ(ξ−1) t T T λ (s)λ(s)ds −ξ 0t rs ds 0 2 e E −ξ λ (s) dW s . = e 0 42
t
(6.62)
From equation(6.51) we have
πtξ
E F (t)
t
= e−ξ
´t 0
rs ds
. Now we can obtain equation(6.59) as follow:
ˆ . T 0 λ (s) dW s = e E F t E −ξ 0 ˆ t. ´ ´ ξ(ξ−1) t T T (t) λ (s)λ(s)ds [F (t) ,−ξ 0. λT (s)dW s ]t 0 2 = e λ (s) dW s E F −ξ e 0 t ˆ . n o o ´ n 0 (t) 0 T ξ 0t −λT (s)[W,F (t) ] + ξ−1 λ (s)λ(s) ds T 2 s = e F , W s − ξλ (s) dW s E ξ(ξ−1) 2
´t
λT (s)λ(s)ds
(t)
0
= m(t) ×
t
(t) Λt ,
(6.63)
where in the last steps we used equation (6.52). Finally equation (6.60) can be obtained by taking conditional expectation from equation(6.59). Finally the following theorem will give us the optimal solutions in this case.
Theorem 6.8. Consider the market de ned in de nition2.1 special case (i). Then: (i) Consider the Merton's problem with the total utility as: J(u) = E
(u)
WT
γ
γ
i.e. the case of power utility with no consumption. Then the optimal portfolio is given by: ut =
−1 −1 1 (T ) ΣTt Σt bt − γσ t 1−γ
(6.64)
(ii) Consider the Merton's problem with the total utility as: ˆ J(c, u) = E
T
(u,c) ct Wt
γ dt +
γ
0
(u,c) WT
γ
γ
i.e. the case of power utility with consumption. Then the optimal portfolio and optimal consumption are given by: u =
−1 1 ΣΣT b 1−γ (´ T ) (s) (s) (T ) (T ) m(s)Λ σ (t) ds + m(T )Λ σ (t) γ −1 t t t − ΣT , ´T (s) (T ) 1−γ m(s)Λ ds + m(T )Λ t t t
(6.65)
(t)
ct = ´ T t
m(t) × Λt (s)
(T )
m(s) × Λt ds + m(T ) × Λt
.
where the m(t) and Λ(τt ) are de ned in equation (6.61) with ξ =
43
(6.66)
γ γ−1
.
Proof.
(i) Note that by the same reasoning as in the proof of part(i) of theorem(6.3), equations
(6.33) and (6.35), we will obtain:
γ−1
w η = h i E πTξ
,
(0,Z (λ) ,λ)
(T )
(λ)
πt W t where
γ . By using the de nition of γ−1
ξ=
(λ) (0,Z (λ) ,λ) πt Wt
(T )
ˆ
Λt
(6.67)
= wΛt ,
(6.68)
, equation (6.61), we have:
t ) ξΛ(T s
T (s) − λ (s) dW s
(T )
σ = w 1+ 0 ˆ t T (λ) = w+ ξπs(λ) Ws(0,Z ,λ) σ (T ) (s) − λ (s) dW s .
(6.69)
0 And we obtain
0 W , π (λ) W (C,Z,λ) t (0,Z (λ) ,λ) (T ) at , = ξ σ (t) − λ (t) . (λ) (λ) (0,Z ,λ) πt Wt
(6.70)
Finally by using equation (4.35) we nd the optimal portfolio:
ΣΣT
u =
−1
b + ΣT
−1
a(C
(λ) ,Z (λ) ,λ)
−1 (T ) b + ξ ΣT σ − Σ−1 b −1 −1 (T ) = (1 − ξ) ΣΣT b + ξ ΣT σ . 1 −1 ΣT Σ−1 b − γσ (T ) . = 1−γ ΣΣT
=
−1
(6.71)
(ii) By a similar argument as what we used to obtain equations (6.39) and (6.45), we will have:
γ−1
η= (C (λ) πt Wt
w γ γ ´ E 0T πtγ−1 dt+πTγ−1
)
(λ) ,Z (λ) ,λ
=η
Also by using lemma (6.7) we have:
ˆ
Et
T
πsξ ds
+
πTξ
1 γ−1
ˆ
T
=
t
Et
h´
T t
,
πsξ ds
(6.72)
+
πTξ
i
.
(6.73)
h i Et πsξ ds + Et πTξ
t
ˆ
T
=
(s)
(T )
m(s)Λt ds + m(T )Λt .
(6.74)
t Note that:
ˆ
T
m(s) ×
d .
ˆ
Λ(s) . ds
T
m(s) × d Λ(s) ds − m(τ ) × Λ(t) τ τ dτ τ ˆ T T (s) (s) = ξm(s)Λτ σ (τ ) − λ (τ ) ds dW τ =
τ
τ ) −m(τ ) × Λ(τ τ dτ 44
(6.75)
So we may continue our calculations in equation (6.74) as follow:
ˆ Et
T
πsξ ds
+
πTξ
ˆ
t
T
=
m(s)ds + m(T ) ˆ t ˆ T (s) m(s) × Λ. ds d + . 0 τ ˆ t (T ) d m(T ) × Λt + 0 ˆ t ˆ T ) m(s)ds + m(T ) − m(τ ) × Λ(τ = τ dτ 0 0 ˆ t ˆ T T (s) (s) ξm(s)Λτ σ (τ ) − λ (τ ) ds dW τ + 0 τ ˆ t T ) ξm(T )Λ(T σ (T ) (τ ) − λ (τ ) dW τ + τ 0 ˆ T ˆ t ) m(s)ds + m(T ) − m(τ ) × Λ(τ = τ dτ 0 0 ˆ t ˆ T (s) ) (T ) m(s)Λ(s) (τ ) ds + m(T )Λ(T (τ ) + ξ τ σ τ σ 0
0
τ
ˆ
T
πsξ ds
−Eτ
+
πTξ
T λ (τ ) dW τ .
(6.76)
τ We may conclude that:
h (C (λ) ,Z (λ) ,λ) at ,
W,π
(λ)
(λ) (λ) W (C ,Z ,λ)
i0 t
(C (λ) πt Wt
)
(λ) ,Z (λ) ,λ
ii0 h´ T W , Et t πsξ ds + πTξ h´ i t = T ξ ξ Et t πs ds + πT (´ T ) (s) (T ) m(s)Λt σ (s) (t) ds + m(T )Λt σ (T ) (t) t = ξ − λ (t) . ´T (s) (T ) m(s)Λt ds + m(T )Λt t h
By using equation (4.35) we will nd the optimal portfolio of equation (6.65):
−1 (C (λ) ,Z (λ) ,λ) b + ΣT a −1 = (1 − ξ) ΣΣT b (´ T ) (s) (s) (T ) (T ) m(s)Λ σ (t) ds + m(T )Λ σ (t) −1 t t t +ξ ΣT . ´T (s) (T ) m(s)Λ ds + m(T )Λ t t t
u =
ΣΣT
−1
45
(6.77)
To obtain the optimal consumption of equation (6.66), we may proceed as follow:
ct =
πCt (η)
πt Wt
1
=
1
η γ−1
´
T t
η γ−1 πtξ (s)
(T )
m(s) × Λt ds + m(T ) × Λt
(t)
= ´T t
m(t) × Λt (s)
(T )
m(s) × Λt ds + m(T ) × Λt
where in the last step we used equation (6.59).
46
,
(6.78)
Chapter 7 Conclusions and Recommendations In this project, we considered the classical portfolio choice problem, also known as Merton's problem, when the opportunity set is stochastic.
The two main approaches to the problem,
namely the stochastic control theory and the martingale (or duality) approach, were discussed along with an extended literature study in chapters 1, 3, and 4.
Then we proposed a new
approach to portfolio choice problem, called the
in chapter 5.
chapter 6, we solve three special cases:
direct approach,
Finally, in
logarithmic utility in an incomplete market, power
utility in the original Merton's setting, and power utility in a market with Gaussian term structure.
For the logarithmic case we provided three dierent approaches.
For the power
utility cases, we nd out that the optimal solutions dier in the following ways,
•
In the Merton setting, the optimal portfolio only includes the myopic term, while the excess hedging demand term (containing the covariance between zero-coupon bonds and assets) enters in the optimal portfolio for the Gaussian term structure case.
•
In the Merton setting the optimal portfolio is deterministic whether we consider consumption or not, but for the Gaussian term structure case the optimal portfolio is deterministic when there is no consumption, and stochastic otherwise.
As the potential line of research based on the work conducted in this project, we propose the following topics:
•
Considering speci c stochastic factor models We may consider stochastic factors to model asset predictability, stochastic volatility, and interest rates. For example one may take other term-structure models such as C.I.R. or a stochastic volatility model such as Heston model. Alternatively one may try to consider more general market settings such as quadratic asset returns (which includes both C.I.R. and Heston model, see [61]) or HJM model (for example see [73]). See chapters 1 and 3 for more references.
•
Adapting the direct approach to alternative preference criteria As mentioned earlier, the traditional
static
choice of the utility function and investment
horizon is not realistic. it has long been recognized by economists that preferences may not be Intertemporally separable.
In particular, the utility associated with the choice
of consumption at a given date is likely to depend on past choices of consumption. For example high past consumption generates a desire for high current consumption. According to Browning [13], this idea dates back to the 1890 book `Principles of Economics' by
47
Alfred Marshall. Generalizations of standard time-separable preferences that have been
recursive or stochastic dierential utility (see, for example, [32, 76, 77]), habit formation criterion (see [67]), and forward performance criterion (see [68, 87]). One suggested include
may try to adapt the idea of the direct approach to these alternative formulations.
•
Providing validity conditions for the direct approach As mentioned in chapter 5, we have only checked the rst order optimality conditions. So, technically, the results obtain by the direct approach can only be considered as candidates for the optimal solution.
One may try to provide su cient conditions, or ideally the
minimal conditions, under which these candidate solutions are indeed optimal.
•
Extending the direct approach to general semimartingale markets One may try to extend the direct approach to a general semimartingale market.
The
main tools in developing the direct approach are properties of stochastic exponentials, which can still be used in a semimartingale setting. Nonetheless, such extensions would be more involved and obtainting validity conditions would be more demanding.
48
Appendix A Stochastic Exponential and Logarithms De nition A.1. exponential of
Let
X,
X=
(1) (k) Xt , . . . , X t
denoted by
E (X)
T
be a
k×1
semimartingale. Then the stochastic
is de ned as the unique solution of the following SDE
ˆ
t
Y =1+
diag (Y u− ) dX u .
(A.1)
0
T (1) (k) Y = Yt , . . . , Yt
De nition A.2.
Let
the two processes
Y (k)
by
Log (Y ),
and
(k)
Y−
be a
k×1
semimartingale such that, for all
do not vanish. Then the stochastic logarithm of
is de ned as
ˆ Log (Y ) ,
.
(diag (Y u− ))−1 dY u .
Y,
k,
denoted
(A.2)
0 Equivalently,Log (Y
Remark
A.3
.
)
is the unique process
X
satisfying
Y = diag (Y 0 ) E (X).
Y = diag (Y 0 ) E (X). We usually process ϕ we have: ˆ T ˆ T T dY ϕ , ϕT dX. Y 0 0
Suppose that
for a given predictable
denote
dX
by
dY , for example Y
(A.3)
In a similar way, we de ne:
Note that we have
d [ϕ, Y ] , d [ϕ, X] . Y dY = diag (Y ) dX , so we also have: dX = (diag (Y ))−1 dY , d [ϕ, X] = (diag (Y ))−1 d [ϕ, Y ] .
(A.4)
(A.5)
Proposition A.4. Let X and Y be two continuous semimartingale, and α be a constant. Then: (i) E (X) × E (Y ) = e[X,Y ] E (X + Y ) (ii) E(X) = e[Y ]−[X,Y ] E (X − Y ) E(Y ) α(α−1) (iii) E (X)α = e 2 [X] E (αX)
49
Proof.
Note that we have:
dE (X) = E (X) dX ˆ d[E (X)] = d[ E (X) dX] = E (X)2 d[X] Similar equations hold for
Y.
We also have:All the cases are straightforward application of Itô's
formula. (i) We have:
d (E (X) E (Y )) = E (X) dE (Y ) + E (Y ) dE (X) + d [E (X) , E (Y )] dE (Y ) dE (X) d [E (X) , E (Y )] = E (X) E (Y ) + + E (Y ) E (X) E (X) E (Y ) = E (X) E (Y ) {dY + dX + d [X, Y ]} = E (X) E (Y ) d (X + Y + [X, Y ]) So we obtain:
E (X) E (Y ) = E (X + Y + [X, Y ]) = e[X,Y ] E (X + Y ) (ii) Again by Itô's formula:
dE (X) E (X) dE (Y ) 1 E (X) d[E (Y )] E (X) d[E (X) , E (Y )] E (X) = − + 2 −2 d E (Y ) E (Y ) E (Y ) E (Y ) 2 E (Y ) E (Y )2 E (Y ) E (X) E (Y ) E (X) dE (X) dE (Y ) d[E (Y )] d[E (X) , E (Y )] − = − + E (Y ) E (X) E (Y ) E (X) E (Y ) E (Y )2 E (X) {dX − dY + d[Y ] − d[X, Y ]} = E (Y ) Hence:
E (X) = E (X − Y + [Y ] − [X, Y ]) = e[Y ]−[X,Y ] E (X − Y ) E (Y )
(iii) By Itô's formula:
1 dE (X)α = αE (X)α−1 dE (X) + α(α − 1)E (X)α−2 d[E (X)] 2 dE (X) α(α − 1) d[E (X)] α = E (X) α + E (X) 2 E (X)2 α(α − 1) = E (X)α αdX + d[X] 2 Finally we obtain:
α(α−1) α(α − 1) E (X) = E αX + [X] = e 2 [X] E (αX) 2 α
50
Appendix B Other Results Lemma B.1. (variant of Bay's rule)- Let Z = (Zt ) be a positive martingale with Z0 = 1 under measure P. De ne a new measure Q by dQ = ZT . Then a process X = (Xt ) is a Q-(local) dP martingale if and only if
ZX
is a
P
(local) martingale.
Theorem B.2. (change of measure)- Suppose, under measure P, Z = E (Y ) is a positive = ZT . Then a martingale for some semimartingale Y = (Yt ), and de ne measure Q by dQ dP process
Proof.
X = (Xt )
is a
Q-local
From lemmaB.1,
X
X + [X, Y ]
martingale if and only if
is a
Q
local martingale if and only if
is a
ZX
P-local is a
martingale.
P-local
martingale.
But we have
d(ZX) = X− dZ + Z− dX + d[X, Z], The rst term in the right-hand-side is a local martingale, since local martingale if and only if Note that we have:
V
ˆ
Hence
V
Z
ˆ Z− dX + [X, Z] =
ˆ Z− dX +
ZX is a De ne U , X+[X, Y ].
is a martingale. So
Z− dX+[X, Z] is a local martingale.
ˆ
=
U =
V ,
´
ˆ Z− d[X, Y ] =
Z− dU
1 dV Z−
is a local martingale if and only if
U , X + [X, Y ]
is a local martingale.
Theorem B.3. (Linear SDE) Consider the following SDE: ( dr t = (θ(t) + A(t)r t ) dt + ∆(t)dW t r 0 is given
(B.1)
Where W = (Wt ) is a k-dimensional Brownian motion, and A (.), θ (.) and ∆ (.) are deterministic matrix functions of appropriate dimensions. Then subject to some stability conditions on the coe cients, the unique strong solution of equation(B.1)is given by: ˆ t ˆ t −1 −1 r t = Υ(t) r 0 + Υ (u)θ(u)du + Υ (u)∆(u)dW u . 0
(B.2)
0
Where Υk×k (t) is the solution of the following matrix dierential equations: ( ˙ Υ(t) = A(t)Υ(t), Υ(0) = Ik×k . 51
(B.3)
Furthermore the mean andi covariance functions of r, namely m(t) = E [rt ] and V (t) = h E (rt − m(t)) (rt − m(t))T are the solution of the linear equations: ˙ m(t) = A(t)m(t) + θ(t), ˙ V (t) = A(t)V (t) + V (t)AT (t) + ∆(t)∆T (t),
(B.4)
which have the following solutions:
n o ´t m(t) = Υ(t) m(0) + 0 Υ−1 (s)θ(s)ds , n o (B.5) ´t T V (t) = Υ(t) V (0) + 0 Υ−1 (u)∆(u) (Υ−1 (u)∆(u)) du ΥT (t). h i Finally the auto-correlation function of r, i.e. ρ(s, t) = E (rs − m(s)) (rt − m(t))T is given
by:
ˆ
s∧t −1
ρ(s, t) = Υ(s) V (0) +
−1
Υ (u)∆(u) Υ (u)∆(u)
T
du ΥT (t).
(B.6)
0
Theorem B.4. Consider the one-factor Hull-White extension of Vasicek model: drt = (ϑ(t) − art ))dt + δdWt
Also assume that initial bond prices P M (0, T ), and initial instantaneous forward-rate curve, f M (0, T ) are given.Then we have: (i)The short-rate rt is explicitly given by: ˆ
−a(t−s)
ˆ
t
t
ϑ(u)du + δ e−a(t−u) dWu s ˆ t s = (rs − ω(s)) e−a(t−s) + ω(t) + δ e−a(t−u) dWu
rt = rs e
+
−a(t−u)
e
s
Where: ω(t) , f M (0, t) +
2 δ2 1 − e−at 2 2a
(ii)rt is Gaussian with mean and variance given by:
(iii)
´t s
Es [rt ] = (rs − ω(s)) e−a(t−s) + ω(t) δ2 Vars [rt ] = 1 − e−2a(t−s) 2a ru du|Fs is also Gaussian with (conditional) mean and variance given by: h´ i M t P (0,s) Es s ru du = B(s, t) (rs − ω(s)) + log P M (0,t) + 12 (V (0, t) − V (0, s)) h´ i Vars st ru du = V (s, t)
where: 1 1 − e−a(T −t) a δ2 2 −a(t−s) 1 −2a(t−s) 3 V (s, t) , 2 t − s + e − e − a a 2a 2a
B(s, t) ,
52
(B.7)
(iv) For s < t we have: i h ´t P (s, t) , Es e− s ru du = A(s, t)e−B(s,t)rs
where:
P M (0, t) A(s, t) = M e P (0, s)
n o 2 f M (0,s)B(s,t)− δ4a (1−e−2as )B 2 (s,t)
53
Bibliography [1] Ait-Sahalia, Y., and Brandt, M. Variable selection for portfolio choice.
Finance 56, 4 (2001), 12971351.
[2] Arrow, K. J.
Aspects of the theory of risk-bearing.
Journal of
Yrjo Jahnsson lectures. Saatio,
Helsinki, 1965. [3] Bates, D. Post-87 crash fears in the s&p 500 futures option market.
metrics 94, 1-2 (2000), 181238.
[4] Becherer, D.
Journal of Econo-
The numeraire portfolio for unbounded semimartingales.
Stochastics 5, 3 (2001), 327341.
Finance and
[5] Benzoni, L., Collin-Dufresne, P., and Goldstein, R. S. Portfolio choice over the life-cycle when the stock and labor markets are cointegrated.
the Journal of Finance lxii,
5 (2007), 21232167. [6] Borell, C. asset prices.
Monotonicity properties of optimal investment strategies for log-brownian
Mathematical Finance 17, 1 (2007), 143153.
[7] Bouchard, B., and Touzi, N. solutions.
Weak dynamic programming principle for viscosity
submitted for publication
(2009).
[8] Brandt, M. Estimating portfolio and consumption choice: A conditional euler equations approach.
Journal of Finance 54, 5 (1999), 16091645.
[9] Brennan, M. The role of learning in dynamic portfolio decisions.
1, 3 (1997), 295306.
European nance review
[10] Brennan, M., Schwartz, E., and Lagnado, R. Strategic asset allocation.
Economic Dynamics and Control 21, 8-9 (1997), 13771403.
journal of
[11] Brennan, M. J., and Xia, Y. Stochastic interest rates and the bond-stock mix.
pean Finance Review 4, 2 (2000), 197210.
[12] Brennan, M. J., and Xia, Y.
Finance 57, 3 (2002), 12011238.
Dynamic asset allocation under in ation.
Euro-
Journal of
[13] Browning, M. A simple nonadditive preference structure for models of household behavior over time.
The Journal of Political Economy 99, 3 (1991), 607637.
[14] Campbell, J., and Cochrane, J. By force of habit: A consumption-based explanation of aggregate stock market behavior.
Journal of Political Economy 107, 2 (1999), 205251. 54
[15] Campbell, J., and Viceira, L. Consumption and portfolio decisions when expected returns are time varying.
Quarterly Journal of Economics 114, 2 (1999), 433495.
[16] Campbell, J. Y., and Viceira, L. M. Who should buy long-term bonds?
Economic Review 91, 1 (2001), 99127.
[17] Campbell, J. Y., and Viceira, L. M.
Long-Term Investors.
Strategic Asset Allocation: Portfolio Choice for
Oxford University Press, 2002.
[18] Canner, N., Mankiw, N., and Weil, D. An asset allocation puzzle.
nomic Review 87, 1 (1997), 181191.
[19] Carmona, R. E.
American
Indierence pricing.
Princeton University Press, 2009.
[20] Castaneda-Leyva, N., and Hernandez-Hernandez, D.
Optimal consumption-
investment problems in incomplete markets with stochastic coe cients.
on Control and Optimization 44, 4 (2005), 13221344. [21] Chacko, G., and Viceira, L. M.
American Eco-
SIAM Journal
Dynamic consumption and portfolio choice with
stochastic volatility in incomplete markets.
Review of Financial Studies 18,
4 (2005),
13691402. [22] Choulli, T., Stricker, C., and Li, J. Minimal hellinger martingale measures of order q.
Finance and Stochastics 11
(2007), 399427.
[23] Constantinides, G. M. A theory of the nominal term structure of interest rates.
Review of Financial Studies 5, 4 (1992), 531552.
The
[24] Cox, J. C., and Huang, C. F. Optimal consumption and portfolio policies when asset prices follow a diusion process.
Journal of Economic Theory 49
[25] Cox, J. C., and Huang, C.-f.
A variational problem arising in nancial economics.
Journal of Mathematical Economics 20
(1991), 465487.
[26] Cox, J. C., Ingersoll, J. E., and Ross, S. A. interest rates.
(1989), 3383.
Econometrica 53, 2 (1985), 385407.
A theory of the term structure of
[27] Crandall, M. G., Ishii, H., and Lions, P.-L. User's guide to viscosity solutions of second order partial dierential equations.
27, 1 (1992), 167.
Bulletin of the American Mathematical Society
[28] Deelstra, G., Grasselli, M., and Koehl, P.-F. Optimal investment strategies in a cir framework.
journal of Applied Probability 37
(2000), 936946.
[29] Detemple, J., Garcia, R., and Rindisbacher, M. A monte carlo method for optimal portfolios.
The Journal of Finance 58, 1 (2003), 401446.
[30] Detemple, J., and Rindisbacher, M.
Closed-form solutions for optimal portfolio
selection with stochastic interest rate and investment constraints.
15, 4 (2005), 539568.
55
Mathematical Finance
[31] Duffie, D.
nance,
Intertemporal asset pricing theory.
Handbook of The Economics of Fi-
In
G. Constantinides, M. Harris, and R. Stulz, Eds., vol. 2. Elsevier North-Holland,
Amsterdam, 2003, pp. 639742. [32] Duffie, D., and Lions, P.-L. Pde solutions of stochastic dierential utility.
Mathematical Economics 21
Journal of
(1992), 577606.
[33] Ekeland, I., and Taflin, E.
A theory of bond portfolios.
Probability 15, 2 (2005), 12601305.
[34] Fama, E., and Schwert, G. Asset returns and in ation.
5, 2 (1977), 115146.
The Annals of Applied
Journal of Financial Economics
[35] Ferson, W. E., and Harvey, C. R. The risk and predictability of international equity
The Review of Financial Studies 6, 3 (1993), 527566.
returns.
[36] Fleming, W. H., and Hernandez-Hernandez, D. An optimal consumption model with stochastic volatility.
Finance and Stochastics 7
[37] Fleming, W. H., and Soner, H. M.
tions, second edition ed.
(2003), 245262.
Controlled Markov Processes and Viscosity Solu-
Springer, 2006.
[38] French, K., Schwert, G., and Stambaugh, R. Expected stock returns and volatility.
Journal of Financial Economics 19, 1 (1987), 329.
[39] Glosten, L. R., Jagannathan, R., and Runkle, D. E. On the relation between the expected value and the volatility of the nominal excess return on stocks.
The Journal of
[40] Goll, T., and Kallsen, J. Optimal portfolios for logarithmic utility.
Stochastic Pro-
Finance 48, 5 (1993), 17791802. cesses and Their Applications 89
(2000), 3148.
[41] Goll, T., and Kallsen, J. A complete explicit solution to the log-optimal portfolio problem.
The Annals of Applied Probability 13, 2 (2003), 774799.
[42] Harvey, C. R.
Time-varying conditional covariances in tests of asset pricing models.
Journal of Financial Economics 24, 2 (1989), 289317.
[43] He, H., and Pearson, N. D.
Consumption and portfolio policies with incomplete
markets and short-sale constraints: The in nite dimensional case.
Theory 54
Journal of Economic
(1991), 259304.
[44] Heaton, J., and Lucas, D.
Macroeconomic Dynamics 1
Market frictions, savings behavior, and portfolio choice.
(1997), 76101.
[45] Karatzas, I., and Kardaras, C. The numeraire portfolio in semimartingale nancial models.
Finance and Stochastics 11
(2007), 447493.
[46] Karatzas, I., Lehoczky, J. P., and Shreve, S. E. Optimal portfolio and consumption decisions for a 'small investor' on a nite horizon.
Optimization 25, 6 (1987), 15571586.
56
SIAM Journal on Control and
[47] Karatzas, I., Lehoczky, J. P., Shreve, S. E., and Xu, G.-L. Martingale and duality methods for utility maximization in an incomplete market.
Optimization 29, 3 (1991), 702730.
[48] Karatzas, I., and Shreve, S. E.
SIAM Journal on Control and
Methods of Mathematical Finance.
Springer, 1998.
[49] Karoui, N. E., Peng, S., and Quenez, M. C. Backward stochastic dierential equations in nance.
Mathematical Finance 7, 1 (1997), 171.
[50] Kim, T. S., and Omberg, E. Dynamic nonmyopic portfolio behavior.
cial Studies 9, 1 (1996), 141161.
[51] Kimball, M. S. Precautionary saving in the small and in the large.
Review of Finan-
Econometrica 58, 1
(1990), 5373. [52] Korn, R., and Kraft, H. stochastic interest rates.
A stochastic control approach to portfolio problems with
SIAM Journal on Control and Optimization 40, 4 (2001), 1250
1269. [53] Korn, R., and Kraft, H. on the stability of continuous-time portfolio problems with stochastic opportunity set.
Mathematical Finance 14, 3 (2004), 403414.
[54] Kraft, H. Optimal portfolios and heston's stochastic volatility model: an explicit solution for power utility.
Quantitative Finance 5, 3 (2005), 303313.
[55] Kramkov, D., and Schachermayer, W. the asymptotic elasticity of utility functions and optimal investment in incomplete markets.
The Annals of Applied Probability 9,
3
(1999), 904950. [56] Kramkov, D., and Schachermayer, W. Necessary and su cient conditions in the problem of optimal investment in incomplete markets.
13, 4 (2003), 15041516.
The Annals of Applied Probability
[57] Kramkov, D., and Sirbu, M. On the two-times dierentiability of the value functions in the problem of optimal investment in incomplete markets.
16, 3 (2006), 13521384.
Annals of Applied Probability
[58] Landsberger, M., and Meilijson, I. Demand for risky nancial assets: A portfolio analysis.
Journal of Economic Theory 50, 1 (1990), 204213.
[59] Lions, P.-L. Optimal control of diusion processes and hamilton-jacobi-bellman equations part 2 : viscosity solutions and uniqueness.
Equations 8, 11 (1983), 12291276.
Communications in Partial Dierential
[60] Lions, P.-L. Optimal control of diustion processes and hamilton-jacobi-bellman equations part i:
the dynamic programming principle and application.
Partial Dierential Equations 10, 8 (1983), 11011174.
[61] Liu, J. Portfolio selection in stochastic environments.
20, 1 (2007), 139.
57
Communications in
The Review of Financial Studies
[62] Malamud, S., and Trubowitz, E. The structure of optimal consumption streams in general incomplete markets.
Mathematics and Financial Economics 1
[63] Markowitz, H. Portfolio selection.
(2007), 129161.
The Journal of Finance 7, 1 (1952), 7791.
[64] Merton, R. C. Lifetime portfolio selection under uncertainty: the continuous-time case.
The Review of Economics and Statistics 51, 3 (1969), 247257.
[65] Merton, R. C. Optimum consumption and portfolio rules in a continuous-time model.
Journal of Economic Theory 3, 373-413 (1971), 373.
[66] Mnif, M. Portfolio optimization with stochastic volatilities and constraints: An application in high dimension.
Appl Math Optim 56
(2007), 243264.
[67] Munk, C. Portfolio and consumption choice with stochastic investment opportunities and habit formation in preferences.
Journal of Economic Dynamics and Control 32
(2008),
35603589. [68] Musiela, M., and Zariphopoulou, T. The backward and forward dynamic utilities and their associated pricing systems:
The case study of the binomial model.
preprint
(2003). [69] Neave, E. H. Multiperiod consumption-investment decisions and risk preference.
of Economic Theory 3, 1 (1971), 4053.
Journal
[70] Pagan, A. R., and Schwert, G. W. Alternative models for conditional stock volatility.
Journal of Econometrics 45, 1-2 (1990), 267290.
[71] Pham, H. Smooth solutions to optimal investment models with stochastic volatilities and portfolio constraints. [72] Pliska, S. R.
Applied Mathematics and Optimization 46, 1 (2002), 5578.
A stochastic calculus model of continuous trading: Optimal portfolios.
Mathematics of Operations Research 11, 2 (1986), 371382.
[73] Ringer, N., and Tehranchi, M. Optimal portfolio choice in the bond market.
and Stochastics 10
[74] Ross, S. A. applications.
Finance
(2006), 553573.
Some stronger measures of risk aversion in the small and the large with
Econometrica 49, 3 (1981), 621638.
[75] Sangvinatsos, A., and Wachter, J. Does the failure of the expectations hypothesis matter for long-term investors?
Journal of Finance 60, 1 (2005), 179230.
[76] Schroder, M. D., and Skiadas, C. Optimal lifetime consumption-portfolio strategies under trading constraints and generalized recursive preferences.
their Applications 108
Stochastic Processes and
(2003), 155202.
[77] Schroder, M. D., and Skiadas, C. Lifetime consumption-portfolio choice under trading constraints, recursive preferences, and nontradeable income.
their Applications 115, 1 (2005), 130.
58
Stochastic Processes and
[78] Scruggs, J. T. Resolving the puzzling intertemporal relation between the market risk premium and conditional market variance: A two-factor approach.
Journal of Finance 53,
2 (1998), 575603. [79] Sorensen, C.
Dynamic asset allocation and xed income management.
Financial and Quantitative Analysis 34, 4 (1999), 513531.
[80] Stoikov, S., and Zariphopoulou, T. hedgeable risks and under cara preferences. [81] Stutzer, M.
[82] Touzi, N.
Optimal investments in the presence of un-
IMA Volume Series in press
Portfolio choice with endogenous utility:
Journal of Econometrics 116
Journal of
(2009).
a large deviations approach.
(2003), 365386.
Stochastic control problems, viscosity solutions, and application to nance.
Lecture Notes Scuola Normale Superiore
[83] Wachter, J.
(2002), Pisa.
Risk aversion and allocation to long-term bonds.
Theory 112, 2 (2003), 325333.
Journal of Economic
[84] Wachter, J. A. Portfolio and consumption decisions under mean-reverting returns: An exact solution for complete markets.
37, 1 (2002), 6391.
The Journal of Financial and Quantitative Analysis
Stochastic Controls: Hamiltonian Systems and HJB Equations, vol. 43 of Stochastic Modelling and Applied Probability. Springer, New York, 1999.
[85] Yong, J., and Zhou, X. Y.
[86] Zariphopoulou, T. A solution approach to valuation with unhedgeable risks.
and Stochastics 5
Finance
(2001), 6182.
[87] Zariphopoulou, T. Optimal asset allocation in a stochastic factor model - an overview and open problems. [88] Zitkovic, G.
Preprint
(2009).
Utility theory-historical perspectives.
Finance. in press, 2009.
59
In
Encyclopedia of Quantitative
A DISSERTATION SUBMITTED FOR THE DEGREE OF Master of Science in Applied Mathematics (Financial Engineering) UNDER SUPERVISION OF Prof. A. Bagchi and Prof. F. Jamshidian
Abstract This project is concerned with the continuous-time portfolio choice problem, also known as Merton's problem, when the opportunity set is stochastic (e.g. when the interest rate and/or volatility is stochastic). There are two main approaches for solving continuous-time portfolio problems: the classical stochastic control approach and the so called martingale approach. The main contribution of this project is to develop a new approach, called the
direct approach.
Unlike the stochastic control approach, it is not based on the Markov state assumption and can be extended to the general semimartingale market (though we have not tried to do so). Its advantage over the martingale approach is that the direct approach, as its name suggests, is dealing with the primal problem
directly.
So, unlike the martingale approach, the completeness
or incompleteness of the market has not so much aect on it. Furthermore we are able to obtain the general form of the optimal portfolio policy directly, and not through a dual problem.
Contents 1 Introduction
2
2 Preliminaries
6
2.1
The market model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.2
Consumption pairs, wealth processes and self- nancing strategies . . . . . . . . .
8
2.3
Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3 Stochastic Control Theory
11
4 Martingale or Duality Approach
14
4.1
Proof of the Martingale Approach . . . . . . . . . . . . . . . . . . . . . . . . . .
5 The Direct Approach 5.1
15
22
Proof of the Direct approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Examples
24
32
6.1
Logarithmic Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
6.2
Power Utility
35
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1
Original Merton Setting
6.2.2
Complete Gaussian Market
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36 40
7 Conclusions and Recommendations
47
A Stochastic Exponential and Logarithms
49
B Other Results
51
Bibliography
54
1
Chapter 1 Introduction∗ A fundamental concern for investors is the problem of portfolio optimization when they trade between a risk-free asset and a large number of risky assets. One of the most important decisions many people face is the choice of a portfolio of assets for retirement savings. These assets may be held as a supplement to de ned-bene t public or private pension plans; or they may be accumulated in a de ned contribution pension plan, as the major source of retirement income. In either case, a dizzying array of assets is available. Institutional investors also face complex decisions. Some institutions invest on behalf of their clients, like pension funds and insurance companies. Others, such as foundations and university endowments, are similar to individuals in that they seek to nance a long-term stream of discretionary spending.
The investment
options for these institutions have also expanded enormously since the days when a portfolio of government bonds was a norm. Modern nance theory is often thought to have started with the mean-variance analysis of Markowitz in the early 1950s (see [63]). This made the portfolio choice theory the original subject of modern nance. He introduced the mean-variance as a criterion for portfolio selection. It is the criterion mostly used by practitioners in spite of its unrealistic hypothesis such as normal returns, single period, etc. The continuous-time portfolio problem has its origin in the pioneering work of Merton [64, 65]. It is concerned with nding the optimal investment strategy of an investor. In other words, how much of which security she should hold at every time instant between now and a time horizon
T,
to maximize her expected utility from intermediate consumption and accumulated
wealth at the end of the time horizon. In the classical Merton problem the investor can allocate her money into a risk-less savings account and
d
dierent risky stocks. Using the methods of
stochastic optimal control, Merton derived a nonlinear partial dierential equation (Bellman equation) for the value function of the optimization problem. He also produced the closed-form solution of this equation for the special cases of power, logarithmic, and exponential utility functions. A drawback of his approach, however, was the assumption of a constant investment opportunity set (i.e.
constant or deterministic factors).
This made the model unrealistic,
speci cally for long-term investors. There are two main approaches for solving continuous-time portfolio problems: the classical stochastic control approach and the so called martingale approach. The stochastic control approach, which has been used since the seminal work of Merton, is based on the requirement of Markov state processes and involves solving a highly nonlinear PDE, the so called ∗
This literature study is partly excerpted from Zariphopoulou [87].
2
Hamilton-Jacobi-Bellman (HJB) equation.
The strength of this approach is in its access to
the well-developed theory of PDEs and the corresponding numerical techniques.
Using this
approach, the cases of special utilities (namely, the exponential, power and logarithmic) have been extensively analyzed. In these cases, convenient scaling properties reduce the associated HJB equation to a more tractable quasilinear one. However, when the utility function is general, very little is known about the maximal expected utility as well as the form and properties of the optimal policies once the log-normality assumption is relaxed and correlation between the stock and the factor is introduced. This is despite the Markovian nature of the problem at hand, the advances in the theories of fully nonlinear PDEs and stochastic control, and the computational tools that exist today. Speci cally, general results on the validity of the Dynamic Programming Principle, regularity of the value function, existence and veri cation of optimal feedback controls, representation of the value function and numerical approximations are still lacking. A more recent approach to the problem of expected utility maximization, which permits us to avoid the assumption of Markov asset prices, is based on duality characterizations of portfolios provided by the set of martingale measures.
For the case of a complete nancial
market, where the set of martingale measures is a singleton, this
martingale methodology
was
developed by Pliska [72], Cox and Huang [24, 25] and Karatzas, Lehoczky and Shreve [46]. Considerably more di cult is the case of incomplete nancial models.
It was studied in a
continuous-time diusion model by He and Pearson [43] and by Karatzas, Lehoczky, Shreve and Xu [47]. The central idea here is to solve a dual variational problem and then to nd the solution of the original problem by convex duality, similarly to the case of a complete model. This powerful approach is applicable to general market models and yields elegant results for the value function and optimal consumption and terminal wealth. However, the optimal portfolio must be then characterized via martingale representation results for the optimal wealth process, so little can be said about the structure and properties of the optimal investments. Another weakness of this approach is the lack of appropriate numerical methods. The lack of rigorous results for the value function when the utility function is general limits our understanding of the optimal policies. Informally speaking, the optimal portfolio consists of two components.
The rst is the so-called myopic portfolio and has the same functional
form as the one in the classical Merton problem.
The second component, usually referred
to as the excess hedging demand, is generated by the stochastic factor.
Conceptually, very
little is understood about this term. In addition, the sum of the two components may become zero which implies that it is optimal for a risk averse investor not to invest in a risky asset with positive risk premium. A satisfactory explanation for this counter-intuitive phenomenon, related to the so-called market participation puzzle (see [5, 18, 44]), is also lacking. Besides these di culties, there are other issues that limit the development of an optimal
static
investment theory in complex market environments.
One of them is the
choice of the
utility function at the speci c investment horizon.
Direct consequences of this choice are,
from one hand, the lack of exibility to revise the risk preferences at other times and, from the other, the inability to assess the performance of investment strategies beyond the prespeci ed horizon. Addressing these limitations has been the subject of a number of studies and various approaches have been proposed. With regards to the horizon length, the most popular alternative has been the formulation of the investment problem in
(0, +∞)
and incorporating
either intermediate consumption or optimizing the investors long-term optimal behavior (see, among others, [48, 81]).
Investment models with random horizon have also been examined
3
([22]). The revision of risk preferences has been partially addressed by recursive utilities (see, for example, [32, 76, 77]). Recently, Musiela and Zariphopoulou developed a forward performance criterion which addresses both issues of the horizon length and revision of risk preferences (see [68, 87]). Let us now focus on the main subject of this project, the use of stochastic factor models in continuous-time portfolio choice problems, and brie y discuss the existing body of work. Stochastic factors have been used in portfolio choice to model asset predictability, stochastic volatility, and interest rates. The predictability of stock returns was rst discussed in [34, 35, 38]; see also [9, 10, 14, 15]. More complex models were analyzed in [1, 8]. The role of stochastic volatility in investment decisions was studied in [3, 21, 38, 39, 42, 70, 78], and others. Models that combine predictability and stochastic volatility were analyzed, among others, in [50, 54, 61, 86, 71]. In a dierent modeling direction, stochastic factors have been incorporated in asset allocation models with stochastic interest rates (see, for example, [11, 12, 16, 23, 26, 28, 30, 75, 79, 83]). From the technical point of view, the analysis is not much dierent. However, various technically interesting questions arise (see, for example, [52, 54, 73]). Classical textbooks on the subject include [17, 31, 48], among others. More speci cally, Korn and Kraft ([52]), Zariphopoulou [86], Pham [71], and Fleming and Hernández-Hernández [36] used stochastic control approach to handle the optimal consumption and asset allocation problems with stochastic opportunity set. However they took some restrictive assumptions on stochastic factors as well as the market price of risk, which excludes models such as Heston's stochastic volatility model. Kramkov and Schachermayer ([55, 56]) provided minimal conditions for the validity of martingale approach on a general semimartingale market. Korn and Kraft ([53]) provided some counter examples to highlight the fact that uncritical application of the two main approaches of solving continuous-time portfolio problems can lead to wrong conclusions if only the necessary and not the su cient conditions of the main results are checked. Chacko and Viceira [21] considered recursive utility (including power utility) over intermediate consumption.
They assumed a stochastic volatility model where the reciprocal
of volatility follows a mean-reverting square-root process which is instantly correlated with stock returns.
They derived analytic expressions for the optimal consumption and portfolio
policies which where exact for the case of power utility and approximate for the case of recursive utility. Castaneda and Hernandez ([20]) used a combination of martingale approach and stochastic control theory to nd explicit solutions for power and logarithmic utility functions. Kraft ([54]) proved a veri cation result which covers Heston's stochastic volatility model for the power utility. Ekeland and Ta in ([33]) studied the problem of optimal portfolio choice in a bond market described in the general HJM framework. They proved the existence of an optimal portfolio in two cases: when the driving Wiener process is nite-dimensional and when the Wiener process is in nite dimensional but the market price of risk is deterministic. Ringer and Tehranchi ([73]) considered the same problem, but with a Markovian HeathJarrowMorton model of the interest rate term structure driven by an in nite-dimensional Wiener process. They gave su cient conditions for the existence and uniqueness of an optimal trading strategy. Karatzas and Kardaras ([45]) introduced the concept of whose wealth appears
better
numeraire portfolio, a trading strategy
when compared to the wealth generated by any other strategy, in
the sense that the ratio of the two processes is a supermartingale. They derived necessary and su cient conditions for the numeraire portfolio to exist. Liu ([61]) solved dynamic portfolio choice problems, up to the solution of an ordinary dierential equation (ODE), when the asset returns are quadratic and the agent has a power utility function.
4
He also considered three
special cases of his model: a pure bond portfolio problem where the bond returns is described by quadratic term structure model (which includes Gaussian and CIR models), a pure stock portfolio problem when the volatility follows Heston model, and a mixed bond-stock portfolio problem with quadratic term structure and Heston stochastic volatility. The main contribution of this project is developing a new approach to the continuous-time optimal consumption and terminal wealth problem with stochastic opportunity set. It will be called the
direct approach.
It is based on probabilistic arguments, and it can potentially be
extended to the general semimartingale market, though we have not tried to do so.
Hence,
unlike the stochastic control approach, it is not based on the Markov state assumption and it can be used to obtain general results. Its advantage over the martingale approach is that the direct approach, as its name suggests, is dealing with the primal problem
directly.
So, unlike
the martingale approach, the completeness or incompleteness of the market has not so much aect on it. Furthermore we are able to obtain the general form of the optimal portfolio policy directly, and not through the dual problem. The rest of the report is organized as follows. In chapter 2 we de ned the market speci cations, derived some elementary results, and formalized the portfolio choice problem. Chapter 3 is a literature review about stochastic control approach to portfolio choice. In chapter 4 we present, axiomatically, the martingale approach for incomplete markets. Chapter 5, on generalities of the direct approach, contains the main results of the project. In chapter 6, we considered three special cases: the case of logarithmic utility under general speci cation of the market, the case of power utility under Black-Scholes type market (the original Merton setting), and the case of power utility with Gaussian term structure. Finally in chapter 7 we have summarized the main results and suggested some topics for further research. For the sake of completeness, we have included, in the appendices, some results on stochastic exponentials and logarithms, change of measure, and Gaussian term structure models.
5
Chapter 2 Preliminaries 2.1 The market model We start by de ning the market. This de nition will be used throughout chapter 4, chapter 5 and section 6.1.
But in chapter 3 and section 6.2, we will take some special cases of the
de nition which will be de ned separately. Note that the market is allowed to be
stochastic and adapted price processes are without jump.
the coe cients are all
(but
not necessarily Markov
incomplete,
processes), and the
De nition 2.1. (General market model) Consider a stochastic basis Ω, F, (Ft )t∈[0,T ] , P Ft
is the ltration generated by
k
independent Brownian motions
The market consists of a bank account process
B = (Bt ),
(e.g. bonds, stock, etc.) represented by the price process take
n ≤ k.
W =
, where T .
(1) (k) Wt , . . . , Wt
n other zero-dividend assets T (1) (n) A = At , . . . , At . We always and
Assume that the price processes are positive continuous semi-martingales (Itô
processes) with the following dynamics:
´. ´. µ ds + 0 Σs dW s t , At = A 0E 0 s ´t Bt = e 0 rs ds .
(2.1)
r, the short rate, is an adapted and integrable process. µ, the drift term in real measure, n × 1 adapted and integrable vector process. Finally, for all t ∈ [0, T ], Σt is an n × k > adapted and almost surely of rank n for all t ∈ [0, T ], and Σt Σt is integrable.
Here
is an
Remark
2.2
.
Note that if
n = k,
then the market is complete. In this case
Σ−1 t
exists for all
t ∈ [0, T ]. In the next theorem we parametrize all equivalent martingale measures (EMMs) and state price densities (SPDs) of the market.
Theorem 2.3. (Parametrization of EMMs and SPDs) Consider the market of de nition 2.1. T (k) For any EMM Q, there exist a unique predictable process λ = λ(1) , . . . , λ satisfying the t t system Σλ = µ − r1n×1 ,
6
(2.2)
such that Q can be written as (λ)
dQ dP
= ´ZT , . = E − 0 λ> s dW s .
Z (λ)
(2.3)
Furthermore the corresponding SPD is given by πt = e
−
´t 0
rs ds
ˆ . > λs dW s , E − 0
and the process
f W
(2.4)
t
de ned by
ˆ
t
λs ds,
ft , Wt + W
(2.5)
0
is a Q-Brownian motion. Proof.
Any measure
Q,
equivalent to
P,
Z (λ) for some predictable process
Zt , Et
can be written as:
(λ)
dQ dP
=´ZT . = E − 0 λ> s dW s ,
> (1) (k) λ = λt , . . . , λ t .
(2.6)
To see this we may start by de ning the
dQ
. Then by the representation property of W , the stochastic logarithm dP ´. > of Zt can be represented by Log (Z) = − λ dW s for some predictable process λ. 0 s Z (λ) ). Theorem B.2 in the Equation (2.4) follows from the de nition of SPD (i.e. π , B h i martingale
appendix implies that follows that
f W
f W
is in fact a
de ned by equation (2.5) is a
Q-martingale.
Since
f W
= t,
it
Q-Brownian motion (this result can also be obtained by the Girsanov
theorem).
e , A , and note that Q is an EMM if the process A B e can be found as follow: A
Consider the discounted price processes
e A
is a
Q-martingale.
The
e = A
Q-dynamics
1 A B
= e
−
´.
0 ru du
ˆ E
of
ˆ
.
.
µu du +
Σu dW u ˆ . ˆ . = E (µu − ru 1n×1 ) du + Σu dW u 0 ˆ0 . ˆ . f u − λu du = E (µu − ru 1n×1 ) du + Σ u dW 0 0 ˆ . ˆ . f = E (µu − ru 1n×1 − Σu λ) du + Σ u dW u . 0
0
0 It follows that
e A
is a
Q-martingale
0 only if equation (2.2) holds.
7
(2.7)
n < k (incomplete market case), then we may use equation (2.2) to express n λ in terms of (k − n) other elements of λ, and in this way we nd a parametrization in terms of (k − n) market price of risk processes. On the other hand, if n = k (the
Note that if elements of of EMMs
complete market), then equation(2.2) gives us the unique market price of risk process as:
λt = Σ−1 t (µt − rt 1n×1 ) , Σ−1 t bt .
(2.8)
2.2 Consumption pairs, wealth processes and self- nancing strategies Next, we rede ne the well-established concept of self- nancing strategies. We will use equation(2.10) (u,c) of remark 2.5 frequently throughout the chapters 4, 5, and 6. Also note that we used W
W
for the wealth process, while
De nition 2.4.
w
Take
is used for the Brownian motion.
process for which the stochastic integrals
c = (ct )t∈[0,T ]
well-de ned. Also let process
W (u,c) = W
(u,c)
(u,c) Wt
> (1) (n) ut , . . . , u t be an adapted t∈[0,T ] ´. > ´. are u (diag (A))−1 dA, and 0 (1 − u> 1n×1 ) dB B 0
as the initial wealth.
be a non-negative predictable process. Note that for the
, de ned by
t∈[0,T ]
ˆ
ˆ
.
−1
>
u (diag (A))
, wE
(u,c)
Wt
≥ 0
dA + 0
We also de ne
u,
SF (w)
2.5
.
We call
dB (1 − u 1n×1 ) − B
W (u,c)
(u, c), wealth w .
c
given initial
From equation (2.1) we have
ˆ
.
cs ds ,
(2.9)
0
the wealth process corresponding to a
the proportional consumption rate process
as the set of all pairs
consumption rate process
Remark
t ∈ [0, T ].
for all
self- nancing strategy
.
>
0 we have
u =
Let
where
u
c,
and initial wealth
w.
is a self- nancing strategy for the
(diag (A))−1 dA = µu du + Σu dW u .
So equation
(2.9) can be rewritten as:
W
(u,c)
ˆ
ˆ
.
u> s
ˆ
.
(1 −
.
−
cs ds ˆ . ˆ . > > = wE us (µs − rs 1n×1 ) + rs − cs ds + us Σs dW s .
= wE
(µs ds + Σs dW s ) +
u> s 1n×1 )rs ds
0
0
0
(2.10)
0
0 Note that we may re-write equation(2.9) as
(u,c) n n (i) P P (i) dAt (i) dWt t ut (i) + (1 − ut ) dB − ct dt, (u,c) , Bt Wt At i=1 i=1 (u,c) W0 = w, Now we may interpret
(i)
ut
as the percentage of wealth invested in the
as the percentage investment in the bank account, and of percentage of wealth.
8
c
(2.11)
i-th
asset,
(1 −
n P
(i)
ut )
i=1 as the rate of consumption in terms
Also note that we may alternatively consider the total rate consumption (u,C) (u,C) a little misuse of notation, the wealth process W = Wt by: t∈[0,T ]
ˆ (u,C) Wt
and de ne, with
t
−1 Ws(u,C) u> dA , w+ s (diag (A)) 0 ˆ t ˆ t dBs (u,C) > Cs ds. − + Ws (1 − us 1n×1 ) Bs 0 0
(2.12)
(u,C)
≥ 0 for all t ∈ [0, T ], we call u a self- nancing strategy for the total consumption C . Obviously we would have W (u,C) ≡ W (u,c) , if we de ne
Then if rate
C
Wt
ct ,
Ct (u,C)
So from now on, we denote the wealth process by total consumption. Whether
c
.
(2.13)
Wt
W (u,c)
for both cases of proportional and
is a proportional or total consumption will be known from the
context.
consumption pair, which formalize the consumption behavior of an entity. We identify those consumption pairs which can be nanced by a self nancing strategy as aordable consumption pairs. We will also need the concept of a
De nition 2.6.
consumption pair
(C, Z) consisting of an adapted non´T negative total consumption rate process C = (Ct )t∈(0,T ] with Cs ds < ∞, and an FT -measurable 0 non-negative random variable describing terminal lump-sum consumption Z . We denote the set of all consumption pairs by C . A
is an ordered pair
De nition 2.7. For a consumption pair (C, Z), initial wealth w and a self- nancing strategy u (u,C) (for the total consumption C ), we say that (C, Z, u) is budget-feasible at w if WT ≥ Z a.s. A consumption pair (C, Z) is aordable with initial wealth w , if there exist a self- nancing strategy u
such that
(C, Z, u)
is budget-feasible at
pairs aordable with initial wealth
w.
We denote by
C(w)
the set of all consumption
w.
2.3 Problem Formulation The traditional criterion for optimal portfolio choice has been based on maximal expected utility (for the historical perspective see [88]). The tradition is either assuming an additively h´ i T time-separable utility function for intermediate consumption of the form E u(t, C )dt (in t 0
optimal consumption problem ), or utility of terminal wealth only E [U (WT )] (the portfolio choice problem with this criterion is also called an asset h´ i T allocation problem ), or the sum of these two components E 0 u(t, Ct )dt + U (WT ) . this case the problem is usually called the
On the other hand, it is worth to mention that it has long been recognized by economists that preferences may not be Intertemporally separable. In particular, the utility associated with the choice of consumption at a given date is likely to depend on past choices of consumption. According to Browning [13], this idea dates back to the 1890 book `Principles of Economics'
9
by Alfred Marshall.
For example high past consumption generates a desire for high current
consumption. Generalizations of standard time-separable preferences that have been suggested
recursive or stochastic dierential utility (see, for example, [32, 76, 77]), habit formation criterion (see [67]), and forward performance criterion (see [68, 87]). include
In this project we adapt the traditional criterion of expected utility of terminal wealth and time-separable utility of intermediate consumption. Speci cally we de ne
utility functionals
as
follows.
De nition 2.8.
J : C → R is de ned by: ˆ T u(t, Ct )dt + U (Z) , J(C, Z) , E
A total utility functional
(2.14)
0 with the following assumption on functions u(., .) and U (.): + - u : [0, T ] × R → R is continuous, and for each t ∈ [0, T ], and concave. + - U : R →
R
u(t, .) : R+ → R
is increasing
is increasing and concave.
u(., .) or U (.) is non-zero. Furthermore, either U is strictly concave or zero, t ∈ [0, T ], u(., t) is strictly concave or zero. refer to the function u (., .) as the consumption utility function, and the function U (.)
- At least one of or for all We as the
terminal utility functions.
We may now give the formal de nition of Merton's problem.
Problem 2.9.
(Merton's Problem) Consider a total utility functional
ˆ J(C, Z) , E
T
u(t, Ct )dt + U (Z) .
(2.15)
0 Then for the initial wealth
w,
we de ne Merton's problem as:
sup
(u,c)
J(cW (u,c) , WT
(u,c)∈SF (w)
10
) .
(2.16)
Chapter 3 Stochastic Control Theory∗ In this chapter we present the stochastic control approach without going into details. For the sake of simplicity, we only consider two assets throughout this chapter: a risky stock and a bank account. As already pointed out in the rst chapter, the underlying assumption of the stochastic control approach is assuming a Markov state processes.
More speci cally we should restrict
our de nition of market as follows. The stock price process
B = (Bt )t≥0
S = (St )t≥0
and the bank account
have the following dynamics:
dSt St dBt Bt The stochastic factor
Y = (Yt )t≥0
(1)
= µ (Yt ) dt + σ (Yt ) dWt , = r (Yt ) dt.
(3.1)
is assumed to satisfy:
p (1) (2) . dYt = b (Yt ) dt + d (Yt ) ρdWt + 1 − ρ2 dWt Here
(1) (2) W t = Wt , Wt
Next we de ne the
is a standard Brownian motion.
value function V (w, y, t; T ) as ˆ
V (w, y, t; T ) =
(3.2)
sup E (u,c)∈SF (w)
T
u(s, cs Ws(u,c) )ds
+
(u,c) (u,c) U (WT )|Wt
= w, Yt = y .
(3.3)
t
As solution of a stochastic optimization problem, the value function is expected to satisfy the Dynamic Programming Principle (DPP), namely for all
V (w, y, t; T ) =
sup E (u,c)∈SF (w)
h
V
s ∈ (t, T ),
Ws(u,c) , Ys , s; T
(u,c) |Wt
i
= w, Yt = y .
(3.4)
This is a fundamental result in optimal control and has been proved for a wide class of optimization problems. For a detailed discussion on the validity and strongest forms of the DPP in problems with controlled diusions, we refer the reader to [37]. Key issues are the measurability and continuity of the value function process as well as the compactness of the set of admissible controls. ∗
It is worth mentioning that a proof speci c to the problem at hand has not been
This chapter is mainly excerpted from Zariphopoulou [87].
11
produced to date. Recently, a weak version of the DPP was proposed by Bouchard and Touzi [7] where conditions related to measurable selection and boundness of controls are relaxed. Besides its technical challenges, the DPP exhibits two important properties of the value function process.
Speci cally,
V (w, y, s; T ), s ∈ [t, T ],
is a supermartingale for an arbitrary
investment strategy and becomes a martingale at an optimum (provided certain integrability conditions hold). One may, then, view
V (w, y, s; T ) as the intermediate (indirect) utility in the
relevant market environment. It is worth noticing, however, that the notions of utility and risk aversion for times
t ∈ [0, T )
are tightly connected to the investment opportunities the investor
has in the speci c market.
Observe that the DPP yields a backward in time algorithm for (u,c) the computation of the maximal utility, starting at expiration with U WT and using the martingale property to compute the solution for earlier times. For this reason, we refer to this formulation of the optimal portfolio choice problem as backward. Fundamental results in the theory of controlled diusions yield that if the value function is smooth enough then it satis es the HJB equation,
Vt + bVy + 21 d2 Vyy 1 2 2 2 + sup w u σ V + {w [(µ − r) u + r] − C} V + wuρσdV + u (t, C) = 0. ww w wy 2 (u,C)∈SF (w)
(3.5) Moreover, optimal policies may be constructed in a feedback form from the rst-order conditions in the HJB equation, provided that the candidate feed-back process is admissible and the wealth SDE has a strong solution when the candidate control is used. The latter usually requires further regularity on the value function. In the reverse direction, a smooth solution of the HJB equation that satis es the appropriate terminal and boundary (or growth) conditions may be identi ed with the value function, provided the solution is unique in the appropriate sense. These results are usually known as the
veri cation theorem
and we refer the reader to [37, 85] for a general
exposition on the subject. In maximal expected utility problems, it is rarely the case that the arguments in either direction of the veri cation theorem can be established. Indeed, it is very di cult to show a priori regularity of the value function, with the main di culties coming from the lack of global Lipschitz regularity of the coe cients of the controlled process with respect to the controls and from the non-compactness of the set of admissible policies.
It is, also, very di cult to
establish existence, uniqueness and regularity of the solutions to the HJB equation.
This is
caused primarily by the presence of the control policy in the volatility of the controlled wealth process which makes the classical assumptions of global Lipschitz conditions of the equation with regards to the non linearities fail.
Additional di culties come from state constraints
and the non-compactness of the admissible set. Regularity results for the value function (3.4) for general utility functions have not been obtained to date except for the special cases of homothetic preferences (see, for example, [36, 54, 66, 71, 86]).
The most general result in
this direction, and in a much more general market model, was recently obtained by Kramkov and Sîrbu [57] where it is shown that the value function is twice dierentiable in the spatial argument but without establishing its continuity. Some answers to the questions related to the characterization of the solutions to the HJB equation may be given if one relaxes the requirement to have classical solutions. An appropriate class of weak solutions turns out to be the so called viscosity solutions ([27, 60, 59], and [82]). The analysis and characterization of the value function in the viscosity sense has been carried out for the special cases of power and exponential utility (see, for example, [86]).
12
However,
proving that the value function is the unique viscosity solution of (3.5) has not been addressed. A key property of viscosity solutions is their robustness (see [59]). If the HJB has a unique viscosity solution (in the appropriate class), robustness can be used to establish convergence of numerical schemes for the value function and the optimal feedback laws.
Such numerical
studies have been carried out successfully for a number of applications. However, for the model at hand, no such studies are available. Numerical results using Monte Carlo techniques have been obtained in [29] for a model more general than the one herein. Important questions arise on the dependence, sensitivity and robustness of the optimal feedback portfolio in terms of the market parameters, the wealth, the level of the stochastic factor and the risk preferences. Such questions are central in nancial economics and have been studied, primarily in simpler models in which intermediate consumption is also incorporated (see, among others, [2, 51, 58, 69], and [74]). For diusion models with and without a stochastic factor, qualitative results can be found in [29, 50, 61, 84], and, recently, in [6] (see, also [62] for a general incomplete market discrete model). However, a qualitative study for general utility functions and/or arbitrary factor dynamics has not been carried out to date. Let's have a review on the existing results for the most frequently used utilities, namely, the exponential, power and logarithmic ones.
They have convenient homogeneity properties
which, in combination with the linearity of the wealth dynamics in the control policies, enable us to reduce the HJB equation to a quasilinear one. Under a
distortion transformation
(see, for
example, [86]) the latter can be linearized and solutions in closed form can be produced using the Feynman-Kac formula. The smoothness of the value function and, in turn, the veri cation of the optimal feedback policies follows easily. Multi-factor models for these preferences have been analyzed by various authors. The theory of BSDE has been successfully used to characterize and represent the solutions of the reduced HJB equation (see [49]). The regularity of its solutions has been studied using PDE arguments by [71] and [66], for power and exponential utilities, respectively. Finally, explicit solutions for a three factor model can be found in [61].
13
Chapter 4 Martingale or Duality Approach The Martingale method has been given increasing attention since it was conducted by Pliska [72], Cox and Huang [24, 25] and Karatzas, Lehoczky and Shreve [46].
Martingale method
allows us to solve the problems of utility maximization in a very elegant manner. However, the Martingale method is not omnipotent. When the market is incomplete, traditional Martingale method will be problematic.
He and Pearson [43], and Karatzas, Lehoczky, Shreve and Xu
[47] generalized the approach to incomplete markets. The central idea here is to solve a dual variational problem and then to nd the solution of the original problem by convex duality. In a general incomplete market, the dual problem is a stochastic variational problem where the control variables are the consumption process
c,
the terminal wealth
Z,
and market price
of risk processλ. Namely,
h´ i T J(C, Z) , E 0 u(t, Ct )dt + U (Z) subject to : h´ i T (λ) (λ) E 0 πt Ct dt + πT Z = w, n o −1 −1 λ = Σ ΣΣT b + Σ> ΣΣ> Σ − Ik×k a(C,Z,λ) , (C, Z) ∈ C, λ is an MPR.
sup
(4.1)
see remark 4.8 below. For general semimartingale markets and general utility functional, necessary and su cient conditions for validity of the martingale approach has been proposed, see [55, 56]. These conditions guarantee that the duality gap is zero (i.e. the optimal value of the primal and dual problem are the same), and that the optimal consumption pair
(c, Z) given by
the dual problem is actually aordable. However they only provide the existence of the optimal portfolio
u,
and to nd the portfolio policy one should solve a representation problem.
The situation is much more tractable once we assume that the market is complete. In this case the market price of risk is unique, hence the dual problem would be only in terms of the consumption pair
(c, Z).
But then we can solve this
static problem directly to nd the optimal
consumption and terminal wealth. Speci cally the optimal consumption pair is,
Ct∗ , I(ηπt , t) for t ∈ (0, T ) , Z ∗ , IF (ηπT ). where
IF (.)
and
I(., t) are the inverses of U 0 (x) and
14
∂ u(x, t), ∂x
(4.2)
π = (πt ) is the state price density
process, and the constant
η
is the unique solution of the following equation:
ˆ
T
E πT IF (ηπT ) +
πτ I(ηπτ , τ )dτ = w.
(4.3)
0 See remark 4.10 below. This is actually the original martingale approach. Yet again, to obtain the optimal portfolio we need to solve a representation problem. In the next subsection we present the Martingale approach in details.
4.1 Proof of the Martingale Approach The essence of the martingale approach in incomplete markets, also known as duality approach, is in remark 4.8 and theorem 4.9. In summary the proof goes like this:
•
First we identify a necessary and su cient condition under which an arbitrary consump-
(C, Z)
tion pair
is aordable. It will be done in two steps, lemma 4.6 and proposition
4.7.
•
Then we relate Merton's problem to the dual problem containing only the consumption pairs and not the optimal portfolio
u
(see equation 4.26).
•
Then in theorem 4.9, we identify conditions under which we can solve this dual problem.
•
Finally we consider the case of complete markets in remark4.10.
To express the martingale approach, we need the following de nitions and lemmas.
De nition 4.1. risk process
λ
Assume that we are given a consumption pair (C, Z) and a market-price of (λ) (C,Z,λ) (along with the corresponding SPD π ). Then we de ne the process W
as:
(λ) (C,Z,λ) πt W t
, Et
ˆ (λ) πT Z
T
πτ(λ) Cτ dτ
+
.
(4.4)
t
Lemma 4.2. (Inada condition) We say that a strictly concave, increasing and dierentiable function F : R+ → R satis es Inada conditions if inf F 0 (x) = 0 and supF 0 = +∞. If F : R+ → x x R satis es Inada conditions, then the function IF , which is the inverse of F 0 , is well de ned as a strictly decreasing continuous function on (0, ∞) which image is (0, ∞). De nition 4.3. (Condition A)
U (x) is dierentiable on (0, ∞), strictly concave and satis es Inada conditions. Either (for all t) u(x, t) is zero or (for all t) u(x, t) is dierentiable on (0, ∞), strictly concave and satis es Inada conditions. Also for all η > 0 and any SPD π we assume: ˆ T E πT IF (ηπT ) + πτ I(ηπτ , τ )dτ < ∞. (4.5) Either
U (x)
is zero or
0 Here
IF (.)
and
I(., t)
are the inverses of
U 0 (x)
and
15
∂ u(x, t). ∂x
Lemma 4.4. Consider problem 2.9. Assume that there exists a unique constant η (λ) > 0 such that: ˆ (λ) (λ) E πT IF (η (λ) πT ) +
condition A
holds, then for any MPR λ,
T
πτ(λ) I(η
(λ) πτ(λ) , τ )dτ
= w.
(4.6)
0
Here I(., t) and IF (.) are the inverses of the derivatives of u(., t) and U (.), respectively. Proof.
De ne
ˆ (λ) (λ) f (η) , E πT IF (ηπT ) +
T
πτ(λ) I(ηπτ(λ) , τ )dτ
.
(4.7)
0
Condition A along with lemma 4.2 imply that one or both of IF (.) and I (., t) (for all t ∈ (0, T )) are strictly decreasing continuous functions on
(0, ∞)
which images are
w>0 by η (λ).
inherits these two properties, we can conclude that for any which satis es
f (η) = w.
We denote this unique solution
(0, ∞).
f (.) η>0
Since
there exist a unique
We need to prove the numeraire invariance property of our de nition of budget feasibility (de nition 2.7).
Lemma 4.5. (numeraire invariance) Letπ = (πt ) be an arbitrary de ator and w be the initial wealth. A strategy (C, Z, u) is budget feasible at w if and only if : ´t (u,C) (u,C) > t ∈ [0, T ] π W = π w + πs Ws us (diag (πs As ))−1 d (πA)s t 0 t 0 d(πB) ´t ´t (u,C) > s + 0 πs Ws 1 − us 1n×1 πs Bs − 0 πs Cs ds ≥ 0 (u,C) πT WT ≥ πT Z a.s.
Proof.
The proof is a simple application of product rule.
By de nition
(C, Z, u)
(4.8)
is budget
feasible if and only if:
(u,C) ´ t (u,C) > W = w + Ws us (diag (As ))−1 dAs t ∈ [0, T ] t 0 dB ´ t ´ t (u,C) > + 0 Ws 1 − us 1n×1 B − 0 Cs ds ≥ 0 (u,C) WT ≥ Z a.s.
(4.9)
The second conditions in equation (4.8) are equivalent to the second equations above. To show that the rst condition in equation(4.8) is implied by the rst condition in equation(4.9), we apply the product rule to obtain:
d Note that:
(u,C) πt Wt
(u,C)
= πt dWt
(u,C)
+ Wt
dπt + d π, W (u,C) t
ˆ . −1 (u,C) (u,C) > d π, W = d π, Ws us (diag (As )) dAs t =
0 (u,C) > Wt ut (diag (At ))−1
16
d[π, A]t
(4.10)
(u,C) in equation(4.10), we will obtain: d π, W (u,C) and dWt dBt (u,C) > (u,C) −1 > = πt Wt ut (diag (At )) dAt + Wt 1 − ut 1n×1 − Ct dt Bt
By substituting for
d πW (u)
t
(u,C)
+Wt =
=
(u,C)
dπt + Wt
−1 u> d[π, A]t t (diag (At ))
−1 {πt dAt + diag (At ) 1n×1 dπt + d[π, A]t } u> t (diag (At )) {πt dBt + Bt dπt } (u,C) +Wt 1 − u> − πt Ct dt t 1n×1 Bt d (πB)t (u,C) (u,C) > 1 1 − u> Wt ut (diag (At ))−1 d (πA)t + Wt − πt C (4.11) n×1 t dt t Bt (u,C)
Wt
which in turn, will give us the rst condition in equation(4.8). Finally to show the converse, i.e. that the rst condition in equation(4.9) is also implied (π) by the rst condition in equation(4.8), we start by de ning the process as At = πt A t , (π) (π) (π) Bt = πt Bt ,Ct = πt Ct , and Z = πT Z . Then equation(4.8) is equivalent to assuming (π) (π) (π) (π) that (C , Z , u) is budget budget feasible at w with price processes given by At and Bt . 1 Now by considering the de ator and using the same argument we used form equation(4.10) πt to (4.11), we will reach the rst condition in equation (4.9). As mentioned above, the main idea of martingale approach is to identify conditions under which an arbitrary consumption pair
(C, Z) is aordable.
The following lemma is the rst step
towards this goal.
Lemma 4.6. Assume we are given a consumption pair (C, Z) and a market-price of risk process λ. Then (C, Z) is aordable with initial wealth w if and only if : (i) ˆ T
(λ)
(λ)
πt Ct dt + πt Z ≤ w.
E
(4.12)
0
(n) (ii)- There exist a predictable process u = u(1) t , . . . , ut
ΣTt ut − λt =
T
t∈[0,T ] 0 W , π (λ) W (C,Z,λ) t . (λ) (C,Z,λ) πt W t
satisfying: (4.13)
Furthermore if these conditions hold, then u is the self- nancing strategy for the consumption pair (C, Z). Proof.
(i). Assume (C, Z) to be aordable, i.e. there u such that (C, Z, u) is budget feasible at w. By using lemma4.5
First we prove the necessity of condition
exist a self- nancing strategy for de ator
π,
we will obtain:
´t (u,C) (u,C) > = w + 0 πs W s us (diag (πs As ))−1 d (πA)s t ∈ [0, T ] πt Wt d(πB) ´t ´t (u,C) > s + 0 πs W s 1 − us 1n×1 πs Bs − 0 πs Cs ds ≥ 0 (u,C) πT W T ≥ πT Z a.s.
17
De ne a process
N
by:
ˆ Nt ,
(u,C) πt W t
ˆ
t
+
πs Cs ds
(4.14)
ˆ
0 t
= w+
−1 Ws(u,C) u> s (diag (As ))
t
Ws(u,C) 1 − u> s 1n×1
d (πA)s + 0
0
π (λ) A
Note that by de nition,
π (λ) B
and
are (local) martingales, so
N
d (πB)s .(4.15) Bs
is a non-negative
local martingale, and hence a super martingale. The supper-martingale property give us the desired equation (4.12):
ˆ
T
E πT Z +
πt Ct dt ≤ E [NT ] ≤ N0 = w.
(4.16)
0
(ii)
To prove the necessity of condition we may proceed as follow. From remark (2.5), C (C,Z,λ) as the proportional W is the wealth process of a self- nancing strategy u with c , W (C,Z,λ) consumption and w as the initial wealth, if and only if:
(C,Z,λ) Wt
ˆ = wE
ˆ
.
0 Z (λ) , where B properties of stochastic integrals we may write:
=
u> u Σu dW u
.
(4.17)
0
Recall from remark (2.3) that
π (λ) W (C,Z,λ) =
.
> uu (µu − ru 12×1 ) + ru − cu du + π ,
Z (λ) = E −
´.
λ> dW s 0 s
.
So by using the
Z (λ) (C,Z,λ) W B ´ . E − 0 λ> s dW s ´.
B e 0 rs ds 0 ˆ . ˆ . > > × wE uu (µu − ru 12×1 ) + ru − cu du + uu Σu dW u 0
0
w [− ´ . λTs dW s ,´ . uTs Σs dW s ] 0 = e 0 B0 ˆ . ˆ . > > > ×E us (µs − rs 1n×1 ) − cs ds + us Σs − λs dW s 0 0 ˆ . ˆ . > w > > = E us (µs − rs 12×1 − Σs λs ) − cs ds + us Σs − λs dW s B0 0 0 ˆ . ˆ . w > > E − cs ds + us Σs − λs dW s . (4.18) = B0 0 0 Hence we obtain:
d π (λ) W (C,Z,λ)
t
(λ)
(C,Z,λ)
> −ct dt + u> t Σt − λt dW t (λ) (λ) (C,Z,λ) > Σ − λ = −πt Ct dt + πt Wt u> t t t dW t . = πt W t
On the other hand from equations (4.4) and the martingale representation property of
(4.19)
W,
we
have
d π (λ) W (C,Z,λ)
(λ)
+ πt Ct dt = t 18
(λ) (C,Z,λ) 0 π W , W t dW t .
(4.20)
From the last two results we can conclude
(λ)
(C,Z,λ)
πt Wt
which is equation(4.13). Finally to prove the converse, suppose that conditions
W
(u,c)
ˆ , wE
(λ) (C,Z,λ) 0 > = π W , W Σ − λ , u> t t t t
(i)
and
(ii)
ˆ
.
u> u
hold. Then by de ning
.
u> u Σu dW u
(µu − ru 12×1 ) + ru − cu du +
,
0
0
(ii)
(C,Z,λ) we may invert the proof of necessity of condition , to conclude that W ≡ W (u,c) . Now (C,Z,λ) we may use the de nition of W along with condition to prove that u is indeed a self- nancing strategy and
(C, Z, u)
(i)
is budget feasible.
In the following proposition we improve the result of the previous lemma by making the
u.
aordability condition free of any self- nancing strategy
Also we give an expression for the
corresponding self- nancing strategy, if a consumption pair is aordable.
Proposition 4.7. A consumption pair (C, Z) is aordable with initial wealth w if and only if for a speci c market price of risk λ satisfying −1 λ , Σ> ΣΣT (µ − r1n×1 ) o n −1 + Σ> ΣΣ> Σ − Ik×k
we have
ˆ
>
E
(λ) πt Ct dt
+
W , π (λ) W (C,Z,λ)
0 ! t
,
(λ) (C,Z,λ) π t Wt
(λ) πt Z
(4.21)
≤ w.
(4.22)
0
Furthermore, if this condition holds, the corresponding self- nancing strategy is given by: −1 u = ΣΣ>
Proof.
( µ − r1n×1 + Σ
0 !) W , π (λ) W (C,Z,λ) t (λ)
(C,Z,λ)
.
0
De ne
(C,Z,λ)
at
,
[W,π(λ) W (C,Z,λ) ]t (λ)
πt
(C,Z,λ)
Wt
and
(4.23)
πt W t
bt , µ−r1n×1 .
Then from condition
(4.6), and theorem (2.3) we have the following system for vectors
λ
and
(ii) of lemma
u:
( (C,Z,λ) ΣTt ut − λt = at Σt λ t = b t . By solving for
Note that rank
(4.24)
λ from the rst equation and substituting in the second equation we will obtain (C,Z,λ) Σt ΣTt ut − at = bt . (4.25) ΣΣT
=
rank (Σ)
= n,
so the
n×n
matrix
rearranging the last result we will obtain equation (4.23).
ΣΣT
is invertible.
By substituting for
u
Now by
in the rst
equation of the system above, we will obtain equation (4.21). Hence the necessary and su cient condition of lemma (4.6) can be restated as this proposition.
19
Remark
4.8
.
Note that from proposition (4.7), Merton's problem is equivalent to the following
variational problem:
h´
i
T 0
J(C, Z) , E u(t, Ct )dt + U (Z) subject to : h´ i T (λ) (λ) E 0 πt Ct dt + πT Z = w, o n −1 −1 Σ − Ik×k a(C,Z,λ) , λ = Σ ΣΣT b + ΣT ΣΣT (C, Z) ∈ C, λ is an MPR.
sup
(4.26)
This problem, generally known as the dual of problem , can be de ned even in a general semimartingale setting (see [55, 56] ). Now we are ready to prove the main result.
Theorem 4.9. (Martingale Approach) Consider problem 2.9 and assume that holds. De ne (λ) (λ) Ct , I(η (λ) πt , t) for t ∈ (0, T ) , (λ)
Z (λ) , IF (η (λ) πT ), 0
(C,Z,λ)
at
,
(λ)
πt
(C,Z,λ)
Wt
(4.27)
. Also de ne bt , µ − r1n×1 , and
where IF (.) and I(., t) are the inverses of U 0 (x) and [W ,π(λ) W (C,Z,λ) ]t
condition A
∂ u(x, t) ∂x
so that (λ)
(C,Z,λ)
dπt Wt
(λ)
(C,Z,λ)
= πt Wt
(C,Z,λ)
at
dW t .
(4.28)
If an MPR λ satis es >
λ=Σ
> −1
ΣΣ
o n (λ) (λ) > > −1 b + Σ ΣΣ Σ − Ik×k a(C ,Z ,λ) ,
(4.29)
then, under that speci c MPR λ, the optimal consumption is Ct(λ) and the optimal terminal wealth is Z (λ) . Furthermore the corresponding optimal self- nancing strategy is given by: > −1
u = ΣΣ
(C (λ) ,Z (λ) ,λ)
b + Σa
,
(4.30)
and we have W (u,C) = Z (λ) . Proof.
Consider the following problem: sup
J(C, Z) , E
subject to h´ T E 0
h´ T 0
(λ) πt Ct dt
(C, Z) ∈ C, λ is
a
+
i u(t, Ct )dt + U (Z)
(λ) πT Z
= w, given MPR.
Since there is not enough constraint to make the consumption pair
λ,
(4.31)
i
this problem is a relaxation of Merton's problem.
(C, Z)
aordable under
This means that the solution of this
problem gives an upper bound for the Merton's optimal solution. Now the main idea is that if
20
this solution of this relaxation problem satis es equation (4.21) as well, then it would be the solution for Merton's problem. First we try to nd the solution of equation (4.31). The Lagrangian is:
ˆ
T
L (C, Z, η) , E
u(t, Ct )dt + U (Z) ˆ T (λ) (λ) −η E πt Ct dt + πT Z − w . 0
(4.32)
0 To check the rst order conditions we rst nd the directional derivatives:
i h´ T (λ) 0 u (t, C + εx ) − ηπ x dt , t t t t 0 h i (λ) d L (C, Z + εy, η) = E U 0 (Z + εy) − ηπt y , dε h´ i T (λ) (λ) d L (C, Z, η) = E 0 πt Ct dt + πt Z − w, dη
d L (C dε
+ εx, Z, η) = E
(4.33)
x = (xt ) is an arbitrary predictable process, y is an arbitrary FT -measurable variable, and η is a positive number. The rst order conditions give us the solution, where
d | L (C, Z, η) dη η=0
= 0 =⇒ η = η (λ) , (λ) =⇒ Ct = Ct , =⇒ Z = Z (λ) .
d | L (C + εx, Z, η) = 0 dε ε=0 d | L (C, Z + εy, η) = 0 dε ε=0
random
(4.34)
Note that we only checked the necessary conditions of optimality, and not the su cient (second order) conditions. On the other hand equation (4.21), the su cient condition for aordability of
C (λ) , Z (λ)
,
translates into equation (4.29). So, as already mentioned, if λ also satis es this condition, then C (λ) , Z (λ) is also the optimal consumption pair in Merton's problem. Finally equation (4.30) follows directly from proposition (4.7).
Remark
4.10
.
Note that if the market is complete, i.e. n = k , then Σ would be invertible and λ = Σ−1 b. In this case equation (4.29) is automatically
the market price of risk would be
satis ed. This means that the relaxed problem in the proof of theorem (4.9) is equivalent to the Merton problem, and the solution is given by equations (4.27) and (4.30). Furthermore we may simplify equation (4.30) as follow:
u = ΣΣ>
−1
b + Σ>
21
−1
a(C
(λ) ,Z (λ) ,λ)
.
(4.35)
Chapter 5 The Direct Approach In this chapter we present an new approach for tackling the problem of portfolio choice.
It
has the strong aspects of both existing approaches: like the stochastic control theory, it deals with the problem in its original form, and like the martingale approach, it uses probabilistic argument which can be extended to general market settings. Recall that from equation (2.10) we have:
W
(u,c)
ˆ
ˆ
.
= wE
u> s
.
u> s Σs dW s
(µs − rs 1n×1 ) + rs − cs ds +
0
.
(5.1)
0
The main idea of the direct approach is to use the representation above to convert Merton's problem into
ˆ sup u∈P, c∈P +
Here
P
is the space of
n
T
E
(u,c) u(t, ct Wt )dt
+
(u,c) U (WT )
.
(5.2)
0
dimensional predictable vector processes, and
P + is
the space of non-
negative predictable processes. Then to nd a candidate for the optimal control, we may check the rst order optimality conditions:
(
Where the Lagrangian
L
d | L(u + v, c) = 0, d =0 d | L(u, c + y) = 0. d =0
(5.3)
is of the form,
ˆ
T
L (u, c) , E
(u,c) u(t, ct Wt )dt
+
(u,c) U (WT )
.
(5.4)
0
c ≥ 0, we should consider the casehof zero consumption separately. i (u,c) Lagrangian as L (u) , L (u, 0) = E U (WT ) , and the rst order
Also because of the constraint In this case we de ne the
d | L(u + v) = 0 (we also de ne W (u) , W (u,0) ). d =0 In the remainder of this chapter, we will show that the rst order conditions stated above
condition would be
can be converted into a set of equations that can be solved to obtain the optimal solutions. Namely, for the case of zero consumption we will have:
−1 ut = Σt Σ> t
( (µt − rt 1n×1 ) + Σt 22
0 ) W , V (u) t (u)
Vt
,
(5.5)
and for the general case we have:
(u,c) = Vt
(u,c) (u,c) t, ct Wt Wt , 0 [W ,V (u,c) ]t > −1 (µt − rt 1n×1 ) + Σt . ut = Σt Σt (u,c) ∂ u ∂x
(5.6)
Vt
Here the processes
V (u)
(u,c)
Vt
and
, Et
h´
T t
V (u,c) are
de ned as follows,
h i (u) (u) (u) Vt , Et U 0 WT WT , i (u,c) (u,c) (u,c) (u,c) ∂ 0 W . u s, c W c W ds + U W s s s s T T ∂x
(5.7)
For the main result see theorem 5.6, and equations 5.52 and 5.55. Note that the main hurdle [W,V u ]0t in implementing this approach is to nd explicit expressions for in the case of zero Vtu 0
consumption, and
V
(u,c)
and
[W,V (u,c) ]t
for the general case. (u,c) Vt We should mention that we have not looked at conditions which guarantee the optimality
of the candidates, i.e.
the counterparts of veri cation results in stochastic control approach
(see [37, 85]) or the necessary and su cient condition of martingale approach (see [55, 56]). As pointed out by Korn and Kraft [53], these results are crucially important and uncritical application of any method for continuous-time portfolio optimization can be misleading in the case of a stochastic opportunity set. We have proposed nding such conditions for validity of the direct approach as a future line of research (see chapter 7). Besides these technically challenging issues, there are a number of interesting questions on the economic properties of the optimal portfolios. From (5.5) and (5.6) one sees that the optimal portfolio consists of two terms, namely,
um = t
Σt Σ> t
−1
(µt − rt 1n×1 ) ,
uht =
Σt Σ> t
−1
Σt
[W , V ∗ ]0t . Vt∗
(5.8)
The rst component is known as the myopic investment strategy. It corresponds functionally to the investment policy followed by an investor in markets in which the investment opportunity set remains constant through time. The myopic portfolio is always positive for a nonzero market price of risk. The second term is called the excess hedging demand. It represents the additional investment caused by the presence of the stochastic factor. It does not have a constant sign, for the > −1 signs of the correlation coe cient Σt Σt Σt , and the volatility of the optimal V ∗ process are not de nite. The excess hedging demand term vanishes in the uncorrelated case; and when the volatility of the stochastic factor process is zero (the latter case can be reduced to the classical Merton one). Finally, the excess hedging demand term also vanishes for the case of logarithmic utility (see section 6.1 and [10]). It is worth to mention that despite the nomenclature
hedging demand,
a rigorous study
for the precise characterization and quanti cation of the risk that is not hedged has not been carried out. Indeed, in contrast to derivative valuation where the notion of imperfect hedge is well de ned, such a notion has not been established in the area of investments (see [80] for a special case).
23
As a nal observation, note that total allocation in the risky assets might become zero even if the risk premium is not zero. This phenomenon, related to the so called market participation puzzle, appears at rst counter intuitive, for classical economic ideas suggest that a risk averse investor should always retain nonzero holdings in an asset that oers positive risk premium. We refer the reader to, among others, [5, 18, 44]. In the next section we will present in detail the argument used to obtain the main results of the direct approach.
5.1 Proof of the Direct approach We start by identifying
(u+v,c) (u,c+y) d d | W and | W , which in some sense, are the building d =0 t d =0 t
blocks of our proof.
Lemma 5.1. Let u be a self- nancing strategy for the proportional consumption rate c. Then for all locally bounded predictable processes v and y we have ˆ t ˆ t d (u+v,c) (u,c) T T T v s µs − rs 1n×1 − Σs Σs us ds + |=0 Wt = Wt v s Σs dW s , (5.9) d 0 0 ˆ t d (u,c+y) (u,c) |=0 Wt = −Wt ys ds. (5.10) d 0
Proof.
De ning
Yt
Yt
as
ˆ t ˆ tn o > (uu + v u )T Σu dW u . , (uu + v u ) (µu − ru 1n×1 ) + ru − cu du +
(5.11)
0
0 From equation(2.10) we have
(u+v,c)
Wt
d (u+v,c) |=0 Wt d
1
= wE (Y )t = weYt − 2 [Y ]t . So we 1 (u,c) d = Wt |=0 Yt − [Y ]t . d 2
may conclude:
(5.12)
From equation(5.11), after interchanging the order of dierentiation and integration, we would obtain
ˆ
d |=0 Yt = d
ˆ
t
v Tu 0
t
v Tu Σu dW u .
(µu − ru 1n×1 ) du +
(5.13)
0
Also from equation(5.11) it follows that
ˆ
[Y ]t =
t T
(uu + v u ) Σu dW u ˆ
0 t
(uu + v u )T Σu d[W u ]ΣTu (uu + v u )
= ˆ0 t
(uu + v u )T Σu (Ik×k du) ΣTu (uu + v u )
= ˆ0 t
(uu + v u )T Σu ΣTu (uu + v u ) du.
= 0
24
(5.14)
By dierentiating from the last result we would have
d |=0 [Y ]t = d
ˆ
t
2v Tu Σu ΣTu uu du.
(5.15)
0
Equation(5.9) follows by substituting from equation (5.13) and (5.15) into equation (5.12). For obtaining equation (5.10) we may start by de ning
ˆ Yet ,
ˆ
t
uTu
t
uTu Σu dW u .
(µu − ru 1n×1 ) + ru − (cu + yu ) du +
0
(5.16)
0
Similar to equation (5.12) we have:
But
h i d 1 (u,c+y) (u,c) d |=0 Wt = Wt |=0 Yet − Ye . d d 2 t h i ´t d d e = − y du , and | Y | Ye = 0. Hence we obtain equation d =0 t d =0 0 u
(5.17)
(5.10).
t
In the following two lemmas, for the case of zero consumption, we nd an expression for d | L(u + v) in terms of the martingale V (u) of equation (5.51). d =0
Lemma 5.2. For the special case of zero consumption, de ne the Lagrangian as: h i (u) L (u) , E U (WT ) ,
(5.18)
where W (u) = W (u,0) . Then for all locally bounded predictable processes v , we have: h d (u) (u) 0 WT |=0 L(u + v) = E U WT d ˆ T ˆ T T v s µs − rs 1n×1 − Σs Σs us ds + × 0
Proof.
T
v Ts Σs dW s
(5.19)
0
By dierentiating equation (5.18), and interchanging the order of dierentiation with
expectation, we would have
d d (u+v) |=0 L(u + v) = E |=0 U (WT ) d d (u) d (u+v) 0 = E U (WT ) |=0 WT . d Now by substituting for
(5.20)
(u+v) d from equation (5.9) we will get the result. | W d =0 T
Lemma 5.3. For the special case of zero consumption, we de ne the martingaleV (u) as: (u)
Vt
h i (u) (u) , Et U 0 WT WT .
(5.21)
Then for all locally bounded predictable processes v , we have d |=0 L(u + v) = E d
ˆ
ˆ
T
Vs(u) v Ts
µs − rs 12×1 −
0
Σs ΣTs us
T
v Ts Σs d
ds + 0
W,V
(u)
s
.
(5.22)
25
Proof.
By considering the fact that
(u) (u) VTu = U 0 WT WT ,
equation (5.19) can be rewritten as
ˆ T ˆ T d T u T T v s Σs dW s . |=0 L(u + v) = E VT v s µs − rs 12×1 − Σs Σs us ds + d 0 0
(5.23)
By integration by parts we have:
VTu ˆ =
ˆ 0
T
ˆ
T
v Ts
µs − rs 12×1 −
Σs ΣTs us
ds + ˆ T µs − rs 12×1 − Σs Σs us ds +
T
v Ts Σs dW s
0 T
Vsu v Ts Σs dW s Vsu v Ts 0 0 ˆ t ˆ T ˆ t T T T v s Σs dW s dVtu v s µs − rs 12×1 − Σs Σs us ds + + 0 0 0 ˆ T v Ts Σs d [W , V u ]s . +
(5.24)
0 Note that, subject to some technical conditions, the second and third terms in the right-handside are zero-mean martingales. Now by taking expectation we will get the result. The following two lemmas are the counterparts of lemmas 5.2 and 5.3, for the case of non-zero consumption.
Lemma 5.4. For the general case with consumption, de ne the Lagrangian as: ˆ L (u, c) , E
T
(u,c) u(t, ct Wt )dt
+
(u,c) U (WT )
.
(5.25)
0
Then for all locally bounded predictable processes v , we would have: d |=0 L (u + v, c) d ˆ t ˆ T ˆ t ∂ (u,c) (u,c) T T T = E u t, ct Wt ct Wt v s µs − rs 12×1 − Σs Σs us ds + v s Σs dW s dt 0 ∂x 0 0 ˆ T ˆ T (u,c) (u,c) 0 T T T WT v s µs − rs 12×1 − Σs Σs us ds + v s Σs dW s , (5.26) +E U WT 0
0
and for all locally bounded predictable processes y, we have: d |=0 L (u, c + y) d ˆ T ˆ t ∂ (u,c) (u,c) = E u t, ct Wt Wt y t − ct ys ds dt 0 ∂x 0 ˆ T (u,c) (u,c) 0 −E U WT WT ys ds . 0
26
(5.27)
Proof.
By dierentiating equation (5.25), and interchanging the order of dierentiation with
expectation and then with integration, we would have
d |=0 L (u + v, c) = E d
ˆ
T
d d (u+v,c) (u+v,c) |=0 u(t, ct Wt )dt + |=0 U (WT ) d 0 d ˆ T d ∂ (u,c) (u+v,c) u t, ct Wt ct |=0 Wt dt = E d 0 ∂x (u,c) d (u+v,c) 0 +E U (WT ) |=0 WT . d
Now by substituting for
(5.28)
(u+v,c) (u+v,c) d d | W and | W from equation (5.9) we will get d =0 t d =0 T
equation (5.26). For obtaining equation (5.27), by dierentiating equation (5.25) we would have:
ˆ
T
d d (u,c+y) (u,c+y) |=0 u(t, (ct + yt ) Wt )dt + |=0 U (WT ) d 0 d ˆ T d ∂ (u,c) (u,c+y) = E u t, ct Wt ct |=0 Wt dt d 0 ∂x ˆ T ∂ (u,c) (u,c+y) u t, ct Wt yt Wt dt (5.29) +E 0 ∂x (u,c) d (u,c+y) 0 +E U (WT ) |=0 WT . (5.30) d
d |=0 L (u, c + y) = E d
Now substituting for
(u,c+y) (u,c+y) d d | W and | W from equation(5.10) would result in d =0 t d =0 T
equation (5.27).
Lemma 5.5. De ne the processV (u,c) by Vs(u,c)
ˆ , Es s
T
∂ (u,c) (u,c) (u) (u) 0 WT . u t, ct Wt ct Wt dt + U WT ∂x
(5.31)
Then for all locally bounded predictable processes v , we would have: d |=0 L(u + v, c) = E d
ˆ
ˆ
T
Vs(u,c) v Ts
µs − rs 12×1 −
Σs ΣTs us
v Ts Σs d
ds +
0
T
W,V
0
and for all locally bounded predictable processes y, we have: d |=0 L (u, c + y) = E d
Proof.
ˆ
T
∂ (u,c) (u,c) (u,c) u t, ct Wt Wt − Vt yt dt . ∂x
0
(u,c)
s
,
(5.32)
(5.33)
To derive equation (5.33), we note that by equation (5.27)
d |=0 L (u, c + y) = E d
ˆ 0
T
∂ (u,c) (u,c) u t, ct Wt Wt yt dt − I − II, ∂x
27
(5.34)
where
ˆ t ∂ (u,c) (u,c) u t, ct Wt ct Wt I , E ys ds dt 0 ∂x 0 ˆ T (u,c) (u,c) 0 ys ds . II , E U WT WT ˆ
T
(5.35)
0 For
I
we may write
ˆ
T
ˆ
t
∂ (u,c) (u,c) E I = u t, ct Wt ct Wt ys dsdt ∂x 0 0 ˆ Tˆ T ∂ (u,c) (u,c) = E u t, ct Wt ct Wt ys dtds ∂x 0 s ˆ Tˆ T ∂ (u,c) (u,c) E Es = u t, ct Wt ct Wt ys dtds ∂x 0 s ˆ T ˆ T ∂ (u,c) (u,c) ys Es = E ct Wt dt ds . u t, ct Wt s ∂x 0 h i (u,c) (u,c) (u,c) 0 For handling II , rst we de ne a martingale Gs , Es U W T WT . integration by parts to the de nition of II gives us ˆ T (u,c) ys ds II = E GT 0 ˆ T ˆ T ˆ t (u,c) (u,c) ys Gs ds + ys ds dGt = E 0 0 0 ˆ T (u,c) ys Gs ds , = E
(5.36)
Then applying
(5.37)
0 where we used the fact that, subject to some technical conditions, the second term in the second line is a martingale so its expectation vanishes. Now substituting for
I
and
II
in equation (5.34)
will result in equation (5.33). Similarly, to derive equation (5.32), we will start by equation (5.26)
d |=0 L (u + v, c) = I + II + III, d
(5.38)
where
ˆ
ˆ t ∂ (u,c) (u,c) T T I , E u t, ct Wt ct Wt v s µs − rs 12×1 − Σs Σs us ds dt (5.39) 0 0 ∂x ˆ T ˆ t ∂ (u,c) (u,c) T II , E u t, ct Wt ct Wt v s Σs dW s dt (5.40) 0 ∂x 0 h (u,c) (u,c) III , E U 0 WT WT ˆ T ˆ T T T T v s Σs dW s . (5.41) × v s µs − rs 12×1 − Σs Σs us ds + T
0
0 28
Expression
III
can be handled by following the same argument as in the proof of lemma 5.3,
which gives us
ˆ
ˆ
T
Ves(u,c) v Ts µs − rs 12×1 − Σs ΣTs us ds +
III = E 0
T
v Ts Σs d 0
where
i h (u,c) (u,c) (u,c) 0 e WT . Vs , Es U W T
For handling
I,
ˆ
T
h
W , Ve (u,c)
i
,
(5.42)
s
(5.43)
we follow a calculation similar to equation (5.36)
ˆ
t
∂ (u,c) (u,c) T T u t, ct Wt ct Wt v s µs − rs 12×1 − Σs Σs us dsdt E ∂x 0 0 ˆ Tˆ T ∂ (u,c) (u,c) T T E u t, ct Wt ct Wt v s µs − rs 12×1 − Σs Σs us dtds ∂x s 0 ˆ Tˆ T ∂ (u,c) (u,c) T T dtds E Es u t, ct Wt ct Wt v s µs − rs 12×1 − Σs Σs us ∂x 0 s ˆ T ˆ T ∂ (u,c) (u,c) T T Es E u t, ct Wt ct Wt dt v s µs − rs 12×1 − Σs Σs us ds (. 5.44) s ∂x 0
I = = = =
For handling
II ,
rst we de ne:
Vs(u,c,t)
, Es
∂ (u,c) (u,c) u t, ct Wt ct Wt ∂x
(5.45)
Now integration by parts would give us
ˆ t ∂ (u,c) (u,c) T v s Σs dW s u t, ct Wt ct Wt E ∂x 0 ˆ t (u,c,t) T = E Vt v s Σs dW s 0 ˆ t ˆ t ˆ s ˆ t (u,c,t) T T (u,c,t) T (u,c,t) = E Vs v s Σs dW s + v u Σu dW u dVs + v s Σs d W , V s 0 0 0 0 ˆ t v Ts Σs d W , V (u,c,t) s , = E (5.46)
0
where we assume that the rst two integrals in the third line are zero mean martingales. We
29
may now proceed as follow:
ˆ
II = = = =
ˆ t ∂ (u,c) (u,c) T E u t, ct Wt ct W t v s Σs dW s dt ∂x 0 0 ˆ T ˆ t T (u,c,t) v s Σs d W , V. dt E s 0 0 ˆ T ˆ t (u,c,t) 0 T dsdt v s Σs W , V. E s 0 0 ˆ T ˆ T T (u,c,t) 0 E v s Σs W , V. dtds s 0 s "ˆ 0 # ˆ T
T
T
v Ts Σs
= E ˆ
V.(u,c,t) dt
W, .
0 T
v Ts Σs d
= E
ˆ
ds s
T
V.
W,
(u,c,t)
dt .
.
0
(5.47)
s
Finally we put everything back together to obtain equation (5.32):
d |=0 L (u + v, c) = I + II + III d ˆ T ˆ Es = E 0
ˆ
T
+E 0
ˆ
T
+E 0
ˆ +E ˆ = E
T
T
∂ (u,c) (u,c) u t, ct Wt ct Wt dt s ∂x × v Ts µs − rs 12×1 − Σs ΣTs us ds ˆ T T (u,c,t) v s Σs d W , V. dt . s (u,c) > T e Vs v s µs − rs 12×1 − Σs Σs us ds h i > (u,c) v Σs d W , Ve s
0 T
ˆ
(5.48)
(5.49)
s
T
∂ (u,c) (u,c) (u,c) (u,c) 0 WT u t, ct Wt ct Wt dt + U WT Es 0 s ∂x ×v Ts µs − rs 12×1 − Σs ΣTs us ds ˆ T ˆ T h i T (u,c,t) (u,c) e +E v s Σs d W , V. dt + d W , V s 0 . s ˆ T = E Vs(u,c) v Ts µs − rs 12×1 − Σs ΣTs us ds 0 ˆ T T (u,c) +E v s Σs d W , V , (5.50) s 0
Where in the last step we used the fact that
(u,c)
Vs
Finally we are ready to prove the main result.
30
=
´T s
(u,c,t)
Vs
(u,c) dt + Ves .
Theorem 5.6. (the Direct Approach) Consider problem 2.9. For the case of zero consumption, de ne the martingale V (u) by h
(u) Vt
, Et U
0
(u) WT
(u) WT
i
.
(5.51)
Then a self- nancing strategy u (with no consumption) satis es the rst-order optimality conditions, i.e. dd |=0 L(u + v) = 0 for all bounded predictable process v , if and only if −1 Σt ΣTt
ut =
[W , V u ]0t (µt − rt 1n×1 ) + Σt . Vtu
(5.52)
For the general case (i.e. nonzero consumption) de ne the processV (u,c) by ˆ
(u,c) Vt
T
, Et t
∂ (u,c) (u,c) (u,c) (u,c) 0 WT . u s, cs Ws cs Ws ds + U WT ∂x
(5.53)
Then the pair (u, c) satis es the rst-order optimality conditions, i.e. (
d | L(u + v, c) = 0 d =0 d | L(u, c + y) = 0 d =0
(5.54)
for all bounded predictable process v and y, if and only if (u,c) = Vt
∂ u ∂x
(u,c) t, ct Wt
(u,c)
Wt
, 0 [W ,V (u,c) ]t T −1 (µt − rt 1n×1 ) + Σt . ut = Σt Σt (u,c)
(5.55)
Vt
Proof.
First consider the case of zero consumption. By lemma (5.3) the rst order condition
ˆ
becomes
T
E
v Ts
Vsu
µs − rs 12×1 −
Σs ΣTs us
+ Σs [W , V
u 0 ]s ds
= 0,
(5.56)
0
for all bounded
predictable
processes
v.
Since the coe cient in the braces is predictable, we
may conclude that it must vanish:
Vsu µs − rs 12×1 − Σs ΣTs us + Σs [W , V u ]0s = 0,
for all
0 < s < T.
(5.57)
This last result is equivalent to equation (5.52). For the general case (with nonzero consumption) we use lemma (5.5) to rewrite the rst order conditions as:
E
h´
for all bounded
T 0
E n
h´ n T 0
(u,c)
v Ts Vs
o i (u,c) − Vt yt dt = 0, 0 o i µs − rs 12×1 − Σs ΣTs us + Σs W , V (u,c) s ds = 0,
∂ u ∂x
(u,c)
t, ct Wt
predictable process v and y.
(u,c)
Wt
(5.58) (5.59)
Again since all terms in the braces are predictable,
they must vanish:
∂ u ∂x
(u,c)
Vs
(u,c)
(u,c)
(u,c)
− Vt = 0, for all 0 < t < T, 0 µs − rs 12×1 − Σs ΣTs us + Σs W , V (u,c) s = 0, for all 0 < s < T. t, ct Wt
Wt
This last system is equivalent to equation (5.55).
31
(5.60)
Chapter 6 Examples The most frequently used utilities are the exponential, power and logarithmic ones. Exponential case case has been extensively studied not only in optimal investment models but, also, in indierence pricing where valuation is done primarily under exponential preferences (see [19] for a concise collection of relevant references). However, a well known criticism of the exponential utility is that the optimal portfolio does not depend on the investors wealth.
While
this property might be desirable in asset equilibrium pricing, it appears to be problematic and counter intuitive for investment problems. Here we focus on logarithmic and power utilities.
6.1 Logarithmic Utility In this section, we consider the general market of de nition 2.1 and the logarithmic utility functional of the following form,
ˆ
T
log (Ct ) dt + log (Z) .
J(C, Z) , E
(6.1)
0 The logarithmic utility plays a special role in portfolio choice.
As it will be shown in this
section, the optimal portfolio is myopic, namely,
u = ΣΣT
−1
(µ − r1n×1 ) ,
This is a well-known fact, and the associated myopic portfolio, also know as the
portfolio,
(6.2)
growth optimal
has been extensively studied in the general semimartingale market settings (see, for
example, [4] and [45]). The associated optimal wealth is the so-called "numeraire portfolio". It has also been extensively studied, for it is the numeraire with regards to which all wealth processes are supermartingales under the historical measure (see, among others, [40] and [41]). Using the general results of chapter 4 and 5 would be an overkill for the case of logarithmic utility, since there is a much more simpler way.
Actually the idea of the direct approach (u,c) stem from this solution method. Note that from equation(2.10) we have Wt = wE (Y )t = Yt − 21 [Y ]t we , where,
Yt ,
´t 0
´t u> ) + rs − cs ds + 0 u> s (µs − rs 1n×1 s Σs dW s , ´t > > [Y ]t = 0 us Σs Σs us ds.
32
(6.3)
The optimum criterion can be expanded as follow,
ˆ
T
E
(u,c) log(ct Wt )dt
0
ˆ = E
ˆ
T
log (ct ) dt + 0
+
T
(u,c) log(WT )
(u,c) log(Wt )dt
(6.4)
+ log
(u,c) WT
.
(6.5)
0
Now we may with the second term above to obtain,
h i (u,c) E log(WT ) 1 = log(w) + E YT − [Y ]T 2 ˆ T 1 > > > = log(w) + E − us Σs Σs us + us (µs − rs 1n×1 ) + rs − cs ds . 2 0
(6.6)
The third term can also be written in the form,
ˆ
T
E 0
ˆ
T
ˆ
T
− T log(w) 1 [Y ]t dt 2
Yt dt − 0 0 ˆ T ˆ t 1 > > > = E − us Σs Σs us + us (µs − rs 1n×1 ) + rs − cs dsdt 2 0 0 ˆ T ˆ t > us Σs dW s dt +E 0 0 ˆ T ˆ T 1 > > > = E − us Σs Σs us + us (µs − rs 1n×1 ) + rs − cs dtds 2 0 s ˆ T ˆ t > + E us Σs dW s dt 0 0 ˆ T 1 > > > (T − s) − us Σs Σs us + us (µs − rs 1n×1 ) + rs − cs ds , = E 2 0 ´t > the last step we assume u Σ dW s to be a martingale. Substituting back 0 s s = E
where in
(u,c) log(Wt )dt
(6.7)
the last
two results in the optimal criterion gives us,
ˆ
T
E
(u,c) log(ct Wt )dt
+
(u,c) log(WT )
0
= (1 + T ) log (w) ˆ T 1 > > > +E (1 + T − s) − us Σs Σs us + us (µs − rs 1n×1 ) + rs ds 2 0 ˆ T +E {log(cs ) − (1 + T − s)cs } ds .
(6.8)
0 Now it su ces to maximize the quadratic integrand
1 > u 7→ − u> Σs Σ> s u + u (µs − rs 1n×1 ) , 2 33
(6.9)
and the integrand
c 7→ {log(c) − (1 + T − t)c}.
Finally elementary calculus will give us the
optimal portfolio as of equation (6.2) and the optimal consumption as,
1 . 1+T −t
ct =
(6.10)
Applying the direct approach is also straightforward. (u,c) and V :
(u,c) Vt
, Et
h´ T t
h
Vtu
0
(u) WT
(u) WT
First we nd the martingales
i
= 1, , Et U i (u) (u,c) (u,c) (u) ∂ 0 W = 1 + T − t. u s, c W c W ds + U W s s s s T T ∂x
(6.11)
0
V (u)
0 = W , V (u,c) t = 0. So for the case of zero consumption, by using t equation (5.52), the optimal portfolio would be: Obviously
W , V (u)
ut = Σt ΣTt
−1
(µt − rt 1n×1 ) ,
(6.12)
which is the same as what we found earlier by martingale approach. For the case of non-zero consumption, equation (5.55) will give use the same portfolio strategy as above, and the optimal consumption as:
(u,c)
Vt
=
∂ 1 (u,c) (u,c) u t, ct Wt Wt =⇒ ct = , ∂x 1+T −t
(6.13)
which is what we have already found in equation (6.10). Finally let's take the martingale approach. Besides proving that the optimal portfolio and consumption is of the form (6.2) and (6.10), we will also prove that the optimal market price of risk, and the optimal wealth process are given by:
λ = ΣT ΣΣT w (1 + T − t) ´0t = e (1 + T )
(C (λ) ,Z (λ) ,λ) Wt
−1
b,
(6.14)
−1
rs +(µs −rs 1n×1 )T (Σs ΣT s)
ˆ
.
(µ − r1n×1 )
×E
T
(µs −rs 1n×1 ) ds
−1 Σs ΣTs
Σs dW s
0 We start by nding an expression for I(y, t) = IF (y) = y1 and we obtain:
E
η (λ) ˆ
(λ) πT IF (η
(λ) (λ) πT )
of lemma 4.4. Note that
T
πτ(λ) I(η
+
(λ) πτ(λ) , τ )dτ
∂ u (t, x) ∂x
1+T . η (λ)
=
0 Then equation (4.6) will give us
(λ)
Ct
η (λ) =
.
(6.15)
t
= U 0 (x) =
1 , so x
(6.16)
1+T , and we have: w
1
(λ)
= I(η (λ) πt , t) =
η
1
(λ)
Z (λ) = IF (η (λ) πT ) =
(λ) (λ) πt (λ)
η (λ) πT
34
= =
w (λ)
(1 + T ) πt w
(λ)
(1 + T ) πT
.
,
(6.17)
(6.18)
The next step is to derive the process
(λ)
(C (λ) ,Z (λ) ,λ)
πt Wt
(λ)
(λ)
π (λ) W (C ,Z ,λ) . From equation (4.4) we may conclude: ˆ T (λ) (λ) (λ) (λ) πτ Cτ dτ = Et πT Z + t
w (1 + T − t) = . (1 + T ) But then
0 W , π (λ) W (C,Z,λ) t = 0,
(6.19) 0
so
a(C
(λ) ,Z (λ) ,λ)
,
[W ,π(λ) W (C,Z,λ) ]t (λ)
πt
(C,Z,λ)
Wt
=0
and equation (4.29)
λ as in equation (6.14). Now we may use theorem (4.9) to conclude that, only for (λ) (λ) is the optimal consumption pair. To nd equation (6.10), particular MPR λ, Ct , Z
would give us this
we observe that
(λ)
ct = Now by substituting for
Ct
(C (λ) ,Z (λ) ,λ) Wt
(C (λ) ,Z (λ) ,λ)
(λ)
w
=
πt W t
(1 +
(λ) (C (λ) ,Z (λ) ,λ) T ) πt Wt
.
(6.20)
in the denominator from equation (6.19) we will
obtain the result. Equation (6.2) directly follows from equation (4.30). Finally by substituting (λ) for πt from equation (2.4) into equation (6.19) we would have:
(C (λ) ,Z (λ) ,λ)
Wt
And substituting for
=
w (1 + T − t) (λ)
(1 + T ) πt w (1 + T − t) ´ = ´. − 0t ru du (1 + T ) e E − 0 λTu dW u ˆ . w (1 + T − t) ´0t (rs +λTs λs )ds T λs dW s . e E = (1 + T ) 0 t λ
(6.21)
from equation (6.14) will give us equation (6.15).
6.2 Power Utility In this section we take the power utility functional, de ned as:
ˆ
T
J(C, Z) , E 0
Ctγ Zγ dt + . γ γ
(6.22)
Then we consider the problem originally solved by Merton, and redrive the solution. After that we consider the case where the interest rate is stochastic, and follows a multi-factor Gaussian model. The optimal portfolio in the two cases dier in the following ways (compare theorem 6.3 and theorem 6.8),
•
In the Merton setting, the optimal portfolio only includes the myopic term, while the excess hedging demand term (containing the covariance between zero-coupon bonds and assets) enters in the optimal portfolio for the Gaussian term structure case.
•
In the Merton setting the optimal portfolio is deterministic whether we consider consumption or not, but for the Gaussian term structure case the optimal portfolio is deterministic when there is no consumption, and stochastic otherwise.
35
6.2.1
Original Merton Setting
Consider a market with deterministic coe cients, de ned as follow.
De nition 6.1. A market with deterministic coe cients is de ned as a special case of de nition 2.1 where
r, µ,
and
Σ
are deterministic functions of time, and
n = k
(i.e.
the market is
complete). In this case we write:
rt = r(t) µt = µ(t) Σt = Σ(t).
(6.23)
After deriving a useful result in lemma 6.2, we will identify Merton's solution in theorem 6.3.
Lemma 6.2. Consider the market de ned in de nition6.1 and let λ(t) and π = (πt ) be the unique market-price-of-risk and SPD identi ed in theorem 2.3, equations (2.8) and (2.4), respectively . Then for any constant ξ we have: πtξ = m(t) × Λt h i Es πtξ = m(t) × Λs
for all
t,
(6.24)
for all
s < t,
(6.25)
where the deterministic function m(t) and the martingale Λ = (Λt ) are de ned as: ´
t ξ−1 T λ (s)λ(s)}ds 2 m(t) , eξ 0 {−r(s)+ , ´. T Λ , E −ξ 0 λ (s) dW s .
Proof.
(6.26)
By using proposition A.4 (part iii) of appendix A, we obtain equation (6.24) as follow:
πtξ
´ ˆ . ξ T − 0t ru du = e E − λ (s) dW s 0
t
ˆ . ξ T −ξ 0 r(s)ds = e E − λ (s) dW s 0 t ˆ . ´t ´. T ξ(ξ−1) T [− −ξ 0 r(s)ds λ (s)dW ] s t 0 = e E −ξ λ (s) dW s e 2 0 t ˆ . ´t ξ−1 T λT (s) dW s . = eξ 0 {−r(s)+ 2 λ (s)λ(s)}ds E −ξ ´t
0
(6.27)
t
To obtain equation (6.25), we only need to take conditional expectation from the last result, and use the fact that
m(.)
is deterministic and
Λ
is a martingale:
h i Es πtξ = Es [m(t) × Λt ] = m(t)Es [Λt ] = m(t) × Λs .
36
(6.28)
Theorem 6.3. Consider the market de ned in de nition6.1 . Then: (i) Consider the Merton's problem with the total utility as: J(u) = E where
(u) WT
γ ,
γ
(6.29)
γ > 0 is a constant. Then the optimal portfolio is deterministic and is given by: u=
−1 1 ΣΣT (µ − r1n×1 ) 1−γ
(6.30)
(ii) Consider the Merton's problem with the total utility as: ˆ J(c, u) = E
T
(u,c)
ct Wt
γ
γ
0
dt +
(u,c)
WT
γ
γ
i.e. the case of power utility with consumption. Then the optimal portfolio is the same as the case of no consumption (i.e. equation 6.30). The optimal proportional consumption is also deterministic and is given by: ct = ´ T t
m(t)
m(s)ds + m(T )
,
(6.31)
where the function m(t) is de ned as: ´
t ξ−1 T m(t) , eξ 0 {−r(s)+ 2 λ (s)λ(s)}ds ,
with ξ = Proof.
γ γ−1
(6.32)
.
As pointed out in remark 4.10, since the market is complete, equation (4.29) holds
automatically. We only need to nd an expression for
(λ) (λ) a(C ,Z ,λ) ,
and then equations (4.27)
and (4.35) will give us the optimal solution. Also note that since the market is complete and −1 the unique MPR is λ = Σ (µ − r1n×1 ), we may drop the function argument (λ) in η (λ), (λ) π ,etc. (i) To apply the martingale approach we start with the function
IF (y) = y
1 γ−1 .
So
η
η
of lemma 4.4. We have
can be derived as follow:
E [πT IF (ηπT )]
=
h γ i 1 η γ−1 E πTγ−1 = w γ−1
w =⇒ η = h γ i E πTγ−1
.
(6.33)
(0,Z (λ) ,λ) de ned in theorem 4.9. Note that by taking The next goal is to nd an expression h ifor a ξ γ ξ , γ−1 in lemma 6.2 we have Et πT = m(T ) × Λt , from which we may also conclude: h i h i E πTξ = E0 πTξ = m(T ) × Λ0 = m(T ). 37
(6.34)
π (λ) W (0,Z
Now consider the martingale
(λ)
(0,Z (λ) ,λ)
πt Wt
(λ) ,λ)
of equation (4.4),
, Et [πT IF (ηπT )] h γ i 1 = η γ−1 Et πTγ−1 w = (m(T ) × Λt ) m(T ) = w × Λt ˆ . γ T λ (s) dW s . = wE − γ−1 0 t
(6.35)
This last result gives us
(0,Z (λ) ,λ)
a
,
W , π (λ) W (C,Z,λ)
0 t
(λ) (0,Z (λ) ,λ) πt Wt
γ λ (t) γ−1 γ = − Σ−1 b. γ−1
= −
(6.36)
Finally by using equation (4.35) we nd the optimal portfolio:
(ii) Here
u(x, t) = U (x) =
ΣΣT
−1
xγ , so γ
b + ΣT
−1
(λ)
(λ)
a(C ,Z ,λ) −1 −1 γ = ΣΣT b − ΣT Σ−1 b γ−1 −1 1 = ΣΣT b. 1−γ
u =
(6.37)
1
I(y, t) = IF (y) = y γ−1 .
ˆ
Hence equation 4.6 would become:
T
w = E
πt I(ηπt , t)dt + πT IF (ηπT )
0
ˆ
T
1 γ−1
= E
πt (ηπt ) dt + πT (ηπT ) ˆ T γ γ 1 γ−1 γ−1 γ−1 = η E πt dt + πT ,
1 γ−1
0
(6.38)
0 And we obtain
η
as:
γ−1
w η = h´ γ γ i T γ−1 γ−1 E 0 πt dt + πT To nd an expression for
(λ) (λ) a(C ,Z ,λ) ,
(6.39)
rst note that from equation 6.25 of lemma 6.2, we
38
have
Et πsξ = m(s) × Λt ˆ Et
t < s. So: ˆ T h i ξ ξ Et πsξ ds + Et πTξ = πs ds + πT t ˆ T = m(s) × Λt ds + m(T ) × Λt t ˆ T m(s)ds + m(T ) × Λt . =
for all
T
t
(6.40)
t Now by applying the product rule we obtain:
ˆ
T
πsξ ds
Et
+
πTξ
ˆ
T
m(s)ds + m(T ) ˆ T ˆ t Λτ d m(s)ds + m(T ) + 0 . τ ˆ t ˆ T m(s)ds + m(T ) dΛτ + 0 τ ˆ T ˆ t = m(s)ds + m(T ) − Λτ m (τ ) dτ 0 0 ˆ t ˆ T − m(s)ds + m(T ) Λτ ξλT (τ ) dW τ 0 τ ˆ t ˆ T Λτ m (τ ) dτ m(s)ds + m(T ) − = 0 0 ˆ t ˆ T ξ ξ Eτ πs ds + πT ξλT (τ ) dW τ . −
=
0
t
0
(6.41)
τ
In the last step we used equation (6.40). From this last result we may conclude:
ˆ
T
πTξ
πsξ ds
ˆ
T
+ E = m(s)ds + m(T ) 0 0 h h´ ii0 T ξ ξ W , E. . πs ds + πT h´ i t = −ξλ (τ ) T ξ ξ Et t πs ds + πT From the rst equation above we will nd the following explicit form for 1
η γ−1 = E Now consider the process
h´ T
π (λ) W (C
(C (λ) πt Wt
w
πsξ ds 0
+
(λ) ,Z (λ) ,λ)
)
(λ) ,Z (λ) ,λ
πTξ
i = ´T 0
w m(s)ds + m(T )
(6.42)
(6.43)
η:
.
(6.44)
of equation (4.4):
ˆ
T
, Et πt I(ηπt , t)dt + πT IF (ηπT ) t ˆ T 1 ξ ξ = η γ−1 Et πs ds + πT , t 39
(6.45)
where we took
ξ=
γ . By using equation (6.43) we may conclude γ−1
h i0 C (λ) ,Z (λ) ,λ) (λ) ( W,π W
(C (λ) ,Z (λ) ,λ) at ,
t
(C (λ) πt W t
(λ) ,Z (λ) ,λ
)
ii0 h h´ T W , Et t πsξ ds + πTξ i t = −ξλ (τ ) h´ = T ξ ξ Et t πs ds + πT But this is exactly what we found in equation (6.36) for the case of zero consumption. Hence the optimal portfolio would be also given by equation (6.30). To obtain the optimal consumption, we may proceed as follow:
πt Ct = πt I(ηπt , t) 1
= η γ−1 πtξ 1
= η γ−1 × m(t) × Λt ˆ T 1 m(t) = ´T η γ−1 m(s)ds + m(T ) Λt m(s)ds + m(T ) t t ˆ T 1 m(t) ξ ξ = ´T η γ−1 Et πs ds + πT m(s)ds + m(T ) t t m(t) (C (λ) ,Z (λ) ,λ) πt Wt . = ´T m(s)ds + m(T ) t Finally by using the relation
ct =
Ct
(C (λ) ,Z (λ) ,λ) W
(6.46)
, we will get the optimal proportional consump-
t
tion of equation (6.31).
6.2.2
Complete Gaussian Market
Consider the complete Gaussian market de ned bellow.
De nition 6.4.
By a
Gaussian Market
we are referring to the special case of de nition 2.1
where the volatility and the market price of risk process are deterministic, i.e.
λt = λ (t).
Furthermore the short-rate process
r
follows a
k -factor
Σt = Σ(t)
and
Gaussian short rate model.
That is, we have:
rt =
k X
(i)
rt = 11×k r t ,
(6.47)
i=1 where
r = r(1) , . . . , r(k) are independent Gaussian process (i) (i) ft(i) drt = ϑ(i) (t) + α(i) (t) rt dt + δ (i) (t)dW
with the following dynamics:
f or i = 1, . . . , k,
(6.48)
or equivalently,
f t. dr t = (θ (t) + diag (α (t)) r t ) dt + diag (δ (t)) dW Here
ϑ(i) (.), α(i) (.)
and
δ (i) (.)
are given deterministic functions.
40
(6.49)
We will also need to consider
De nition 6.5.
The
τ -maturity
τ -maturity
bonds explicitly.
bond price process
(τ )
Bt
B
(τ )
=
(τ ) Bt
is de ned as:
´τ − t rs ds , EQ e t
(6.50)
T σ (τ ) (t) to be the deterministic volatility of B (τ ) and ρ(A,τ ) = ρ(1,τ ) (t) , . . . , ρ(n,τ ) (t) to be the deterministic instantaneous correlation between the returns of assets A and the τ (τ ) maturity bond B . (τ ) (τ ) The following lemma will introduce a family of martingales F = Ft which play a Also assume
main role in nding the optimal solutions.
Lemma 6.6. For a maturity τ > 0 and an arbitrary constant ξ , de ne a family of martingales (τ ) F (τ ) = Ft as: E F
(τ )
t
−ξ
, Et e
´τ 0
rs ds
(6.51)
Then for the Brownian motion W we have:
Proof.
W , F (τ )
0
= ξσ (τ ) .
First we show that for some deterministic function
(6.52)
f (τ )
we have:
´τ ´τ −ξ 0 ru du Et e−ξ 0 ru du = f (τ )EQ e . t To see this, recall from equation (2.5) that
f u = dW u + λ (u) du. dW
(6.53)
Now from equation(B.2)
of appendix B, we have:
rt
ˆ t ˆ t −1 −1 fu Υ (u)θ(u)du + Υ (u)diag (δ (t)) dW = Υ(t) r 0 + 0 0 ˆ t ˆ t −1 −1 = Υ(t) r 0 + Υ (u)θ(u)du + Υ (u)diag (δ (t)) dW u 0 0 ˆ t Υ−1 (u)diag (δ (t)) λ(u)du +Υ(t) 0
, r˜ t + g(t).
(6.54)
r˜ , 11×k r˜ and g (t) , 11×k g(t). Note that rt = r˜t + g (t). Also, since r and P-dynamics of r˜ are identical, the same is true for r and r˜. So: ´τ ´τ ´τ Et e−ξ 0 ru du = e−ξ 0 g(u)du Et e−ξ 0 r˜u du ´τ ´ −ξ 0τ g(u)du Q −ξ 0 ru du = e Et e .
De ne of
And we may take
f (τ ) , e−ξ
´τ 0
g(u)du
.
41
the
Q-dynamics
(6.55)
´t
r du|Fs is Gaussian, and that its conditional variance s u is deterministic (to see this in a special case, refer to equation B.7 in appendix B). So we may Now it is a well known fact that
write:
E F
(τ )
t
, Et e
−ξ
´τ 0
ru du
´τ −ξ 0 ru du = f (τ )EQ e t ´τ ´t −ξ t ru du = f (τ )e−ξ 0 ru du EQ e t = f (τ )e−ξ = f (τ )e = f (τ )e = f (τ )e Since Vart
´τ t
ru du
−ξ
−ξ
−ξ
´t 0
´t 0
´t 0
´t 0
´τ Q ´τ ξ2 ru du −ξEt ( t ru du)+ 2 Vart ( t ru du)
e
ru du
e
ru du
e
ru du
e
ξ2 −ξ 2
ξ2 −ξ 2
ξ2 −ξ 2
´τ Vart ( t ru du)
e−Et (
´τ Vart ( t ru du)
´τ ξ − t ru du EQ e t
´τ Vart ( t ru du)
Bt
Q
(τ )
´τ t
ξ
ru du)+ 21 Vart (
´τ t
ru du)
ξ
.
(6.56)
is deterministic, we may conclude that:
F (τ ) = ξLog B (τ ) + BV. for some process
BV
(6.57)
of bounded variation. From this we may conclude
0 0 W , F (τ ) t = ξ W , Log B (τ ) t = ξσ (τ ) (t),
(6.58)
which is equation (6.52). The following lemma is a counterpart of lemma 6.2, and will play a crucial rule.
Lemma 6.7. Consider the market de ned in de nition 6.4 and let π = (πt ) be the SPD process. Then for any constant ξ we have: (t)
π ξ = m(t) × Λt , h it Es πtξ = m(t)Λ(t) for s < t, s
(6.59) (6.60)
where the deterministic function m(t) and the family of martingales Λ(t) are de ned as follow: ´t
ξ−1
T
m(t) , eξ 0 (−ξσ (s)+ 2 λ(s)) λ(s)ds , T ´. Λ(t) , E ξ 0 σ (t) (s) − λ (s) dW s .
Proof.
(t)
(6.61)
With a calculation similar to the way we obtain equation(6.27), we have:
πtξ
´ ˆ . ξ T − 0t rs ds = e E − λ (s) dW s 0 tˆ . ´ ´ ξ(ξ−1) t T T λ (s)λ(s)ds −ξ 0t rs ds 0 2 e E −ξ λ (s) dW s . = e 0 42
t
(6.62)
From equation(6.51) we have
πtξ
E F (t)
t
= e−ξ
´t 0
rs ds
. Now we can obtain equation(6.59) as follow:
ˆ . T 0 λ (s) dW s = e E F t E −ξ 0 ˆ t. ´ ´ ξ(ξ−1) t T T (t) λ (s)λ(s)ds [F (t) ,−ξ 0. λT (s)dW s ]t 0 2 = e λ (s) dW s E F −ξ e 0 t ˆ . n o o ´ n 0 (t) 0 T ξ 0t −λT (s)[W,F (t) ] + ξ−1 λ (s)λ(s) ds T 2 s = e F , W s − ξλ (s) dW s E ξ(ξ−1) 2
´t
λT (s)λ(s)ds
(t)
0
= m(t) ×
t
(t) Λt ,
(6.63)
where in the last steps we used equation (6.52). Finally equation (6.60) can be obtained by taking conditional expectation from equation(6.59). Finally the following theorem will give us the optimal solutions in this case.
Theorem 6.8. Consider the market de ned in de nition2.1 special case (i). Then: (i) Consider the Merton's problem with the total utility as: J(u) = E
(u)
WT
γ
γ
i.e. the case of power utility with no consumption. Then the optimal portfolio is given by: ut =
−1 −1 1 (T ) ΣTt Σt bt − γσ t 1−γ
(6.64)
(ii) Consider the Merton's problem with the total utility as: ˆ J(c, u) = E
T
(u,c) ct Wt
γ dt +
γ
0
(u,c) WT
γ
γ
i.e. the case of power utility with consumption. Then the optimal portfolio and optimal consumption are given by: u =
−1 1 ΣΣT b 1−γ (´ T ) (s) (s) (T ) (T ) m(s)Λ σ (t) ds + m(T )Λ σ (t) γ −1 t t t − ΣT , ´T (s) (T ) 1−γ m(s)Λ ds + m(T )Λ t t t
(6.65)
(t)
ct = ´ T t
m(t) × Λt (s)
(T )
m(s) × Λt ds + m(T ) × Λt
.
where the m(t) and Λ(τt ) are de ned in equation (6.61) with ξ =
43
(6.66)
γ γ−1
.
Proof.
(i) Note that by the same reasoning as in the proof of part(i) of theorem(6.3), equations
(6.33) and (6.35), we will obtain:
γ−1
w η = h i E πTξ
,
(0,Z (λ) ,λ)
(T )
(λ)
πt W t where
γ . By using the de nition of γ−1
ξ=
(λ) (0,Z (λ) ,λ) πt Wt
(T )
ˆ
Λt
(6.67)
= wΛt ,
(6.68)
, equation (6.61), we have:
t ) ξΛ(T s
T (s) − λ (s) dW s
(T )
σ = w 1+ 0 ˆ t T (λ) = w+ ξπs(λ) Ws(0,Z ,λ) σ (T ) (s) − λ (s) dW s .
(6.69)
0 And we obtain
0 W , π (λ) W (C,Z,λ) t (0,Z (λ) ,λ) (T ) at , = ξ σ (t) − λ (t) . (λ) (λ) (0,Z ,λ) πt Wt
(6.70)
Finally by using equation (4.35) we nd the optimal portfolio:
ΣΣT
u =
−1
b + ΣT
−1
a(C
(λ) ,Z (λ) ,λ)
−1 (T ) b + ξ ΣT σ − Σ−1 b −1 −1 (T ) = (1 − ξ) ΣΣT b + ξ ΣT σ . 1 −1 ΣT Σ−1 b − γσ (T ) . = 1−γ ΣΣT
=
−1
(6.71)
(ii) By a similar argument as what we used to obtain equations (6.39) and (6.45), we will have:
γ−1
η= (C (λ) πt Wt
w γ γ ´ E 0T πtγ−1 dt+πTγ−1
)
(λ) ,Z (λ) ,λ
=η
Also by using lemma (6.7) we have:
ˆ
Et
T
πsξ ds
+
πTξ
1 γ−1
ˆ
T
=
t
Et
h´
T t
,
πsξ ds
(6.72)
+
πTξ
i
.
(6.73)
h i Et πsξ ds + Et πTξ
t
ˆ
T
=
(s)
(T )
m(s)Λt ds + m(T )Λt .
(6.74)
t Note that:
ˆ
T
m(s) ×
d .
ˆ
Λ(s) . ds
T
m(s) × d Λ(s) ds − m(τ ) × Λ(t) τ τ dτ τ ˆ T T (s) (s) = ξm(s)Λτ σ (τ ) − λ (τ ) ds dW τ =
τ
τ ) −m(τ ) × Λ(τ τ dτ 44
(6.75)
So we may continue our calculations in equation (6.74) as follow:
ˆ Et
T
πsξ ds
+
πTξ
ˆ
t
T
=
m(s)ds + m(T ) ˆ t ˆ T (s) m(s) × Λ. ds d + . 0 τ ˆ t (T ) d m(T ) × Λt + 0 ˆ t ˆ T ) m(s)ds + m(T ) − m(τ ) × Λ(τ = τ dτ 0 0 ˆ t ˆ T T (s) (s) ξm(s)Λτ σ (τ ) − λ (τ ) ds dW τ + 0 τ ˆ t T ) ξm(T )Λ(T σ (T ) (τ ) − λ (τ ) dW τ + τ 0 ˆ T ˆ t ) m(s)ds + m(T ) − m(τ ) × Λ(τ = τ dτ 0 0 ˆ t ˆ T (s) ) (T ) m(s)Λ(s) (τ ) ds + m(T )Λ(T (τ ) + ξ τ σ τ σ 0
0
τ
ˆ
T
πsξ ds
−Eτ
+
πTξ
T λ (τ ) dW τ .
(6.76)
τ We may conclude that:
h (C (λ) ,Z (λ) ,λ) at ,
W,π
(λ)
(λ) (λ) W (C ,Z ,λ)
i0 t
(C (λ) πt Wt
)
(λ) ,Z (λ) ,λ
ii0 h´ T W , Et t πsξ ds + πTξ h´ i t = T ξ ξ Et t πs ds + πT (´ T ) (s) (T ) m(s)Λt σ (s) (t) ds + m(T )Λt σ (T ) (t) t = ξ − λ (t) . ´T (s) (T ) m(s)Λt ds + m(T )Λt t h
By using equation (4.35) we will nd the optimal portfolio of equation (6.65):
−1 (C (λ) ,Z (λ) ,λ) b + ΣT a −1 = (1 − ξ) ΣΣT b (´ T ) (s) (s) (T ) (T ) m(s)Λ σ (t) ds + m(T )Λ σ (t) −1 t t t +ξ ΣT . ´T (s) (T ) m(s)Λ ds + m(T )Λ t t t
u =
ΣΣT
−1
45
(6.77)
To obtain the optimal consumption of equation (6.66), we may proceed as follow:
ct =
πCt (η)
πt Wt
1
=
1
η γ−1
´
T t
η γ−1 πtξ (s)
(T )
m(s) × Λt ds + m(T ) × Λt
(t)
= ´T t
m(t) × Λt (s)
(T )
m(s) × Λt ds + m(T ) × Λt
where in the last step we used equation (6.59).
46
,
(6.78)
Chapter 7 Conclusions and Recommendations In this project, we considered the classical portfolio choice problem, also known as Merton's problem, when the opportunity set is stochastic.
The two main approaches to the problem,
namely the stochastic control theory and the martingale (or duality) approach, were discussed along with an extended literature study in chapters 1, 3, and 4.
Then we proposed a new
approach to portfolio choice problem, called the
in chapter 5.
chapter 6, we solve three special cases:
direct approach,
Finally, in
logarithmic utility in an incomplete market, power
utility in the original Merton's setting, and power utility in a market with Gaussian term structure.
For the logarithmic case we provided three dierent approaches.
For the power
utility cases, we nd out that the optimal solutions dier in the following ways,
•
In the Merton setting, the optimal portfolio only includes the myopic term, while the excess hedging demand term (containing the covariance between zero-coupon bonds and assets) enters in the optimal portfolio for the Gaussian term structure case.
•
In the Merton setting the optimal portfolio is deterministic whether we consider consumption or not, but for the Gaussian term structure case the optimal portfolio is deterministic when there is no consumption, and stochastic otherwise.
As the potential line of research based on the work conducted in this project, we propose the following topics:
•
Considering speci c stochastic factor models We may consider stochastic factors to model asset predictability, stochastic volatility, and interest rates. For example one may take other term-structure models such as C.I.R. or a stochastic volatility model such as Heston model. Alternatively one may try to consider more general market settings such as quadratic asset returns (which includes both C.I.R. and Heston model, see [61]) or HJM model (for example see [73]). See chapters 1 and 3 for more references.
•
Adapting the direct approach to alternative preference criteria As mentioned earlier, the traditional
static
choice of the utility function and investment
horizon is not realistic. it has long been recognized by economists that preferences may not be Intertemporally separable.
In particular, the utility associated with the choice
of consumption at a given date is likely to depend on past choices of consumption. For example high past consumption generates a desire for high current consumption. According to Browning [13], this idea dates back to the 1890 book `Principles of Economics' by
47
Alfred Marshall. Generalizations of standard time-separable preferences that have been
recursive or stochastic dierential utility (see, for example, [32, 76, 77]), habit formation criterion (see [67]), and forward performance criterion (see [68, 87]). One suggested include
may try to adapt the idea of the direct approach to these alternative formulations.
•
Providing validity conditions for the direct approach As mentioned in chapter 5, we have only checked the rst order optimality conditions. So, technically, the results obtain by the direct approach can only be considered as candidates for the optimal solution.
One may try to provide su cient conditions, or ideally the
minimal conditions, under which these candidate solutions are indeed optimal.
•
Extending the direct approach to general semimartingale markets One may try to extend the direct approach to a general semimartingale market.
The
main tools in developing the direct approach are properties of stochastic exponentials, which can still be used in a semimartingale setting. Nonetheless, such extensions would be more involved and obtainting validity conditions would be more demanding.
48
Appendix A Stochastic Exponential and Logarithms De nition A.1. exponential of
Let
X,
X=
(1) (k) Xt , . . . , X t
denoted by
E (X)
T
be a
k×1
semimartingale. Then the stochastic
is de ned as the unique solution of the following SDE
ˆ
t
Y =1+
diag (Y u− ) dX u .
(A.1)
0
T (1) (k) Y = Yt , . . . , Yt
De nition A.2.
Let
the two processes
Y (k)
by
Log (Y ),
and
(k)
Y−
be a
k×1
semimartingale such that, for all
do not vanish. Then the stochastic logarithm of
is de ned as
ˆ Log (Y ) ,
.
(diag (Y u− ))−1 dY u .
Y,
k,
denoted
(A.2)
0 Equivalently,Log (Y
Remark
A.3
.
)
is the unique process
X
satisfying
Y = diag (Y 0 ) E (X).
Y = diag (Y 0 ) E (X). We usually process ϕ we have: ˆ T ˆ T T dY ϕ , ϕT dX. Y 0 0
Suppose that
for a given predictable
denote
dX
by
dY , for example Y
(A.3)
In a similar way, we de ne:
Note that we have
d [ϕ, Y ] , d [ϕ, X] . Y dY = diag (Y ) dX , so we also have: dX = (diag (Y ))−1 dY , d [ϕ, X] = (diag (Y ))−1 d [ϕ, Y ] .
(A.4)
(A.5)
Proposition A.4. Let X and Y be two continuous semimartingale, and α be a constant. Then: (i) E (X) × E (Y ) = e[X,Y ] E (X + Y ) (ii) E(X) = e[Y ]−[X,Y ] E (X − Y ) E(Y ) α(α−1) (iii) E (X)α = e 2 [X] E (αX)
49
Proof.
Note that we have:
dE (X) = E (X) dX ˆ d[E (X)] = d[ E (X) dX] = E (X)2 d[X] Similar equations hold for
Y.
We also have:All the cases are straightforward application of Itô's
formula. (i) We have:
d (E (X) E (Y )) = E (X) dE (Y ) + E (Y ) dE (X) + d [E (X) , E (Y )] dE (Y ) dE (X) d [E (X) , E (Y )] = E (X) E (Y ) + + E (Y ) E (X) E (X) E (Y ) = E (X) E (Y ) {dY + dX + d [X, Y ]} = E (X) E (Y ) d (X + Y + [X, Y ]) So we obtain:
E (X) E (Y ) = E (X + Y + [X, Y ]) = e[X,Y ] E (X + Y ) (ii) Again by Itô's formula:
dE (X) E (X) dE (Y ) 1 E (X) d[E (Y )] E (X) d[E (X) , E (Y )] E (X) = − + 2 −2 d E (Y ) E (Y ) E (Y ) E (Y ) 2 E (Y ) E (Y )2 E (Y ) E (X) E (Y ) E (X) dE (X) dE (Y ) d[E (Y )] d[E (X) , E (Y )] − = − + E (Y ) E (X) E (Y ) E (X) E (Y ) E (Y )2 E (X) {dX − dY + d[Y ] − d[X, Y ]} = E (Y ) Hence:
E (X) = E (X − Y + [Y ] − [X, Y ]) = e[Y ]−[X,Y ] E (X − Y ) E (Y )
(iii) By Itô's formula:
1 dE (X)α = αE (X)α−1 dE (X) + α(α − 1)E (X)α−2 d[E (X)] 2 dE (X) α(α − 1) d[E (X)] α = E (X) α + E (X) 2 E (X)2 α(α − 1) = E (X)α αdX + d[X] 2 Finally we obtain:
α(α−1) α(α − 1) E (X) = E αX + [X] = e 2 [X] E (αX) 2 α
50
Appendix B Other Results Lemma B.1. (variant of Bay's rule)- Let Z = (Zt ) be a positive martingale with Z0 = 1 under measure P. De ne a new measure Q by dQ = ZT . Then a process X = (Xt ) is a Q-(local) dP martingale if and only if
ZX
is a
P
(local) martingale.
Theorem B.2. (change of measure)- Suppose, under measure P, Z = E (Y ) is a positive = ZT . Then a martingale for some semimartingale Y = (Yt ), and de ne measure Q by dQ dP process
Proof.
X = (Xt )
is a
Q-local
From lemmaB.1,
X
X + [X, Y ]
martingale if and only if
is a
Q
local martingale if and only if
is a
ZX
P-local is a
martingale.
P-local
martingale.
But we have
d(ZX) = X− dZ + Z− dX + d[X, Z], The rst term in the right-hand-side is a local martingale, since local martingale if and only if Note that we have:
V
ˆ
Hence
V
Z
ˆ Z− dX + [X, Z] =
ˆ Z− dX +
ZX is a De ne U , X+[X, Y ].
is a martingale. So
Z− dX+[X, Z] is a local martingale.
ˆ
=
U =
V ,
´
ˆ Z− d[X, Y ] =
Z− dU
1 dV Z−
is a local martingale if and only if
U , X + [X, Y ]
is a local martingale.
Theorem B.3. (Linear SDE) Consider the following SDE: ( dr t = (θ(t) + A(t)r t ) dt + ∆(t)dW t r 0 is given
(B.1)
Where W = (Wt ) is a k-dimensional Brownian motion, and A (.), θ (.) and ∆ (.) are deterministic matrix functions of appropriate dimensions. Then subject to some stability conditions on the coe cients, the unique strong solution of equation(B.1)is given by: ˆ t ˆ t −1 −1 r t = Υ(t) r 0 + Υ (u)θ(u)du + Υ (u)∆(u)dW u . 0
(B.2)
0
Where Υk×k (t) is the solution of the following matrix dierential equations: ( ˙ Υ(t) = A(t)Υ(t), Υ(0) = Ik×k . 51
(B.3)
Furthermore the mean andi covariance functions of r, namely m(t) = E [rt ] and V (t) = h E (rt − m(t)) (rt − m(t))T are the solution of the linear equations: ˙ m(t) = A(t)m(t) + θ(t), ˙ V (t) = A(t)V (t) + V (t)AT (t) + ∆(t)∆T (t),
(B.4)
which have the following solutions:
n o ´t m(t) = Υ(t) m(0) + 0 Υ−1 (s)θ(s)ds , n o (B.5) ´t T V (t) = Υ(t) V (0) + 0 Υ−1 (u)∆(u) (Υ−1 (u)∆(u)) du ΥT (t). h i Finally the auto-correlation function of r, i.e. ρ(s, t) = E (rs − m(s)) (rt − m(t))T is given
by:
ˆ
s∧t −1
ρ(s, t) = Υ(s) V (0) +
−1
Υ (u)∆(u) Υ (u)∆(u)
T
du ΥT (t).
(B.6)
0
Theorem B.4. Consider the one-factor Hull-White extension of Vasicek model: drt = (ϑ(t) − art ))dt + δdWt
Also assume that initial bond prices P M (0, T ), and initial instantaneous forward-rate curve, f M (0, T ) are given.Then we have: (i)The short-rate rt is explicitly given by: ˆ
−a(t−s)
ˆ
t
t
ϑ(u)du + δ e−a(t−u) dWu s ˆ t s = (rs − ω(s)) e−a(t−s) + ω(t) + δ e−a(t−u) dWu
rt = rs e
+
−a(t−u)
e
s
Where: ω(t) , f M (0, t) +
2 δ2 1 − e−at 2 2a
(ii)rt is Gaussian with mean and variance given by:
(iii)
´t s
Es [rt ] = (rs − ω(s)) e−a(t−s) + ω(t) δ2 Vars [rt ] = 1 − e−2a(t−s) 2a ru du|Fs is also Gaussian with (conditional) mean and variance given by: h´ i M t P (0,s) Es s ru du = B(s, t) (rs − ω(s)) + log P M (0,t) + 12 (V (0, t) − V (0, s)) h´ i Vars st ru du = V (s, t)
where: 1 1 − e−a(T −t) a δ2 2 −a(t−s) 1 −2a(t−s) 3 V (s, t) , 2 t − s + e − e − a a 2a 2a
B(s, t) ,
52
(B.7)
(iv) For s < t we have: i h ´t P (s, t) , Es e− s ru du = A(s, t)e−B(s,t)rs
where:
P M (0, t) A(s, t) = M e P (0, s)
n o 2 f M (0,s)B(s,t)− δ4a (1−e−2as )B 2 (s,t)
53
Bibliography [1] Ait-Sahalia, Y., and Brandt, M. Variable selection for portfolio choice.
Finance 56, 4 (2001), 12971351.
[2] Arrow, K. J.
Aspects of the theory of risk-bearing.
Journal of
Yrjo Jahnsson lectures. Saatio,
Helsinki, 1965. [3] Bates, D. Post-87 crash fears in the s&p 500 futures option market.
metrics 94, 1-2 (2000), 181238.
[4] Becherer, D.
Journal of Econo-
The numeraire portfolio for unbounded semimartingales.
Stochastics 5, 3 (2001), 327341.
Finance and
[5] Benzoni, L., Collin-Dufresne, P., and Goldstein, R. S. Portfolio choice over the life-cycle when the stock and labor markets are cointegrated.
the Journal of Finance lxii,
5 (2007), 21232167. [6] Borell, C. asset prices.
Monotonicity properties of optimal investment strategies for log-brownian
Mathematical Finance 17, 1 (2007), 143153.
[7] Bouchard, B., and Touzi, N. solutions.
Weak dynamic programming principle for viscosity
submitted for publication
(2009).
[8] Brandt, M. Estimating portfolio and consumption choice: A conditional euler equations approach.
Journal of Finance 54, 5 (1999), 16091645.
[9] Brennan, M. The role of learning in dynamic portfolio decisions.
1, 3 (1997), 295306.
European nance review
[10] Brennan, M., Schwartz, E., and Lagnado, R. Strategic asset allocation.
Economic Dynamics and Control 21, 8-9 (1997), 13771403.
journal of
[11] Brennan, M. J., and Xia, Y. Stochastic interest rates and the bond-stock mix.
pean Finance Review 4, 2 (2000), 197210.
[12] Brennan, M. J., and Xia, Y.
Finance 57, 3 (2002), 12011238.
Dynamic asset allocation under in ation.
Euro-
Journal of
[13] Browning, M. A simple nonadditive preference structure for models of household behavior over time.
The Journal of Political Economy 99, 3 (1991), 607637.
[14] Campbell, J., and Cochrane, J. By force of habit: A consumption-based explanation of aggregate stock market behavior.
Journal of Political Economy 107, 2 (1999), 205251. 54
[15] Campbell, J., and Viceira, L. Consumption and portfolio decisions when expected returns are time varying.
Quarterly Journal of Economics 114, 2 (1999), 433495.
[16] Campbell, J. Y., and Viceira, L. M. Who should buy long-term bonds?
Economic Review 91, 1 (2001), 99127.
[17] Campbell, J. Y., and Viceira, L. M.
Long-Term Investors.
Strategic Asset Allocation: Portfolio Choice for
Oxford University Press, 2002.
[18] Canner, N., Mankiw, N., and Weil, D. An asset allocation puzzle.
nomic Review 87, 1 (1997), 181191.
[19] Carmona, R. E.
American
Indierence pricing.
Princeton University Press, 2009.
[20] Castaneda-Leyva, N., and Hernandez-Hernandez, D.
Optimal consumption-
investment problems in incomplete markets with stochastic coe cients.
on Control and Optimization 44, 4 (2005), 13221344. [21] Chacko, G., and Viceira, L. M.
American Eco-
SIAM Journal
Dynamic consumption and portfolio choice with
stochastic volatility in incomplete markets.
Review of Financial Studies 18,
4 (2005),
13691402. [22] Choulli, T., Stricker, C., and Li, J. Minimal hellinger martingale measures of order q.
Finance and Stochastics 11
(2007), 399427.
[23] Constantinides, G. M. A theory of the nominal term structure of interest rates.
Review of Financial Studies 5, 4 (1992), 531552.
The
[24] Cox, J. C., and Huang, C. F. Optimal consumption and portfolio policies when asset prices follow a diusion process.
Journal of Economic Theory 49
[25] Cox, J. C., and Huang, C.-f.
A variational problem arising in nancial economics.
Journal of Mathematical Economics 20
(1991), 465487.
[26] Cox, J. C., Ingersoll, J. E., and Ross, S. A. interest rates.
(1989), 3383.
Econometrica 53, 2 (1985), 385407.
A theory of the term structure of
[27] Crandall, M. G., Ishii, H., and Lions, P.-L. User's guide to viscosity solutions of second order partial dierential equations.
27, 1 (1992), 167.
Bulletin of the American Mathematical Society
[28] Deelstra, G., Grasselli, M., and Koehl, P.-F. Optimal investment strategies in a cir framework.
journal of Applied Probability 37
(2000), 936946.
[29] Detemple, J., Garcia, R., and Rindisbacher, M. A monte carlo method for optimal portfolios.
The Journal of Finance 58, 1 (2003), 401446.
[30] Detemple, J., and Rindisbacher, M.
Closed-form solutions for optimal portfolio
selection with stochastic interest rate and investment constraints.
15, 4 (2005), 539568.
55
Mathematical Finance
[31] Duffie, D.
nance,
Intertemporal asset pricing theory.
Handbook of The Economics of Fi-
In
G. Constantinides, M. Harris, and R. Stulz, Eds., vol. 2. Elsevier North-Holland,
Amsterdam, 2003, pp. 639742. [32] Duffie, D., and Lions, P.-L. Pde solutions of stochastic dierential utility.
Mathematical Economics 21
Journal of
(1992), 577606.
[33] Ekeland, I., and Taflin, E.
A theory of bond portfolios.
Probability 15, 2 (2005), 12601305.
[34] Fama, E., and Schwert, G. Asset returns and in ation.
5, 2 (1977), 115146.
The Annals of Applied
Journal of Financial Economics
[35] Ferson, W. E., and Harvey, C. R. The risk and predictability of international equity
The Review of Financial Studies 6, 3 (1993), 527566.
returns.
[36] Fleming, W. H., and Hernandez-Hernandez, D. An optimal consumption model with stochastic volatility.
Finance and Stochastics 7
[37] Fleming, W. H., and Soner, H. M.
tions, second edition ed.
(2003), 245262.
Controlled Markov Processes and Viscosity Solu-
Springer, 2006.
[38] French, K., Schwert, G., and Stambaugh, R. Expected stock returns and volatility.
Journal of Financial Economics 19, 1 (1987), 329.
[39] Glosten, L. R., Jagannathan, R., and Runkle, D. E. On the relation between the expected value and the volatility of the nominal excess return on stocks.
The Journal of
[40] Goll, T., and Kallsen, J. Optimal portfolios for logarithmic utility.
Stochastic Pro-
Finance 48, 5 (1993), 17791802. cesses and Their Applications 89
(2000), 3148.
[41] Goll, T., and Kallsen, J. A complete explicit solution to the log-optimal portfolio problem.
The Annals of Applied Probability 13, 2 (2003), 774799.
[42] Harvey, C. R.
Time-varying conditional covariances in tests of asset pricing models.
Journal of Financial Economics 24, 2 (1989), 289317.
[43] He, H., and Pearson, N. D.
Consumption and portfolio policies with incomplete
markets and short-sale constraints: The in nite dimensional case.
Theory 54
Journal of Economic
(1991), 259304.
[44] Heaton, J., and Lucas, D.
Macroeconomic Dynamics 1
Market frictions, savings behavior, and portfolio choice.
(1997), 76101.
[45] Karatzas, I., and Kardaras, C. The numeraire portfolio in semimartingale nancial models.
Finance and Stochastics 11
(2007), 447493.
[46] Karatzas, I., Lehoczky, J. P., and Shreve, S. E. Optimal portfolio and consumption decisions for a 'small investor' on a nite horizon.
Optimization 25, 6 (1987), 15571586.
56
SIAM Journal on Control and
[47] Karatzas, I., Lehoczky, J. P., Shreve, S. E., and Xu, G.-L. Martingale and duality methods for utility maximization in an incomplete market.
Optimization 29, 3 (1991), 702730.
[48] Karatzas, I., and Shreve, S. E.
SIAM Journal on Control and
Methods of Mathematical Finance.
Springer, 1998.
[49] Karoui, N. E., Peng, S., and Quenez, M. C. Backward stochastic dierential equations in nance.
Mathematical Finance 7, 1 (1997), 171.
[50] Kim, T. S., and Omberg, E. Dynamic nonmyopic portfolio behavior.
cial Studies 9, 1 (1996), 141161.
[51] Kimball, M. S. Precautionary saving in the small and in the large.
Review of Finan-
Econometrica 58, 1
(1990), 5373. [52] Korn, R., and Kraft, H. stochastic interest rates.
A stochastic control approach to portfolio problems with
SIAM Journal on Control and Optimization 40, 4 (2001), 1250
1269. [53] Korn, R., and Kraft, H. on the stability of continuous-time portfolio problems with stochastic opportunity set.
Mathematical Finance 14, 3 (2004), 403414.
[54] Kraft, H. Optimal portfolios and heston's stochastic volatility model: an explicit solution for power utility.
Quantitative Finance 5, 3 (2005), 303313.
[55] Kramkov, D., and Schachermayer, W. the asymptotic elasticity of utility functions and optimal investment in incomplete markets.
The Annals of Applied Probability 9,
3
(1999), 904950. [56] Kramkov, D., and Schachermayer, W. Necessary and su cient conditions in the problem of optimal investment in incomplete markets.
13, 4 (2003), 15041516.
The Annals of Applied Probability
[57] Kramkov, D., and Sirbu, M. On the two-times dierentiability of the value functions in the problem of optimal investment in incomplete markets.
16, 3 (2006), 13521384.
Annals of Applied Probability
[58] Landsberger, M., and Meilijson, I. Demand for risky nancial assets: A portfolio analysis.
Journal of Economic Theory 50, 1 (1990), 204213.
[59] Lions, P.-L. Optimal control of diusion processes and hamilton-jacobi-bellman equations part 2 : viscosity solutions and uniqueness.
Equations 8, 11 (1983), 12291276.
Communications in Partial Dierential
[60] Lions, P.-L. Optimal control of diustion processes and hamilton-jacobi-bellman equations part i:
the dynamic programming principle and application.
Partial Dierential Equations 10, 8 (1983), 11011174.
[61] Liu, J. Portfolio selection in stochastic environments.
20, 1 (2007), 139.
57
Communications in
The Review of Financial Studies
[62] Malamud, S., and Trubowitz, E. The structure of optimal consumption streams in general incomplete markets.
Mathematics and Financial Economics 1
[63] Markowitz, H. Portfolio selection.
(2007), 129161.
The Journal of Finance 7, 1 (1952), 7791.
[64] Merton, R. C. Lifetime portfolio selection under uncertainty: the continuous-time case.
The Review of Economics and Statistics 51, 3 (1969), 247257.
[65] Merton, R. C. Optimum consumption and portfolio rules in a continuous-time model.
Journal of Economic Theory 3, 373-413 (1971), 373.
[66] Mnif, M. Portfolio optimization with stochastic volatilities and constraints: An application in high dimension.
Appl Math Optim 56
(2007), 243264.
[67] Munk, C. Portfolio and consumption choice with stochastic investment opportunities and habit formation in preferences.
Journal of Economic Dynamics and Control 32
(2008),
35603589. [68] Musiela, M., and Zariphopoulou, T. The backward and forward dynamic utilities and their associated pricing systems:
The case study of the binomial model.
preprint
(2003). [69] Neave, E. H. Multiperiod consumption-investment decisions and risk preference.
of Economic Theory 3, 1 (1971), 4053.
Journal
[70] Pagan, A. R., and Schwert, G. W. Alternative models for conditional stock volatility.
Journal of Econometrics 45, 1-2 (1990), 267290.
[71] Pham, H. Smooth solutions to optimal investment models with stochastic volatilities and portfolio constraints. [72] Pliska, S. R.
Applied Mathematics and Optimization 46, 1 (2002), 5578.
A stochastic calculus model of continuous trading: Optimal portfolios.
Mathematics of Operations Research 11, 2 (1986), 371382.
[73] Ringer, N., and Tehranchi, M. Optimal portfolio choice in the bond market.
and Stochastics 10
[74] Ross, S. A. applications.
Finance
(2006), 553573.
Some stronger measures of risk aversion in the small and the large with
Econometrica 49, 3 (1981), 621638.
[75] Sangvinatsos, A., and Wachter, J. Does the failure of the expectations hypothesis matter for long-term investors?
Journal of Finance 60, 1 (2005), 179230.
[76] Schroder, M. D., and Skiadas, C. Optimal lifetime consumption-portfolio strategies under trading constraints and generalized recursive preferences.
their Applications 108
Stochastic Processes and
(2003), 155202.
[77] Schroder, M. D., and Skiadas, C. Lifetime consumption-portfolio choice under trading constraints, recursive preferences, and nontradeable income.
their Applications 115, 1 (2005), 130.
58
Stochastic Processes and
[78] Scruggs, J. T. Resolving the puzzling intertemporal relation between the market risk premium and conditional market variance: A two-factor approach.
Journal of Finance 53,
2 (1998), 575603. [79] Sorensen, C.
Dynamic asset allocation and xed income management.
Financial and Quantitative Analysis 34, 4 (1999), 513531.
[80] Stoikov, S., and Zariphopoulou, T. hedgeable risks and under cara preferences. [81] Stutzer, M.
[82] Touzi, N.
Optimal investments in the presence of un-
IMA Volume Series in press
Portfolio choice with endogenous utility:
Journal of Econometrics 116
Journal of
(2009).
a large deviations approach.
(2003), 365386.
Stochastic control problems, viscosity solutions, and application to nance.
Lecture Notes Scuola Normale Superiore
[83] Wachter, J.
(2002), Pisa.
Risk aversion and allocation to long-term bonds.
Theory 112, 2 (2003), 325333.
Journal of Economic
[84] Wachter, J. A. Portfolio and consumption decisions under mean-reverting returns: An exact solution for complete markets.
37, 1 (2002), 6391.
The Journal of Financial and Quantitative Analysis
Stochastic Controls: Hamiltonian Systems and HJB Equations, vol. 43 of Stochastic Modelling and Applied Probability. Springer, New York, 1999.
[85] Yong, J., and Zhou, X. Y.
[86] Zariphopoulou, T. A solution approach to valuation with unhedgeable risks.
and Stochastics 5
Finance
(2001), 6182.
[87] Zariphopoulou, T. Optimal asset allocation in a stochastic factor model - an overview and open problems. [88] Zitkovic, G.
Preprint
(2009).
Utility theory-historical perspectives.
Finance. in press, 2009.
59
In
Encyclopedia of Quantitative
