ROBUST ONE-PERIOD OPTION HEDGING Portfolio ... - CiteSeerX
Abstract. The paper considers robust optimization to cope with uncertainty about the stock return process in one period option hedging problems. The robust approach relates portfolio choice to uncertainty, making more cautious hedges when uncertainty is high. We represent uncertainty by a set of plausible expected returns of the underlying stocks and show that for this set the robust problem is a second order cone program that can be solved efficiently. We apply the approach to find an optimal portfolio to hedge an index option.
Portfolio selection concerns the allocation of wealth to assets such that return is maximized and risk is minimized. The best known mathematical model for portfolio selection is the Markowitz (1952) model. The Markowitz model measures return by the expected value of the random portfolio return and risk by the variance of the portfolio return. The mathematical model is a quadratic programming model. A good reference on portfolio optimization is Zenios (1993). Critics have shown that portfolio optimization is very sensitive to the parameters of the model, in particular to the expected return. The numerical values for the parameters are output from economic models, possibly combined with subjective beliefs. These models are estimated from noisy data and as such subject to statistical and judgemental error. This leads to small inaccuracies in the parameter values. The problem is that small deviations in the parameter values lead to large changes in the optimal portfolio. In classical portfolio optimization the uncertainty in the parameters is not considered and estimates are passed on to the optimization tool as being oracle prophecies. This is also the case for the Markowitz model: the parameter values for expected return and risk are treated as certain. However the parameter values are uncertain and ignoring this leads to a poor actual performance (see e.g. Jobson and Korkie (1981), Michaud (1998, 1989) and Jorion (1986)). Various approaches have been proposed to work around this sensitivity. Statistical literature reports improved parameter estimators; A¨ıt-Sahalia and Brandt (2001) consider direct estimation of portfolio weights; Jagannathan and Ma (2003) impose supplementary ’wrong’ constraints in the optimization model to preclude extreme portfolios (but also opportunities); Horst, Roon and Werker (2002) adjust the coefficient of risk aversion to incorporate estimation uncertainty. 1991 Mathematics Subject Classification. 90C15, 90C20, 90C90, 49M29. JEL codes: C61, G11. Key words and phrases. Robust Optimization, Portfolio Optimization, Nonnegative Cones, Hedging. Corresponding author, Maastricht University, [email protected], The Netherlands Tilburg University, The Netherlands Comments from Peter Schotman and two anonymous referees are gratefully acknowledged. 1
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FRANK LUTGENS, JOS STURM AND ANTOON KOLEN
A growing attention is directed at modeling the parameter uncertainty more explicitly. Bayesian approaches, studied by Black and Litterman (1990, 1992) , Kandel and Stambaugh (1996), P`astor and Stambaugh (2000), Barberis (2000), P`astor (2000) and Cremers (2002) manage uncertainty by a presumption of prior information on the return model. The prior information is combined with the observed data to form the predictive distribution of returns which considers uncertainty. We adopt the approach of an investor who has too little information to form a unique prior on the return model and who considers all plausible return models. On account of the behavioral assumption of uncertainty aversion, the investor considers the minimal expected return over all conceivable priors when evaluating a decision. This robust approach has been studied before in the context of expected utility theory (Gilboa and Schmeidler (1989)), decision theory (Ben-Tal and Nemirovski (1997, 1998), Rustem, Becker and Marty (2000) El Ghaoui, Oustry and Lebret (1998), Goldfarb and Iyengar (2003) and Lutgens (2004)) and finance (Anderson, Hansen and Sargent (1999), Maenhout (1999) and Uppal and Wang (2003)). In the robust optimization framework of Ben-Tal and Nemirovski (1998) and El Ghaoui et al. (1998) a model is formulated with similar structure to the original model, but now the constraints are not only imposed for a single (most likely) instance of parameter values but over a set of (empirically plausible) parameter values. Consequently, the problem is solved assuming worst case behavior of parameter values within this set of plausible parameters. For specific forms of uncertainty and constraints which are linear in the uncertain parameter, the resulting robust models are second order cone programming models, which can be solved efficiently using standard software, such as SeDuMi (Sturm 1999). We use econometric methods to quantify uncertainty in (and empirical plausibility of) the parameters. For financial problems these parameters concern future asset returns, such as the vector of expected returns. Our objective is to use robust optimization for option hedging. Hedging concerns eliminating the risks of a financial position, in our case an option, by constructing a portfolio which, at least, offsets the position. An option is the right but not the obligation to buy or sell its underlying asset for a predetermined price, called the exercise price. Upon maturity, the option’s value derives, as a non-negative piecewise-linear function, from its underlying stock price: depending on the stock price relative to the exercise price, the option is in-the-money and has strictly positive value or is out-of-the-money and has value zero. As a consequence, uncertainty about the underlying asset’s return model affects the option return insofar as the option is in-the-money, i.e. when the option has strictly positive value. Moreover, as the option return is driven by the stock return, we may not consider uncertainty in stock and option returns separately. For example, a long position in both a stock and a put option on this stock have opposite dependence on the stock return; higher stock returns are profitable for the stock investment but have a negative effect on the put option value and vice versa. If disregarded in the robust optimization model, these issues lead to unnecessary conservatism.
ROBUST ONE-PERIOD OPTION HEDGING
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On the other hand, a robust model which accounts for these effects features a form of uncertainty (constraints are non-linear in the uncertain parameters) that complicates solutions and requires new theory. We will explain the difficulty due to options in robust portfolio optimization: section 1 provides an example of a robust portfolio choice problem with one stock and one option; in section 2 we introduce our definitions and notation for the portfolio choice problem with multiple assets and options. We develop a problem transformation which enables us to handle options in robust portfolio optimization: section 3 describes the robust problem and appendix A is devoted to a duality result which enables a transformation of the robust problem. In section 4 we illustrate the methodology for robust option modeling on hedging problems involving options. 1. Example Our ultimate goal is to develop a solution to robustly hedge options with portfolios which may contain the riskfree asset, stocks and options. Yet the ideas of robustness and the problems associated with option modeling are best illustrated by a simpler portfolio optimization example. Consider an investor who can invest in a stock with expected return µ and a call option on that stock. The investment horizon is equal to the time until expiration of the call option. Suppose the investor wishes to choose the portfolio that maximizes the expected return. The solution is to invest all wealth in the asset with the highest expected return. If the option is fairly priced, this will usually be the call option, since this is in essence a levered investment in the stock. Consequently it will exhibit both more risk and a higher expected return. Exceptions could occur if the expected return µ is below the riskfree rate. In that case a long position in the call option could have a perverse return distribution. When the investor is not certain about the expected return on the stock, his optimal portfolio is less trivial. For some values of µ he might want to be long in the call option, for other values the highest expected return requires a short position in the call. Suppose the investor is a robust decision maker. For each portfolio he considers the worst possible value for µ. His optimal portfolio is the best worst case. To put more structure on the uncertainty, suppose that the investor is willing to believe that µ is with certainty in the interval U = [µL , µU ]. To formalize the robust portfolio problem we introduce some further notation. Let w be the fraction of wealth invested in the stock and w d = 1 − w be the fraction of wealth invested in the option. In this example we do not restrict short-selling, so both w and wd can be negative. Maximizing robust expected return is defined as the problem (1)
max min wµ + (1 − w)µd , w
µ∈U
where µd is the expected return of the call. Since the return on an option is a nonlinear function of the return on the underlying stock, the expected return µd will in general also
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FRANK LUTGENS, JOS STURM AND ANTOON KOLEN
be a nonlinear function of µ. The inner minimization is therefore a nonlinear optimization problem in µ. Hence with options we cannot use the standard results in Ben-Tal and Nemirovski (1998) to solve the problem. The solution of the inner minimization problem will depend on w. To solve the robust problem we assume that the return distribution is discrete with a finite number K of possible outcomes. The return distribution is characterized by (2)
rk = µ + ²k ,
k = 1, . . . , K,
P where each ²k occurs with probability πk . We therefore have 0
X. The return on the call is therefore ( −1 if 1 + rk < X/S0 (3) rkd = ark + b if 1 + rk ≥ X/S0 where S0 c0 S0 − X − c 0 b = c0
a =
For a given µ, let I = {k : 1 + rk ≥ X/S0 } describe the set of return outcomes for which the call ends in-the-money, and let I¯ be its complement. The expected return on the call option follows as X µd = πk rkd (4)
k
=
X k∈I
πk (aµ + a²k + b) +
X
πk (−1)
k∈I¯
For some (small) k the option could be out-of-the-money for all values of µ ∈ U. For large values of k it could be in-the-money for all µ ∈ U. Finally, there might exist intermediate values of k such that the option is in-the-money for a sub-interval of U and out-of-the-money for the complement of U.
As long as the set I does not change, the expected return µd is linear in µ. We will exploit the discreteness of the state space in the following way. Start with the lower bound µ 1 = µL and consider the initial set of in-the-money returns I1 = {k : 1 + µ1 + ²k > X/S0 }. Increase µ until at some point, when µ = µ2 , the largest element from I¯1 will be just in-the-money. Define a new set I2 = {k : 1 + µ2 + ²k > X/S0 }. Continue increasing µ until the next ²k switches from the out-of-the-money set to the in-the-money set or until we reach the upper bound µU . This leads to a partitioning of the interval U into J subintervals Uj . Within each
ROBUST ONE-PERIOD OPTION HEDGING
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of the intervals the relation (4) between µd and µ is linear, while the overall relation between µd and µ is piecewise linear with J − 1 breakpoints. With the discretization of the state space and the resulting partitioning of U we can solve the inner minimization problem in (1). Due to the piecewise linear relation, the minimum will be obtained at one of the breakpoints µj (including the endpoint µJ = µU ). For each w we only need to check a finite number of J points µj to find the minimum. We can then solve for the optimal w by linear programming. A numerical example illustrates the procedure. Let S0 = 1 and X = 1.05. The return distribution is approximated by K = 5 different outcomes. Values for πk and ²k in the numerical example are tabulated below: k πk ²k
1 2 3 4 5 0.0625 0.25 0.375 0.25 0.0625 -0.2 -0.1 0 0.1 0.2
For a fair value of the call option we calculate the discounted expected value of the payoffs. Setting the riskfree rate equal to rf = 3%, the current call price is1 1 X πk max (0, S0 (1 + rf + ²k ) − X) = 0.03034 c0 = 1 + rf k
Assume that the uncertainty set is U = [2%, 6%]. For µ1 = 2%, the call option is in-themoney for I1 = {4, 5}. State k = 3 will be just in-the-money for µ2 = 5%. No further states change set for µ < µU = 6%. This implies the partitioning U = U1 ∪ U2 with U1 = [2%, 5%) and U2 = [5%, 6%]. After the partitioning we can write the robust portfolio problem in the linear programming format (5)
max x w,x
subject to (6a) (6b) (6c)
2w − 7.3(1 − w) < x
5w + 23.6(1 − w) < x
6w + 46.3(1 − w) < x
The inequalities follow directly by substituting µ1 = 2%, µ2 = 5% and µ3 = 6% in (1) and (4). With these numerical values the first restriction has a positive coefficient on w, while the second and third have negative coefficients on w. Figure 1 shows the three lines and the optimal solution. The optimal portfolio is at w = 1.08. The associated expected return is x = 2.75%. The optimal portfolio implies that the investor writes call options and invests in the underlying. In this example a mixed portfolio is optimal because the riskfree rate rf is inside the uncertainty interval U. If we would set rf < µL = 2% the optimal portfolio is to go long in the call option and short the underlying stock. 1In
an actual application the price of the option will be observed in the market.
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FRANK LUTGENS, JOS STURM AND ANTOON KOLEN
Figure 2 shows the risk of the portfolio as a function of w. Risk is defined as the standard deviation of the portfolio returns. The portfolio that maximizes the robust expected return is also a portfolio with very limited risk. In the example σ = 10%, whereas the risk of the optimal portfolio is only around 8%. The qualifier ”around” in assessing the risk is on
Figure 1. Robust expected portfolio return
20%
0 −5% 0.5
1
1.25
Notes: The figure shows the expected portfolio return as a function of the investment w in the stock and an implicit investment of 1−w in the call option. The lines correspond to different beliefs about the expected stock return: µ = 2% (dashed), µ = 5% (thin line) and µ = 6% (dash-dot). The thick curve corresponds to a robust evaluation of the expected portfolio return.
Figure 2. Robust portfolio risk 40%
5% 0.75
1
1.25
Notes: The figure shows the portfolio’s standard deviation as a function of the investment w in the stock and an implicit investment of 1 − w in the call option. The lines correspond to different beliefs about the expected stock return: µ = 2% (dashed), µ = 5% (thin line) and µ = 6% (dash-dot). The thick curve corresponds to a robust evaluation of the expected portfolio variance over all µ ∈ [µL , µU ].
ROBUST ONE-PERIOD OPTION HEDGING
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purpose, since the risk is not uniquely defined. The portfolio variance depends on µ: If the option is out of the money rkd is fixed at −100%. Consequently rkd − µd depends on µ and the portfolio variance is a quadratic function of µ. In the remainder of the paper we show how to solve robust portfolio problem involving options in a much more general setting. We allow for multiple assets and options (puts and calls) with various strikes. The general model also allows for options on portfolios of stocks, like index or basket options. A second generalization concerns the uncertainty set U. Instead of independent intervals on the expected returns of each asset, the uncertainty set will be characterized by an elliptical constraint to allow for correlation among the uncertainty in various assets. To handle such complicated uncertainty sets we develop a generalization of the robust transformation for linear constraints which is described in Ben-Tal and Nemirovski (1998). We show that the general model can still be solved efficiently after reformulating the problem as an second order cone optimization problem (SOC). The solution technique is an extension of the interval partitioning in the univariate example. It handles problems with piece-wise linear relations in the uncertain parameter (i.e. robust versions of variance constraints are not considered). We illustrate the approach on two practical applications in section 4. These applications concern option hedging: an investor who has written a call option on a non-tradable index (e.g. the Dow-Jones Eurex Stoxx 50 index) wishes to hedge the option by investing in a few stocks (constituting the index) and options on these stocks. The objective is to find a hedge which has minimal costs.
2. Portfolio returns without uncertainty We consider the problem of allocating wealth over N different assets, possibly including the riskfree asset, and N d options on these assets. Returns and expected returns on the risky assets are denoted by the N -vector r and µ respectively. The (N × N ) covariance matrix of returns is denoted by Σ. The returns on the options depend on the returns of the risky assets. To make this functional form explicit one could use the N d -vector r d (r) to denote option returns. For notational convenience we omit the argument r and use the N d -vector rd . A portfolio is a real N + N d -vector (w; w d ) with elements wj which denote the amount of wealth allocated to the risky asset j and elements wid which denote the wealth allocated to option i. Our aim is to develop robust versions of linear portfolio constraints. An important type of linear constraint is a restriction on the portfolio performance (7)
T
r T w + r d w d ≥ w0 .
Note that a study on such portfolio restrictions implicitly covers hedging problems. An investor may want to hedge an option, e.g. P optionP i with return rid . In this case he strives d for a portfolio (w, w ) with minimal costs ( w + wd ) and which satisfies restriction (7) with w0 = 0 and wid = −1.
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FRANK LUTGENS, JOS STURM AND ANTOON KOLEN
In the remainder of this section, we introduce the notation to specify the option return vector r d . We consider opportunity sets with one period options: we can buy the option, and if we do, keep it until expiration in the next time period. We adopt the usual notation in the financial literature: X denotes the exercise price, S denotes the price of the underlying asset when the option matures, and S0 > 0 denotes the current price of the underlying asset. The return of the underlying asset is denoted r := (S − S0 )/S0 . Thus, X and S0 are known quantities in non-negative real space R+ , whereas S and 1 + r are quantities in R+ that are revealed at the next time epoch. The payoff of a simple call option (the right to buy) with exercise price X is max{0, S −X}. If the call option costs c0 > 0, then its return is ¾ ½ S−X d − 1. rc = max 0, c0 Since S = (1 + r)S0 , we may rewrite rcd as a piece-wise linear function of r: rcd = max{0, ac r + bc } − 1 with ac :=
S0 S0 − X and bc := . c0 c0
with known coefficients ac > 0 and bc . Similarly, the payoff of a simple put option (the right to sell) with exercise price X is max{0, X − S}. If the put option costs p0 > 0, then its return rpd = max{0, ap r + bp } − 1 with ap := −
S0 X − S0 and bp := . p0 p0
Consider N underlying assets (stocks and bonds) with unknown returns r = (r1 , r2 , . . . , rN )T and N d options with returns ( ) N X (8) rid = max 0, bi + aij rj − 1, j=1
for some given bi and aij , i = 1, . . . , N , j = 1, . . . , N d . It is important to observe that (8) defines rd as an explicit function of rs . Call and put options on a single underlying asset k (simple options) correspond to the special case where aij = 0 for j 6= k and aik > 0 or aik < 0 respectively. A basket option, which gives the right to buy (call) or sell (put) a basket of assets Nb ⊆ N for a given price at maturity corresponds to a situation where aij = 0 for j 6∈ Nb and aik > 0 ∀k ∈ Nb if it concerns a call or aik < 0 ∀k ∈ Nb if it concerns a put.
We say that the ith options is in-the-money if 1+rid > 0 and out-of-the-money if 1+rid = 0. The moneyness of the derivatives is determined by the realization of r through relation (8).
For any given realization r, the derivatives {1, 2, . . . , N d } can be partitioned into the set C ⊆ {1, 2, . . . , N d } of derivatives that are in-the-money, and the set C˜ := {1, 2, . . . , N d } ˜ C ∪ C˜ = of derivatives that are out-of-the-money. Conversely, given a partition (C, C), {1, 2, . . . , N d } and C ∩ C˜ = ∅, we let P (C) be the set of returns that support the partition
ROBUST ONE-PERIOD OPTION HEDGING
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C, i.e. (9)
˜ P (C) := {r : 1 + rid > 0 for i ∈ C, 1 + rjd = 0 for j ∈ C}. d
As a matter of notation, we let A ∈ RN ×N denote the matrix with entries aij on the ith row and the jth column. Let AC denote the |C| × N submatrix of A consisting of the rows i ∈ C, where |C| denotes the cardinality of C. Similarly, we let bC ∈ R|C| denote the subvector of b with entries bi , i ∈ C. Thus, after a suitable row permutation we have · ¸ · ¸ AC bC A= ,b= . AC˜ bC˜
By (8) and (9), (10)
P (C) = {r : bC + AC r > 0, bC˜ + AC˜ r ≤ 0}.
Observe that P (C) is a polyhedral set. Furthermore, we have that (11) and (12)
1 + rCd = bC + AC r for r ∈ cl P (C) 1 + rCd˜ = 0 for r ∈ cl P (C).
Hence rd is a linear function of the uncertain parameter r on P (C), where C is an arbitrary d partition of {1, 2, . . . , N d }. Strictly speaking, the set {1, 2, . . . , N d } can be partitioned in 2N ways, but most of these moneyness configurations have an empty set of supporting returns P (C). ˜ that partition the return space into We consider the moneyness configurations (C, C) non-empty sets P (·) (13) F := {C : C ∪ C˜ = {1, 2, . . . , N d }, C ∩ C˜ = ∅, P (C) 6∈ ∅}. We are now able to specify the return on the option investment (14)
T
rd wd = (AC r + bC − ι)T wCd
if r ∈ P (C)
with wCd the sub-vector of w d corresponding to the options in the set C. Note that (14) is linear for each C ∈ F. For a special case with simple call and put options, the configurations are characterized by sorting derivatives on the same underlying according to the exercise price.h Successive i l Xu , for exercise prices X l and X u of the same underlying asset define a return interval X S0 S0 which the call options with X ≤ Xl and the put options with X ≥ Xu are in-the-money and included in C. We combine such intervals of alternative underlying assets to form a partition C of options such that P (C) is non-empty,
(15)
P (C) =
(
) l u XC,j XC,j r: ≤ 1 + rj ≤ , ∀j = 1, . . . , N : , S0,j S0,j
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FRANK LUTGENS, JOS STURM AND ANTOON KOLEN
l u with XC,j and XC,j denoting the successive exercise prices of options on underlying j, which specify the boundaries on the set P (C) along dimension j. If we let Njd denote the number of derivatives on the underlying j, then there are at most Njd + 1 of these intervals Q d for that particular asset and N j=1 (Nj + 1) in total.
3. Linear portfolio restrictions with uncertainty In practice the return distribution is not known precisely and investors rely on estimates which inevitably induce errors in the coefficients r (and consequently also r d (r)) of the constraint. We consider an investor who is concerned about estimation uncertainty and aims at designing portfolios that are robust to this uncertainty. We contextualize the study by considering a robust investor who adopts a multivariate return model (16)
rk = µ + εk
with probability πk ,
k = 1, .., K.
and is concerned about the mean return vector µ. Not all mean return vectors are plausible (according to the robust investor). The subset U of plausible return vectors of the N underlying assets is called the uncertainty set. In this section, we assume that U is the intersection of {µ ∈ RN : µ ≥ −1} with an N -dimensional ellipsoid, i.e. (17)
U = {µ ∈ RN : µ ≥ −1, kA(µ − µ ˆ)k ≤ θ}
where A is a given M × N matrix (typically M = N ), and θ is a given positive scalar constant. For example, µ ˆ is the sample mean part of the ¡ and A is the ¢ upper triangular 2 −1 T T Choleski decomposition of Ω , with Ω = E (ˆ µ − µ)(ˆ µ − µ) such that A A = Ω−1 . We first consider a stylized situation with a singular return distribution:
(18)
r = µ + ²,
²=0
In this case a restriction (7) on the expected portfolio performance reduces to (19)
T
r T w + r d w d + w0 ≥ 0
for all r(= µ) ∈ U,
with the return on the option holdings defined by (14). The option return is a linear function of the unknown parameter r (= µ) on each set P (C) but is non-linear on R N , in particular it is piecewise linear with non-differentiable (adjoint) points given by {cl(P (C 1 )) ∩ cl(P (C2 )), C1 , C2 ∈ F}. The uncertainty set U is not finite and in fact not countable. Hence (19) represents an infinite number of portfolio constraints on the design variables and is inadequate for computational reasons. Ideally, there exists a transformation of the infinite number of constraints 2Indeed,
if the estimator µ ˆ is a sample mean, than µ ˆ is approximately normally distributed with mean µ and a covariance matrix Ω. Letting A be such that AT A = Ω−1 yields approximately a 95% confidence ellipsoid when θ = 2. The eigenvalues of AT A will then be of the same order as the sample size T underlying the computation of µ ˆ.
ROBUST ONE-PERIOD OPTION HEDGING
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(19) to a finite set of constraints which implies an equivalent feasible set for the design variables and which can be incorporated in an efficiently solvable optimization problem. We propose a method to transform (19) into finitely many second order cone restrictions. First, we partition the robust portfolio constraint (19) into finitely many robust linear constraints. As the constraint is linear in µ on the intervals P (C), C ∈ F (see (14)), it is also linear on U(C) := U ∩ P (C)
for all C ∈ Fµ = F ∩ {C : U ∩ P (C) 6= ∅}. The moneyness configurations partition the uncertainty set U into at most |Fµ | ellipsoidal cuts, i.e. ∪C∈Fµ P (C) ⊇ U. Hence (19) is equivalent to (20)
w0 + µT w + (AC µ + bC − ι)T wCd ≥ 0
for all µ ∈ U(C), C ∈ Fµ .
and a solution to (20) satisfies (19) and vice versa.
Secondly, we design a transformation for a linear constraint in the uncertain parameter µ and an uncertainty set which is the intersection of a linear and ellipsoidal uncertainty set. On each ellipsoidal cut U(C), (20) is a linear (affine) function on µ, which we may rewrite as N X d fC,j (w0 , w, wd )T µ ≥ 0 for all µ ∈ U(C). (21) fC,0 (w0 , w, w ) + j=1
Since the coefficients of this function are different for each ellipsoidal cut U(C), we have added a subscript C. In particular, for µ ∈ U(C) (22)
and (23)
fC,0 (w0 , w, wd ) = w0 + (bC − ι)T wCd
fC,j (w0 , w, wd ) = wj +
X
wid aij for j = 1, 2, . . . , n.
i∈C
Moreover fC (w0 , w, wd ) is a N -vector with elements fC,j (w0 , w, wd ), j = 1, .., N , hence T
fCj (w0 , w, wd ) = w + wCd AC Define
and
AC bC − ι A˜ := −AC˜ , ˜b := −bC˜ + ι , ι IN P =
·
0T A
¸
,q=
·
θ −Aˆ µ.
¸
,
By (10) and (17), n o N ˜ ˜ U(C) = µ ∈ R : Aµ + b ≥ 0, P µ + q ∈ SOC .
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FRANK LUTGENS, JOS STURM AND ANTOON KOLEN
Let K(C) := {(f0 , f ) : f0 + f T µ ≥ 0 for all µ ∈ U(C)},
and note that (21) is equivalent to ¸ · fC,0 (w0 , w, wd ) ∈ K(C). (24) fC (w0 , w, wd )
Lemma 1. The set of feasible solutions (w0 , w, wd ) to (24) has an equivalent formulation, fC,0 (w0 , w, wd ) ≥ q T u + ˜bT v fC (w0 , w, wd ) = P T u + A˜T v (25) u ∈ SOC v ≥ 0. The lemma follows from theorems 1 and 2 presented in the appendix A. We obtain a transformation for the robust portfolio constraint (19) if we use (25) for all C ∈ FU . The transformation leads to an equivalent description of the feasible set which is now described by a finite number of second order cone constraints. For simple call and put options, we have at most Njd + 1 exercise intervals [X l , X u ] for asset j, with Njd denoting the Q d options on underlying j. Hence |FU | ≤ N j=1 (Nj + 1). For a fixed number of options, the transformation of (19) by lemma 1 is efficient. As the number of moneyness configurations may grow exponentially the method is not efficient when the number of options is variable. However for practical (hedging) applications, the number of options is limited as we want to hedge an option with only a limited number of financial instruments to keep the monitoring costs low. We will test the approach on some practical instances (see section 4). For some special cases we may simplify solution 25. Lemma 2. If A is non-singular, (21) is equivalent to ˜µ + ˜bT )v ≥ θk(AT )−1 (w + ATC wCd − A˜T v)k w0 + µ ˆT w + (AC µ ˆ + bC )T wCd − (Aˆ (26) v≥0 Proof. Use (22) and (23) to substitute for fC,0 (w0 , w, wd ) and fC (w0 , w, wd ) in (25), w0 + bTC wCd ≥ q T u + ˜bT v w + ATC wCd = P T u + A˜T v (27) u ∈ SOC v ≥ 0 As A is non-singular, we can express the equality in (27), (28)
−1
u2:N +1 = (AT ) (w + ATC wCd − A˜T v).
We substitute for u2:N in the first inequality which yields −1 w0 + bTC wCd ≥ θu1 − (AT ) (w + ATC wCd − A˜T v) + ˜bT v
ROBUST ONE-PERIOD OPTION HEDGING
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As u ∈ SOC, u1 ≥ ku2:N +1 k. We substitute (29)
u1 = ku2:N +1 k
to obtain the largest set of feasible solutions to the first inequality, ˜µ + ˜bT )v w0 + µ ˆT w + (AC µ ˆ + bC )T wCd − (Aˆ (30) −1 −θkAT (w + AT wd − A˜T v)k ≥ 0 C
C
For any feasible solution (w0 , w, wd , v) to (30), with v ≥ 0, we use (28) for u2:N +1 and set u1 = ku2:N +1 k. By construction u ∈ SOC and (w0 , w, wd , v, u) is a feasible solution to (27). Alternatively, if (w0 , w, wd , v, u) is a feasible solution to (27), then (w0 , w, wd , u˜) with u˜ = (˜ u1 , u2:N +1 ), u˜1 ≤ u1 such that u˜1 = ku2:N +1 k, is also feasible to (27) and by (28)-(29) a feasible solution to (30). ¤ If A follows from the Choleski decomposition of Ω−1 , then (26) reduces to (31)
˜µ + ˜bT )v− w0 + µ ˆT wq+ (AC µ ˆ + bC )T wCd − (Aˆ θ (w + ATC wCd − A˜T v)T Ω(w + ATC wCd − A˜T v) ≥ 0. v ≥ 0
T The √ relation (31) is the generalization which extends the robust portfolio return µ w − θ wT Ωw (see Lutgens (2004), chapter 2) to handle options. The specific result is obtained from (31) by setting AC , bC , A˜ and ˜b to zero.
So far we considered a return model with one possible realization r = µ. Alternatively if the constraint is defined for a particular return rk = µ + εk . Constraint (20) changes to (32)
w0 + (µ + εk )T w + (AC (µ + εk ) + bC − ι)T wCd ≥ 0
for all µ ∈ U ∩ Pεk (C),
with (33)
Pεk (C) = {µ : µ + εk ∈ P (C)}.
Moreover, we may consider the expected value of the portfolio under a discrete approximating return distribution (16). The robust version of a linear constraint on the expected value of the portfolio is (34)
T
w0 + µ w +
K X
πk rd (µ + εk ) wd ≥ 0 for all µ ∈ U.
K X
πk (ACˇ (µ + εk ) + bCˇ − ι)T wCd ≥ 0 for all µ ∈ U ∩K k=1 Pεk (Ck ).
k=1
T
In this case we need a finer partitioning of U. Let Ck denote a partition of the options associated with εk . The new, refined, partitioning considers the set C of all combinations ˇ of C1 , C2 , .., CK such that ∩K k=1 Pεk (Ck ) 6= ∅. For each element C = {C1 , C2 , .., CK } in C, K the portfolio return is linear on the polyhedral set ∩k=1 Pεk (Ck ) and (34) is equivalent to imposing, for all elements in C, (35)
T
w0 + µ w +
k=1
14
FRANK LUTGENS, JOS STURM AND ANTOON KOLEN
The constraints (32) and (35) remain linear and each uncertainty set remains an intersection of an ellipsoid and a polyhedral set. Hence we may apply theorems 1 and 2 to obtain a finite number of second order cones which may replace the infinite number of constraints. For a portfolio constraint with Njd options on asset j and K scenarios, the maximum numQ d ber of second order cone and linear constraints after the transformation is O( N j=1 K(Nj + 1)). Only when Njd and K are fixed, this is an ’efficient’ transformation. 4. Applications: Hedging We illustrate the presented methodology on two option hedging problems. Consider an investor who has written an option on the Dow-Jones euro Stoxx 50 index. The Stoxx 50 index is an artificial index and the investor cannot trade in this index. However the investor may trade in the 50 stocks which constitute the index and options on these stocks. Because of the (strong) relation with the index, these stocks and options would be most appropriate to hedge the written option. The investor does not want to monitor 50 stocks and all options on these stocks. Instead he confines to the five most important stocks and options on these stocks. He selects five stocks which have minimal mutual correlations. These are Deutsche Telecom, l’Oreal, RWE, Saint Gobain and Volkswagen. Table 1. Data Price Option data ( July 16th 04) June 15th 04 July 16th 04 number exercise (min) exercise (max) DJ euro Stoxx 50 2716.45 2713.27 58 2000.00 3400.00 Deutsche Telecom 14.65 13.99 30 9.50 19.00 l’Oreal 64.15 60.25 17 55.00 72.50 RWE 39.74 40.62 17 28.00 44.00 Saint Gobain 41.87 39.31 19 36.00 52.50 Volkswagen 37.13 31.99 14 30.00 42.00 Notes: The table presents the prices of the index and the five stocks at the date of portfolio construction (June 15th) and a summary of the option characteristics on these stocks. The data are obtained from Datastream and are denoted in Euro.
Option Hedging. In principle hedging implies finding a portfolio which pays out at least the proceedings of the written option for all possible scenarios. A perfect hedge portfolio may be prohibitively expensive. In our first application we assume that the investor does not want to construct a perfect (100%) hedge, but an almost perfect hedge which yields at least the value of the written option in p, e.g. p = 95%, of the time. The investor constructs a set of future returns which has a cumulative probability mass equal to p. If the investor beliefs that the returns are normally distributed with mean µ
ROBUST ONE-PERIOD OPTION HEDGING
15
and covariance matrix Σ, then such a set of future returns with confidence level p can be described by an ellipsoid U = {r : (r − µ)T Σ−1 (r − µ) ≤ χ2inv (p, N )}
where χ2inv (p, N ) denotes the pth percentile of the chi-square distribution with N degrees of freedom. We use the sample moments over the period January 1998 to June 2004 for µ and Σ. The investor aims for a hedge with minimal costs which covers the written option i ∗ for all returns in the set U; the investor solves X X min wj + wid i={1,..,N d }\i∗
j={1,..,N }
(36)
T
rT w + rd wd ≥ 0
wid∗ = −1
∀r ∈ U
w, wd ≥ 0.
In particular the hedge should cover the written option for the worst-case return in the set U. We may use lemma 1 or lemma 2 to transform the problem into an efficiently solvable second order cone optimization problem. Using lemma 2 we rewrite (36) as X X min wj + wid j={1,..,N }
(37)
i={1,..,N d }\i∗
µT w + (ATC µ + bC )T wCd − (A˜T µ + ˜bT )v− q θ (w + ATC wCd − A˜T v)T Σ(w + ATC wCd − A˜T v) ≥ 0 wid∗ = −1
∀C ∈ FU .
w, wd , v ≥ 0.
where θ = χ2inv (p, N ) and FU is the set of moneyness configurations for the problem. To determine whether a particular moneyness configuration needs to be considered (i.e. is element of FU ), we need to know if the associated interval (see (15)) intersects with the uncertainty set U. In the case of multiple underlying stocks, the intervals are multi-dimensional hypercubes. Each cube is defined by combining N one-dimensional intervals [Lj , Uj ]. Each dimension corresponds to a stock return and the corresponding one-dimensional interval corresponds to a moneyness configuration of options on that stock3. We check for a nonempty intersection of each cube with the set U by solving a second order cone problem: min{(r − µ)T Σ−1 (r − µ)|L ≤ r ≤ U } where L and U are vectors which define the cube. An optimum less than θ indicates a non-empty intersection of the cube and U. In our application we consider five stocks (listed in table 1) and 10 options (closest to at-the-money) on each stock to hedge a call option on the DJ euro Stoxx 50 index with 3As
we need the consider each combination of intervals from different stocks, the number of cubes we need to consider is exponential in the number of options. Only if we fix the number of options, we may call the transformation efficient.
16
FRANK LUTGENS, JOS STURM AND ANTOON KOLEN
exercise price 2750. Hence the problem involves 51 options. We aim for a 95% robust hedge. The problem size is considerable4. Fortunately the problem size of practical applications is typically smaller. An investor only considers a limited number of different options to hedge. Otherwise monitoring costs would become prohibitively large. The costs of hedging the written call option is 3.8 times the costs of that option. The high hedging costs are partly due to a restricted asset universe: one could also consider investment in the riskfree asset and futures on the DJ Stoxx 50 index. The costs are also high due to the high level of required confidence: the hedge will meet the option for all return scenario in U which, given µ, comprises 95%-probability mass. Figure 3 shows the hedging costs for alternative levels of confidence. Observe that the hedging costs gradually increases for confidence levels up to plus-minus 90%, and then soar up. The enormous rise at the end is due to the absence of an approximate duplicate of the option to be hedged (e.g. another call option on the DJ euro Stoxx 50 with a slightly different exercise price). Without such an asset, hedging extreme returns becomes expensive. Figure 3. Robust hedging costs 5
2 1 0
0.1
0.25
0.5
0.75
0.95 1
The figure shows the costs (vertical axis) of hedging an option for various degrees of robustness (horizontal axis). The option to be hedged is a call option on the DJ Stoxx 50 index with exercise price 2750. The asset universe available for the hedge consists of the stocks listed in table 1 and 50 options on these stocks.
As we increase the confidence level of the hedge, we also observe a shift in the portfolio composition. For low levels of confidence the hedge consists exclusively of options, mainly in-the-money call options but also some put options. For large confidence levels, the hedge consists - next to in-the-money call options - for the largest part of stocks (Volkwagen in this case). 4The
size of FU is 12096, the coefficient matrix for SeDuMi has dimensions 145210 × 169406. Problem setup takes 20 minutes and problem solving 45 minutes on a Pentium IV, 512 RAM running Matlab 6.1 (R12.1) and SeDuMi 1.05 (R5).
ROBUST ONE-PERIOD OPTION HEDGING
17
We also compute a hedging strategy based on a scenario approach. We draw 100 random scenarios from the normal distribution and compute the return on the stock and options for each of these scenarios. The associated optimal hedging portfolio is constructed to cover the written option in all scenarios and has minimal cost. In this approach the cost of the hedge is (on average) 0.6. This would imply arbitrage if the investor would interpret the 100 scenarios as comprising all future events. If we increase the number of scenarios to 1000 (10000), the hedging costs rises to 2.21 (3.18) on average but is still below the robust hedging costs corresponding to a 95% hedge. In the same framework, one could also prevent for large losses beyond the p-confidence level. One may want to restrict losses not to exceed w0 . This requires an additional constraint which is similar to (36) but with a righthand-side equal to −w0 and a confidence level of p ≈ 1. Robust Option Hedging. In the second application we consider an investor who is uncertain about the expected return vector. However the investor is willing to believe that the expected return vector is contained in the uncertainty set U = {µ : (µ − µ ˆ)T Ω−1 (µ − µ ˆ) ≤ χ−1 2 (p, N )} ˆ where µ ˆ is the sample mean and Ω = Σ/T (we use T = 60) is an estimate for the uncertainty in µ ˆ. We consider two cases. In the first case the investor uses a discrete return distribution for the asset returns (16) with K scenarios; in the second case he uses a continuous return distribution. The robust hedging problem for a K-scenario return distribution (16) is X X min wj + wid i={1,..,N d }\i∗
j={1,..,N }
(38)
T
rkT w + rkd wd ≥ 0
wid∗
= −1.
∀rk ∈ {µ.εk | µ ∈ U},
∀k = 1, .., K
We use (35) to transform the problem into a second order cone problem. To compute the set of moneyness configurations we follow the same procedure as in the previous application. In this case we need to repeat the procedure for all scenarios k. When the number of stocks is large N > 5 and the number of option per stock is large N > 10, this is time consuming. In practice however we are concerned with hedging by investing in a few (for monitoring reasons) available financial instruments. We consider the problem of hedging an option on the DJ euro Stoxx 50 index by investing in three stocks and ten options on each of these stocks. The return distribution consists of 100 scenarios µ + εk , k = 1, .., 100. The εk were randomly drawn from N (0, Σ). The confidence level of the uncertainty set for µ is set to 95%.
18
FRANK LUTGENS, JOS STURM AND ANTOON KOLEN
In this case the costs of a robust hedge for a call option on the index with exercise price 2750 would be 2.08. The costs of a non-robust hedge based on the same scenarios would be 0.6 (i.e. implying arbitrage). The difference is the cost of robustness to uncertainty in the average returns. As our objective is to hedge, we impose the restriction (36) for each realization ε k . Because we aim to be robust, we also impose the restriction (36) for all reasonable µ. Hence robustness to uncertainty and a hedge on the variability proceed in the same way. By using a discrete return distribution with a finite number of scenarios, we can still see the source of the restrictions (robustness or hedging). However in practice a continuous return distribution is often more appropriate. Under a continuous return distribution, hedging and robustness require to impose the return distribution for all reasonable means and relevant scenarios. If the scenarios are normally distributed and the uncertainty set for the means is ellipsoidal, the relevant set of returns for which the restriction (36) needs to be imposed is again ellipsoidal. The problem reduces to the problem described in the first application, where the ellipsoid is now described by (Σ + Ω) instead of Σ. Hence the effects of uncertainty in µ and variability in the scenarios are aggregated. The resulting problem will be easier to solve. If we take Ω = Σ/T i.e. the ellipsoid is described by (1 + 1/T )Σ. The associated robust hedge costs 4.06 and is slightly more expensive than the hedge which is not robust to uncertainty in the expected returns (previous section). The plot of the hedging costs as a function of the required robustness would be similar to figure 3, yet slightly drawn to the left due to the term 1/T .
5. Conclusion The combination of options and robust (one-period) portfolio optimization leads to constraints which are piecewise linear in the uncertain parameters. Using conic duality results one may transform the robust problem to a finite dimensional second order cone programming problem. Second order cone programs can be solved efficiently, yet the transformation to the second order cone program is only efficient if the number of options is fixed. For many practical applications however, this limitation is not restrictive. We apply the techniques to hedging problems which involve options. In the first application we use the technique as an alternative to a scenario based approach to find an optimal hedge: Instead of using random scenarios as a basis for selecting a hedge, we use a robust approach to the sampling variance of the returns. This leads to a more stable and more relevant hedge. In the second application we also consider a robust approach to uncertainty in the expected returns. This approach should produce a more reliable hedge. It would be interesting future research to test this approach empirically against the standard non-robust hedging approach. Further extensions. For the numerical example of section 1, we argued that a transformation of a robust version of the variance constraint is not trivial. A study on efficient transformations would be an interesting venue for further research. Provided that we can
ROBUST ONE-PERIOD OPTION HEDGING
19
robustly handle variance restrictions we could consider robust mean-variance portfolio choice with options. Another extension to this study would be to consider multi-period portfolio choice with the possibility to buy and sell options at any time. This requires the pricing of options at intermediate time periods. Models for option pricing such as the Black and Scholes (1973) option pricing formula lack precision to be adopted blindly and moreover rely on uncertain parameters such as the future stock price and volatility. Nevertheless option pricing models agree about the dependencies of option prices on different parameters. An approach for robust multi-period portfolio choice that acknowledges the relations between option prices and parameters but also the uncertainty in the parameters could consider an approximation of the option price dynamics and attribute imprecision to uncertainty in the parameters. For example a first order approximation to a call option’s price is ˆ σ ˆ + V(σ − σ (39) c˜(S, σ) = c(S, ˆ ) + δ(S − S) ˆ)
ˆ σ with σ the underlying asset’s standard deviation, c(S, ˆ ) the Black and Scholes (1973) option price and the ’greeks’ δ, Γ and V measure the option’s price sensitivity to stock price, its curvature and stock volatility respectively. This approximation copies (locally) the dynamics of the Black and Scholes (1973) model. A robust approach results if we consider the worst option price (39) over all plausible expected stock returns and standard deviations. A. Duality to achieve standard-form expressions This appendix derives the duality result that we use in section 2. Given a nonempty set D ⊆ Rn , its homogenized cone in Rn+1 is defined as H(D) := cl {(s, y)|s > 0, y/s ∈ D}.
A set K ⊆ Rn is a cone if and only if K 6= ∅ and x ∈ K, t ≥ 0 =⇒ tx ∈ K.
If in addition,
x, y ∈ K =⇒ x + y ∈ K
then K is a convex cone. It is easily verified that H(D) is a cone; if D is convex then H(D) is a convex cone. The dual of a cone K ⊆ Rn is defined as K∗ := {s ∈ Rn |xT s ≥ 0 for all x ∈ K}.
A dual cone is always closed and convex. If K is convex, then the bi-polar relation holds:
(40)
(K∗ )∗ = cl K.
A second order cone (or Lorentz cone) is defined as ¾ ½ q n 2 2 2 SOC = x ∈ R : x1 ≥ x2 + x3 + · · · xn ,
20
FRANK LUTGENS, JOS STURM AND ANTOON KOLEN
where n is the dimension of the second order cone. The interior of the second order cone is denoted int(SOC), i.e. ½ ¾ q n 2 2 int(SOC) = x ∈ R : x1 > x2 + x3 + · · · x2n . In the proof of Theorem 1 below, we need the following technical lemmas. Lemma 3. Let D 6= ∅. It holds that
H(D)∗ = {(f0 , f ) | f0 + f T r ≥ 0 for all r ∈ D}.
The above lemma is a special case of Corollary 1 in (Sturm and Zhang 2003). Lemma 4. Let K ⊆ Rn be a cone and B an m × n matrix. Then {x | Bx ∈ K∗ } = {B T y | y ∈ K}∗ .
For a proof, see relation (17) in (Sturm and Zhang 2003). An important special case is that for two cones K1 and K2 one has
K1∗ ∩ K2∗ = (K1 + K2 )∗ , ¤T £ as obtained by setting B := I, I and K = K1 × K2 . (41)
Lemma 5. Let
˜ + ˜b ≥ 0}. D = {r | P r + q ∈ SOC, Ar
If D 6= ∅ then
˜ + s˜b ≥ 0}. H(D) = {(s, y)|P y + sq ∈ SOC, s ≥ 0, Ay
Proof. From the definition of H(D), it is clear that if (s, y) ∈ H(D) then (42)
˜ + s˜b ≥ 0 P y + sq ∈ SOC, s ≥ 0, Ay
Conversely, suppose that (s, y) satisfies (42). If s > 0 then y/s ∈ D and hence (s, y) ∈ H(D). Suppose now that s = 0. Since D 6= ∅, there exists r ∈ D. Let σ > 0 be arbitrary. We have from the definition of D and (42) that 1 ˜ + 1 y) + ˜b ≥ 0. y) + q ∈ SOC, A(r σ σ Hence (σr + y)/σ ∈ D and (σ, σr + y) ∈ H(D) for all σ > 0. Letting σ ↓ 0 it follows that (0, y) ∈ H(D). ¤ P (r +
Theorem 1. Let ˜ + ˜b ≥ 0} D = {r | P r + q ∈ SOC, Ar
and consider the cone of linear functions that are nonnegative on D, i.e. K = {(f0 , f ) | f0 + f T r ≥ 0 for all r ∈ D}.
ROBUST ONE-PERIOD OPTION HEDGING
If D 6= ∅ then K = cl
½·
21
¾ ¸¯ q T u + ˜bT v + v0 ¯¯ ¯ u ∈ SOC, v ≥ 0, v0 ≥ 0 . P T u + A˜T v
Proof. We have from Lemma 3 that K = H(D)∗ .
Applying Lemmas 5 and 4 respectively, we have ¯· ¯ 1 H(D) = {(s, y)|P y + sq ∈ SOC} ∩ (s, y) ¯¯ ˜ b ¾∗ ½ · ½ · T ¸¯ q u ¯¯ v0 + ˜bT v u ∈ SOC ∩ = P Tu ¯ A˜T v ½
¸· ¸ ¾ 0T s ≥0 y A˜ ¾∗ ¸¯ ¯ ¯ v0 ≥ 0, v ≥ 0 . ¯
n are self-dual cones.) Further using (41) and (40), we (It is well known that SOC and R+ have ½· T ¾ ¸¯ q u + ˜bT v + v0 ¯¯ ∗ H(D) = cl ¯ u ∈ SOC, v ≥ 0, v0 ≥ 0 . P T u + A˜T v
¤
The following theorem states that the closure operator in the above characterization of K is redundant if a Slater condition holds5. Theorem 2. Let ˜ + ˜b > 0} Do := {r | P r + q ∈ int(SOC), Ar
and let K be defined as in Theorem 1. If D o 6= ∅ then ¾ ½· T ¸¯ q u + ˜bT v + v0 ¯¯ K= ¯ u ∈ SOC, v ≥ 0, v0 ≥ 0 . P T u + A˜T v
Proof. Let
Γ=
½·
¸¯ ¾ q T u + ˜bT v + v0 ¯¯ ¯ u ∈ SOC, v ≥ 0, v0 ≥ 0 . P T u + A˜T v
We know from Theorem 1 that K = cl Γ. It remains to show that Γ is closed, i.e. cl Γ = Γ. (k) Let (t, x) ∈ K = cl Γ, and let (u(k) , v (k) , v0 ), k = 1, 2, . . . be a sequence such that (k)
and
u(k) ∈ SOC, v (k) ≥ 0, v0 ≥ 0 for all k = 1, 2, . . .
· 5A
t x
¸
= lim
k→∞
·
(k) q T u(k) + ˜bT v (k) + v0 P T u(k) + A˜T v (k)
¸
.
referee pointed out that this result was also derived by Ben-Tal and Nemirovski (2001), albeit in a different way. Alternatively the result ay be derived by using duality of second order cone programming problems.
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FRANK LUTGENS, JOS STURM AND ANTOON KOLEN
By definition of Γ, such a sequence must exist, because (t, x) ∈ cl Γ. Let r ∈ D o . We have (k) t + rT x = lim q T u(k) + ˜bT v (k) + v + rT (P T u(k) + A˜T v (k) ) k→∞
= ≥
(43)
0
˜ + ˜b)T v (k) + v0(k) lim (P r + q)T u(k) + (Ar
k→∞
˜ + ˜b)T v (k) . lim (P r + q)T u(k) + (Ar
k→∞
˜ + ˜b > 0, we have Since P r + q ∈ int(SOC) and Ar ½ (P r + q)T u > 0 for all u ∈ SOC\{0} (44) n ˜ + ˜b)T v > 0 for all v ∈ R+ \{0}. (Ar
We claim that the sequence {u(k) } is bounded. Indeed, suppose to the contrary that lim supk→∞ ku(k) k = ∞. From (44), it follows that (P r + q)T u(k) > 0, k→∞ ku(k) k and hence, using also (43), we arrive at the impossible inequality lim inf
t + rT x ≥ lim sup(P r + q)T u(k) = ∞. k→∞
Similarly, we can show by contradiction from (43) and (44) that v (k) must be bounded. Hence (k) this sequence {u(k) , v (k) , v0 } has a cluster point (u, v, v0 ), u ∈ SOC, v ≥ 0, v0 ≥ 0, and · ¸ · T ¸ t q u + ˜bT v + v0 = ∈ Γ. x P T u + A˜T v
This concludes the proof.
¤
ROBUST ONE-PERIOD OPTION HEDGING
23
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