CFMLab.com Preprint #4 Pricing under stochastic volatility ...
CFMLab.com Preprint #4
Mar. 18, 2004
© Copyright
Pricing under stochastic volatility: complete solution via decoupled system of Monge–Ampère and Black–Scholes PDEs Srdjan D. Stojanovic Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, U.S.A. http://math.uc.edu/~srdjan/ March 2004
Abstract We have found, at least from the practical point of view, the complete solution of the option pricing problem for underlying securities obeying stochastic volatility price dynamics. Although partial solutions have existed, and in spite of a considerable attention to it, this problem has been open for about 20 years. The pricing problem is reduced to solving an uncoupled system of a Monge–Ampère type PDE and a Black–Scholes type PDE. Résumé Tarifer sous l’hypothèse de volatilité stochastique : solution complète par l’utilisation d’un système découplé d’EDP de Monge-Ampère et de Black-Scholes. Nous trouvons au moins d’un point de vue pratique, la solution complète au problème de tarification des options sur les titres sous-jacents, dont les prix obéissent à une dynamique de volatilité stochastique. Bien que des solutions partielles existent et aient fait l’objet d’une attention certaine, ce problème est resté ouvert depuis 20 ans. Le problème de la tarification se réduit à résoudre un système découplé d’EDP de type Monge-Ampère et Black Scholes.
Version française abrégée Considérons un « marché boursier à volatilité stochastique », consistant d’un titre unique au prix Y +t / à la date t et vérifiant le système suivant d’équations différentielles stochastiques de type Itô
ÅY t ÅÖ t
+ /
+ /
Y +t / +Á+t, Y +t /, Ö+t // ª1 +t, Y +t /, Ö+t /// Å t Y +t/ p+t, Y +t /, Ö+t // Å B1 +t/
q+t, Y +t /, Ö+t // Å t w+t, Y +t /, Ö+t // , U+t, Y +t/,
Ö t Å B1 t + //
+ /
1 U t, Y t , Ö t
r 2 + + / + //
Å B2 t
(0.1)
+ /0
où B1 et B2 représentent des mouvements browniens indépendants, Á est le taux d’appréciation du titre, ª1 est le dividende, p 0 est la volatilité, et v est un facteur scalaire arbitraire (par exemple, la volatilité : p+t, Y, Ö/ Ö ) à direction q, à diffusion w 0 et où 1 U 1 est le coefficient de corrélation entre prix et facteur. Soit également un marché à options constitué d’une option unique sur le titre ci-dessus, de gain fixe X+Y / à la date de maturité T , et au prix V +t, Y +t /, Ö+t // à la date t T , et où V +t, Y, Ö/ est une fonction inconnue a priori. Soit X ! 0, la richesse. L’utilité associée à la richesse est mesurée par une fonction d’utilité de type HARA : \J + X / X 1J s +1 J/ pour J ± +0, /, J 1, et \1 + X / log+ X / .
Considérons une stratégie de couverture du portefeuille par autofinancement 3+t, X , Y, Ö/ 31 , 32 , où 31 est la valeur monétaire de l’investissement dans l’action et 32 est la valeur de l’investissement dans l’option. Etant donné (0.1), et pour une stratégie fixe 3 , il existe une EDS qui caractérise l’évolution de la richesse X +t/ X 3 +t / . La stratégie de portefeuille J -optimale (le problème de Merton), notée 3ÚJ 3ÚJ 1 , 3ÚJ 2 est telle que :
2
Ú
Et, X,Y,Ö \J , X 3J +T /0.
sup Et, X,Y,Ö \J + X 3 +T // 3
(0.2)
DEFINITION 1. Pour tout J ± +0, / , la fonction du prix de l’option VJ +t, Y, Ö/ est définie comme la solution d’une équation Black-Scholes (abstraite) 3ÚJ 2 0. THEOREME 1. Pour tout J ± +0, /, J 1, le prix de l’option correspondant VJ +t, Y, de l’EDP Black-Scholes généralisée VJ +1,0,0/
1 2
cccc Y 2 VJ 0,2,0 +
/
p2 Y w U VJ +0,1,1/ p
fJ 0,0,1 Ár wU q cccccccccccccccccccccccccccccc w2 1 U2 cccccccccccccccc ccccc +
L M M
+
/
+
p
N
/
de condition terminale VJ +T, Y, Ö/
Á rcccc2cc r J cccccccccccccccc 2 2 L + M M
/
p
N
\ ] ] ^
XY +
qJ fJ 2 - cccccccccccccc c J1
/
fJ
\ ] ]
/
^
cccc w2 VJ 0,0,2 Y r ª1 1 2
+
/
+
/
+
//
+
/
0
, et où fJ est une solution appropriée de l’EDP de type Monge-Ampère
w Ár U cccccccccccccccccccccccccccccc +
/
p
1
w2 J cccccccccccccccc cccccc c f 0,0,2 fJ 2 J1 J p2 Y 2 J fJ cccccccccccccccccccccc cc fJ 0,2,0 fJ 2 J1
fJ +0,0,1/ fJ
/
+
+
de condition terminale fJ +T, Y, Ö/
+
est (formellement) solution
(0.3)
VJ +0,0,1/
+
+
/
/
/
+
+
+
/
VJ +0,1,0/ r VJ
Y Á r J r ª1 p wY J U fJ 0,1,0 fJ ccccccccccccccccccccccccccc fJ 0,1,1 cccccccccccccccccccccccccccccccc cccccccccccccccc cccccccc c c c J1 J1 1 1 J cccccccccccc1cc c fJ 1,0,0 fJ cc2cc w2 U2 fJ 0,0,1 2 cc2cc p2 Y 2 fJ 0,1,0 2 p w Y U fJ 0,0,1 J +
Ö
/
/
+
+
/
/
(0.4)
/
+
/
/
fJ +0,1,0/
0
1.
Pour J 1, le prix de l’option correspondant V1 V1 +t, Y, Ö/ est solution de l’EDP Black-Scholes généralisée (0.3) où f1 1, soit pour f1 +0,0,1/ s f1 0 ((0.4) ne sont pas nécessaires dans ce cas). The author wishes to thank very much to Dr. Christelle Viauroux for the French translation.
1 Statement of problem and it's solution Consider a "stochastic volatility stock market," consisting of a single security with price Y +t/ at time t , obeying Itô SDE system Y +t / +Á+t, Y +t /, Ö+t // ª1 +t, Y +t /, Ö+t /// Å t Y +t/ p+t, Y +t /, Ö+t // Å B1 +t/
ÅY t ÅÖ t
+ /
q+t, Y +t /, Ö+t // Å t w+t, Y +t /, Ö+t // , U+t, Y +t/,
(1.1) Ö t Å B1 t 1 U t, Y t , Ö t 2 Å B2 t where B1 and B2 are independent Brownian motions, and where Á is the appreciation rate, ª1 is the dividend rate, p is the volatility, and Ö is an arbitrary scalar factor (for example, volatility: p t, Y, Ö Ö ), with a drift q, diffusion w, and where U is the price/factor correlation. Of course, it is not essential that Ö is scalar. + /
+ //
+ /
r + + / + //
+
+ /0
/
Consider also an associated option market, consisting of a single option on the above underlying, with a fixed payoff X+Y / (for example, Xcall+Y / Max#0, Y k ' , for some strike price k ) at a fixed expiration time T , with a price V +t, Y +t/, Ö+t // at time t T , where function V +t, Y, Ö/ is a priori unknown. Assuming (1.1), it is not difficult to write down an Itô SDE characterizing the evolution of V +t, Y +t /, Ö+t // . The problem is to characterize, and be able to compute, "a fair option price function" V +t, Y,
Ö. /
Let X ! 0 denotes the wealth. We measure utility of wealth via HARA class of utility functions \J + X / X 1J s +1 J/ , for J ± +0, /, J 1, and \1 + X / log+ X / . Parameter J is called risk-aversion.
Consider a self-financing portfolio hedging strategy 3+t, X , Y, Ö/ 31 +t, X , Y, Ö/, 32 +t, X , Y, Ö/ , where 31 is the cash value of the investment into the underlying stock, and 32 is the cash value of the investment into the option. Assuming (1.1), and for a fixed strategy 3 , it is not difficult to write down an Itô SDE characterizing corresponding evolution of the wealth X +t / X 3 +t / (see (2.3) below). The J -optimal portfolio strategy (Merton's problem) is a strategy 3ÚJ 3ÚJ 1 , 3ÚJ 2 , such that sup Et, X,Y,Ö \J + X 3 +T // 3
Ú
Et, X,Y,Ö \J , X 3J +T /0.
(1.2)
3
DEFINITION 1. For every J ± +0, / , the fair option price function VJ +t, Y, Ö/ is defined as a solution of the (abstract) Black–Scholes equation 3ÚJ 2 0. Intuitively, "a fair option price" is such a price for which it is not rational to speculate by investing, long or short, into options. Variants of this definition have been present in the literature already (see, e.g., [11] and references given there). We shall use alternative notations for partial derivatives: for example, 2 V +t, Y, Ö/ s Ö2 V +0,0,2/ +t, Y, Ö/ .
THEOREM 1. For every J ± +0, /, J 1 , the corresponding fair option price VJ (formally) as a solution of a generalized Black–Scholes PDE VJ +1,0,0/
1 2
cccc Y 2 VJ 0,2,0 +
/
p2 Y w U VJ +0,1,1/ p
fJ 0,0,1 Ár wU q cccccccccccccccccccccccccccccc w2 1 U2 cccccccccccccccc ccccc +
L M M
+
/
+
p
N
/
/
fJ
\ ] ] ^
cccc w2 VJ 0,0,2 Y r ª1 1 2
+
/
+
/
VJ +t, Y,
Ö
/
is characterized
VJ +0,1,0/ r VJ (1.3)
VJ +0,0,1/
0
with the terminal condition VJ +T, Y, and where fJ
Ö XY fJ t, Y, Ö /
+
+
Á rcccc2cc r J cccccccccccccccc 2 2 L + M M
/
p
N
(1.4)
/ /
\ ] ] ^
is an appropriate solution of a Monge–Ampère type PDE qJ fJ 2 - cccccccccccccc c J1
w Ár U cccccccccccccccccccccccccccccc +
/
p
1
w2 J cccccccccccccccc cccccc c f 0,0,2 fJ 2 J1 J p2 Y 2 J fJ cccccccccccccccccccccc cc fJ 0,2,0 fJ 2 J1
fJ +0,0,1/ fJ
+
+
/
/
Y Á r J r ª1 p wY J U cccccccccccccccccccccccccccccccc ccc f 0,1,1 cccccccccccccccccccccccccc c fJ 0,1,0 fJ cccccccccccccccccccccccc J1 J1 J 1 1 J cccccccccccc1cc c fJ 1,0,0 fJ cc2cc w2 U2 fJ 0,0,1 2 cc2cc p2 Y 2 fJ 0,1,0 2 p w Y U fJ 0,0,1 J +
+
//
+
/
+
/
+
+
+
/
+
+
/
/
+
+
/
/
+
/
(1.5)
/
/
fJ +0,1,0/
0
with the terminal condition fJ +T, Y, Ö/
1.
(1.6)
For J 1, the corresponding fair option price V1 V1 +t, Y, Ö/ is characterized as a solution of the generalized Black–Scholes PDE (1.3) with f1 1, i.e., with f1 +0,0,1/ s f1 0 ((1.5)–(1.6) is not needed in that case). REMARK 1. Comparing (1.3) with the literature (see, e.g., equation (15) in [6]) we can see that Theorem 1 imples that, for any J ± +0, / , the so-called "market price of volatility risk," which we denote ³J , is given by
³J
w
1 U2 fJ +0,0,1/ s fJ
r
(1.7)
where fJ is an appropriate solution of (1.5)–(1.6). The exact expression for the market price of volatility risk was not known so far. In particular, ³1 0. Analysing (1.3), we can see that fair option price is unique, i.e., it is same for all J ± +0, / , iff ³J s J 0, and therefore iff ³J 0, and assuming furthermore w2 +1 U2 / ! 0, iff fJ +0,0,1/ 0, for all J ± +0, / . It was claimed recently in the literature (see [3, 4]) that for uniqueness of fair option prices it suffices that the Sharpe ratio +Á r/ s p is constant with respect to Ö . This does not seem to be correct, since although the Sharpe ratio may have a dominant effect on the solution of (1.5), and therefore indeed the "uniqueness" of prices under the above condition may hold approximatelly, ++Á r/2 s +2 p2 // fJ 2 is not the only term in (1.5) that may cause fJ +0,0,1/ 0. Indeed, suppose that fJ +0,0,1/ 0; terms p2 Y 2 J fJ +0,2,0/ fJ s +2 +J 1// and p2 Y 2 + fJ +0,1,0/ /2 s 2 remain in (1.5), and unless, which is trivial, the volatility p p+t, Y, Ö/ does not depend on the factor Ö , we cannot expect that fJ +0,0,1/ 0. Numerical computations confirm my findings. REMARK 2. Multiplying equation (1.5) by +J 1/ , sending J 1, and dividing by f1 , one arrives (at least formally) at 1 1 ccccc f1 +0,0,2/ w2 p Y U f1 +0,1,1/ w q f1 +0,0,1/ Y Á f1 +0,1,0/ Y ª1 f1 +0,1,0/ ccccc 2
2
p2 Y 2 f1 +0,2,0/ f1 +1,0,0/
0
(1.8)
which (is a linear equation, and) together with (1.6), is solved, obviously, by f1 1. This argument is not necessary to derive the above result in the case J 1—it follows from the portfolio theory below. REMARK 3. System (1.3)–(1.6) is uncoupled, which is very useful. Furthermore, fJ does not depend on a particular option payoff X .
4
REMARK 4. If w 0, or if U 1, then ³J 0 , and the same fair option price V VJ VJ +t, Y, Ö/ holds for all J ± +0, / . For example, if w 0, the price is characterized as a solution of a (hypoelliptic (see [7], and also [15, 16])) Black–Scholes PDE 1 + / p2 Y +r ª1 / V +0,1,0/ r VJ q VJ +0,0,1/ 0 2 with the terminal condition (1.4). If also q 0, then (1.9) simplifies further to the (usual) Black–Scholes PDE. VJ +1,0,0/
cccc Y 2 VJ 0,2,0
(1.9)
2 Preliminary results in general portfolio theory Consider, quite generally, a set of m factors A+t /
ÅAt
A1 +t /, ..., Am +t / obeying the Itô SDE dynamics
b+t, A+t // Å t c+t, A+t //.Å B+t /
+ /
(2.1)
where B+t / B1 +t /, ..., Bn +t/ is the n-dimensional Brownian motion (so, b+t, A/ , is an m-vector valued function, and c+t, A/ is an m n-matrix valued function). In addition to the factors, consider a set of k tradable assets with prices S +t / S1 +t /, ..., Sk +t / , obeying (non-linear in A ) Itô SDE dynamics S +t / +as +t, A+t // ª+t, A+t /// Å t S +t / Vs +t, A+t //.Å B+t /
ÅS t
+ /
(2.2)
where the vector-valued function as +t, A/ is the k -vector of appreciation rates, ª+t, A/ is the k -vector of dividend rates of the corresponding assets, Vs +t, A/ is the (volatility) k n-matrix valued function. If factor is tradable, then it can be represented also as one of the equations in system (2.2), and vice-versa. We assume Vs .Vs T ! 0. It can be shown that
ÅX
t
+ /
+
X +t /, A+t //.+as +t, A+t// r/ r X +t //Å t 3+t, X +t /, A+t //.Vs +t, A+t //.Å B+t /
3 t, +
(2.3)
and (2.1) and (2.3) form a closed SDE system, to be controlled. In particular, ª is eliminated, while components of ª may still be hidden in (2.1) and (2.3), and more specifically in as and b (see (3.1)). We shall refer to functions as ,
Vs , b, and c as the market coefficients.
The objective of the investor is to, for a given utility function \ (not necessarily HARA), maximize the expected value of the utility of the final total wealth, i.e., to find an optimal hedging strategy 3Ú +t, X , A/ such that:
M t, +
Ú
sup Et, X, A \+ X 3 +T //
X , A/
Et, X, A \, X 3 +T /0.
3
(2.4)
The standard formalism for solving the stochastic control problem (2.4) is to attempt to solve the associated Hamilton–Jacobi–Bellman (HJB) PDE characterizing the value function M M+t, X , A/ :
M
1 M 2 M 3. as r r X ccccccccccc b.´ A M cccc 3.Vs .Vs T .3 cccccccccccc2cc 2 X X 1 M 3.Vs .cT .´ A ccccccccccc cccc Tr c.cT .´ A ´ A M 0 2 X with the terminal condition M T, X , A \ X . After some simplifications, one Max$ ccccccccc 3 t
+
+
/
/
-
1
#
(2.5)
'(
can see that solving the HJB PDE + / + / (2.5) is equivalent to finding an appropriate solution of the Monge–Ampère type PDE:
2 M M 1 M 2 2 M M 2 M cccccccccccc2ccc cccccccccc ccccc cccccccccc c as r . Vs .Vs T 1 . as r r X cccccccccccc2ccc cccccccccc c b.´ A M cccccccccccc2ccc X t 2 X X X X M M M M 1 T T 1 T T 1 cccccccccc c as r . Vs .Vs .Vs .c .´A cccccccccc c ccccc ´ A cccccccccc c .c.Vs . Vs .Vs .Vs .cT .´ A cccccccccc c X X X X 2 1 2 M ccccc cccccccccccc2ccc Tr c.cT .´ A ´ A M 0 2 X -
+
/ +
1
+
/
/ +
/
-
+
1
/
-
1
+
/
-
1
(2.6)
'
#
with the same terminal condition, while the optimal hedging strategy is given by
3Ú t, +
X , A/
ccccMXcc c cccccccc cccccc 2 ccccXccccM2cc
as r/.+Vs .Vs T /
+
1
1 M cccccccc cccccc ´ A ccccccccccc 2 M X cccc Xcccc2cc -
1
.c.Vs T .+Vs .Vs T /1 .
(2.7)
5
As in Merton's case (see also [14, 15]), in the case of HARA utility \J , we seek the solution of (2.6), in the form X , A/ X 1J fJ +t, A/ s +1 J/ , if J ± +0, /, J 1, and in the form M+t, X , A/ f1 +t, A/ log+ X / if J 1. In + the first case f fJ solves
M t,
J cccccccccccccc c 1J
f 1 cccccccccc cccc t 2
f
f 2 ++as r/.+Vs .Vs T /1 .+as r/ 2 r J/
as r/.+Vs .Vs T /1 .Vs .cT
f
-+
1 2
cccc ´ A f .c.Vs T . Vs .Vs T +
1
/
Jb cccccccccccccc c .´ A f 1J
1 Jf cccc cccccccccccccc c Tr c.cT .´ A ´ A f 2 1J
.Vs .cT .´ A f
#
together with the terminal condition f +T, A/
3ÚJ t, +
X , A/
X MM+as r/ L N
(2.8)
1
0
'
1, while the optimal hedging strategy is now given by
´ f ccccccccAccccccJc .c.VsT . J Vs .Vs T \ ] + ]
fJ
/
1
X PJÚ +t, A/.
^
(2.9)
In the case J 1, f1 solves instead a simple linear PDE, which is even irrelevant, since the optimal hedging strategy 3Ú does not depend on f : 1
3Ú1 t, +
X , A/
X +as r/.+Vs .Vs T /1
X PÚ1 +t, A/.
(2.10)
We refer to equation (2.8) as the reduced Monge–Ampère PDE of optimal portfolio hedging. REMARK 5. Of course, (2.6), (2.7), (2.8), (2.9), and (2.10) have much broader significance then being a tool for proving Theorem 1.
3 A sketch of the proof of Theorem 1
We shall consider only the case J ± +0, /, J 1. We apply the general portfolio theory of Section 2 in the case of a "stochastic volatility market" of Section 1, and in particular (2.8) and (2.9). To that end we identify factors A+t/ Y +t /, Ö+t / , tradable assets S +t / Y +t /, V +t, Y +t /, Ö+t // , and the market coefficients as
Vs
Á, ccccV1cc cc12ccc Y 2 V +0,2,0/ p2 Y w U V +0,1,1/ p q V +0,0,1/ cc12ccc w2 V +0,0,2/ Y Á ª1 V +0,1,0/ V +1,0,0/
L M M M M M M N
-
p
0
w U V +0,0,1/ Y p V +0,1,0/
w 1U2 V +0,0,1/
cccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccc cccccccccccccc c V V
\ ] ] ] ] ] ]
Y +Á ª1 /, q , c
,b
V +1,0,0/
L M M M M N
L M M M
N
^
So, we extract the "abstract Black–Scholes" equation 3ÚJ 2 Ú PJ 2
/
+
Yp
1!
(3.1)
0
wU w
\ ] ] r ] ^
.
1 U2
0. Indeed, we compute, using (2.9),
ccccc Y 2 V +0,2,0/ p2 Y w U V +0,1,1/ p ccccc w2 V +0,0,2/ 1 2
1 2
fJ +0,0,1/ r Á wU V Y r ª1 V +0,1,0/ r V q cccccccccccccccccccccccccccccccccc w2 1 U2 cccccccccccccccc cccccc c V +0,0,1/ cccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccccc 2 p fJ J w 1 U2 V +0,0,1/ 2 +
L M M M M
/
+
/
+
/
N
\ ] ] ] ]
\ ] ] ] ]
^
^
+
(3.2)
/
and therefore (1.3) follows. Equation (1.3) is not closed, since fJ f is not known and in fact may depend on V . Equation (1.3) is therefore coupled with the equation (2.8), which now reads as
J w2 cccccccccccccccccc c f 2J2 2 Y J p2 cccccccccccccccccccccc f 0,2,0 2J2
a1 +V / f 2 a2 +V / f +0,0,1/ f Y J pwU ccccccccccccccccccccccccccc J1
f +0,1,1/ f
+
J Y r ª1 r Á Y 0,1,0 cccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccccc f J1 1 J cccccccccccccc c f 1,0,0 f cccc w2 f 0,0,1 2 2 J1
+0,0,2/
/
f
/
+
f
+
/
+
/
+
+
/
1 f +0,1,0/ 2 Y p w U f +0,0,1/ f +0,1,0/ 0 2 where the remaining coefficients a1 +V / and a2 +V / depend on V , and are given by
cccc Y 2 p2
/
f (3.3)
6
a1 +V /
1 2
ccccc 2 r J -
Y 2 V +0,2,0/ p2 2 Y w U V +0,1,1/ p 2 r V 2 q V +0,0,1/ w2 V +0,0,2/
++
2 Y Á V +0,1,0/ 2 Y ª1 V +0,1,0/ 2 V +1,0,0/ Y 2 V +0,2,0/ p3 2 Y w U V +0,1,1/ p2 V +0,0,2/ w2 2 r V 2 q V +0,0,1/ 2 r Y V +0,1,0/ 2 Y ª1 V +0,1,0/ 2 V +1,0,0/ p 2 w r Á U V +0,0,1/ 4 p w2 U2 1 V +0,0,1/ 2 Á r r Á w2 V +0,0,1/ 2 /+
/
+
+
/
// t ,
+
0
/
-+
/ -+
/,
2 Y p w U V +0,1,0/ V +0,0,1/ Y 2 p2 V +0,1,0/ 2 ccccc
1 p +w U V +0,0,1/ Y p V +0,1,0// 2 p 2 r V 2 q V +0,0,1/ w2 V +0,0,2/ 2 Y Á V +0,1,0/ 0
Y 2 V +0,2,0/ p2 2 Y w U V +0,1,1/
+
2 Y ª1 V +0,1,0/ 2 V +1,0,0/
/11 u ,
p2 w2 + U2 1/ V +0,0,1/
2
01
and a2 + V /
1
cccccccccccccccccccccccccccccccc ccccccccccccc 2 r J 1 V 2 q V 0,0,1 J 1 Y 2 V 0,2,0 p2 2 J 1 V 0,0,1 2 Y w U V 0,1,1 p w2 V 0,0,2 2 Y r ª1 V 0,1,0 2 V 1,0,0 /
+
+
+
/
+
+
+
/
/
+
/
/
+
+
/ +
+
/
+
/
/
+
/
(3.5) //
making (1.3) and (3.3) a coupled system. Moreover the coupling (3.4) and (3.5) appears quite challenging (high derivatives of V are involved, and V +0,0,1/ is in the denominator). Now, using (1.3), we express V +1,0,0/
1
cc2cc Y 2 V 0,2,0 +
/
Ár wU f 0,0,1 q cccccccccccccccccccccccccccccc w2 1 U2 cccccccccccccccccc c L M M
+
+
/
+
p
N
cc2cc w2 V 0,0,2 Y r ª1 1
p2 Y w U V +0,1,1/ p /
/
f
\ ] ]
+
/
+
/
V +0,1,0/ r V (3.6)
V +0,0,1/
^
and using (3.6) in (3.4) and (3.5), we get (uncoupled) a1 + V /
a1
1 Á rcccc2cc cccccccccccccccccccccccccccccccc w2 U2 1 f 0,0,1 2 cccc cccccccccccccccc cccccccccccccccccccccc 2 r J 2 2 2
L M M N
+
/
+
p
/
+
/
f
\ ] ]
(3.7)
^
and a2 + V /
a2
+
f
+
J pq J1 +
/
w +r Á/ U/ +J 1/ p w2 + U2 1/ f +0,0,1/ / s ++J 1/ f p/.
(3.8)
Plugging (3.7) and (3.8) into (3.3), and simplifying, we get (1.5).
4 Computational Example
Let Á ,3 Ö 0.10 Ö , ª1 0, p Ö , q 16 +0.12 Ö/ , U 1 s 2, w Ö , r r (Monte–Carlo generated, see, e.g., [15]) price-Y /volatility– Ö trajectories looks like: r
r
r
r
• !!! v
Y 0.5 0.45 0.4 0.35 0.3 0.25
52.5 50 47.5 45 42.5 0.1 0.2 0.3 0.4 0.5 For J1
0.1 0.2 0.3 0.4 0.5
1 s 10, and some time t T , fJ and corresponding fJ
+0,0,1/
s
fJ look like:
.025, T
.5 . One of the
7
t=0.0166667, g=0.1
3 2 1
500 450 400 350 300 0.1
40 0.2 v 0.3
while for J2
0.1
60 80 Y
60 0.2 80 Y v 0.3 100
100, and same time t T , fJ and corresponding fJ
+0,0,1/
s
40
fJ look like
t=0.0166667, g=100
0.24 0.235 0.23 0.1 0.2 v 0.3
60 80 Y
40
-0.1 -0.15 -0.2 0.1
60 0.2 v 0.3 80 Y 100
40
The spread V1s10 +t, Y, Ö/ V100 +t, Y, Ö/ between two corresponding price-functions, for a call-option with strike price k 60 (and expiration T ), as well as the computed option prices V1s10 +t, Y, Ö/ and V100 +t, Y, Ö/ , look like: t=0.0166667, v=0.2
t=0.0166667 0.2 0.15 0.1 0.05 0.1 0.2 v 0.3
60 80 Y
40
8 6 4 2 58 60 62 64
Y
REMARK 6. Once the pricing problem is solved, i.e., V is computed, the hedging problem can be settled quickly either via (constrained) portfolio rule 3Ú w U V +0,0,1/ s p Y V +0,1,0/, V , which is an appropriate analogue of the Black–Scholes hedging, or via (constrained) portfolio rule 3Ú1 +Á r/ s p2 w U V +0,0,1/ s p Y V +0,1,0/ , V , which is the "log-utility" (constrained) portfolio rule. A full scale of analogous (constrained) portfolio rules corresponding to J ± +0, /, J 1, are possible as well, but they are beyond the scope of this note, and will be discuss elsewhere.
5 References [1] [2] [3] [4] [5] [6] [7] [8]
Fleming W. H. and Sheu S. J., Risk-sensitive control and an optimal investment model, Mathematical Finance, 10(2), (2000), 197-213. Gutiérrez C. E., The Monge–Ampère Equations, Birkhäuser, Boston, 2001. Henderson V., Analytical comparisons of option prices in stochastic volatility models, preprint. Henderson V., Hobson D., Howison S., and Kluge T., A comparison of option prices under different pricing measures in a stochastic volatility model with correlation, preprint. Heston S. L., A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, Vol. 6, No. 2, (1993), 327–343. Hobson D. G., Stochastic volatility, D. Hand and S. Jacka (eds.), Applications of Statistics Series, Arnold, London, 1998. Hörmander L., Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. Hull J. and White A., The pricing of options on assets with stochastic volatility, The Journal of Finance, Vol. 42, No. 2 (1987), 281–300.
8
Jonsson M. and Sircar R., Partial hedging in a stochastic volatility environment, Mathematical Finance, Vol. 12, No. 4, (2002), 375–409. Jonsson M. and Sircar R., Optimal investment problems and volatility homogenization approximations, A. Bourlioux and M.J. [10] Gander (eds.), Modern Methods in Scientific Computing and Applications, 255–281, Kluwer, Dordrecht, 2002. Kallsen J., Utility-based derivative pricing in incomplete markets, Mathematical Finance—Bachelier Congress 2000, Geman, [11] H., Madan, D., Pliska, S. R., Vorst, T. (Eds.), Springer, Berlin, 2002. [12] Merton R. C., Optimum consumption and portfolio rules in a continuous-time model, J. Economic Theory, 3, (1971), 373–413.
[9]
[13]
Merton R. C., Continuous-Time Finance, Blackwell, Cambridge, 1992.
[14]
Stein E. M. and Stein J. C., Stock price distributions with stochastic volatility: an analytic approach, The Review of Financial Studies, Vol. 4, No. 4, (1991), 727–752. Stojanovic S., Computational Financial Mathematics using Mathematica®: optimal trading in stocks and options, Birkhäuser, Boston, 2003. Stojanovic S., Optimal momentum hedging via hypoelliptic reduced Monge–Ampère PDEs, to appear in SIAM Journal on Control & Optimization.
[15] [16]
Mar. 18, 2004
© Copyright
Pricing under stochastic volatility: complete solution via decoupled system of Monge–Ampère and Black–Scholes PDEs Srdjan D. Stojanovic Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, U.S.A. http://math.uc.edu/~srdjan/ March 2004
Abstract We have found, at least from the practical point of view, the complete solution of the option pricing problem for underlying securities obeying stochastic volatility price dynamics. Although partial solutions have existed, and in spite of a considerable attention to it, this problem has been open for about 20 years. The pricing problem is reduced to solving an uncoupled system of a Monge–Ampère type PDE and a Black–Scholes type PDE. Résumé Tarifer sous l’hypothèse de volatilité stochastique : solution complète par l’utilisation d’un système découplé d’EDP de Monge-Ampère et de Black-Scholes. Nous trouvons au moins d’un point de vue pratique, la solution complète au problème de tarification des options sur les titres sous-jacents, dont les prix obéissent à une dynamique de volatilité stochastique. Bien que des solutions partielles existent et aient fait l’objet d’une attention certaine, ce problème est resté ouvert depuis 20 ans. Le problème de la tarification se réduit à résoudre un système découplé d’EDP de type Monge-Ampère et Black Scholes.
Version française abrégée Considérons un « marché boursier à volatilité stochastique », consistant d’un titre unique au prix Y +t / à la date t et vérifiant le système suivant d’équations différentielles stochastiques de type Itô
ÅY t ÅÖ t
+ /
+ /
Y +t / +Á+t, Y +t /, Ö+t // ª1 +t, Y +t /, Ö+t /// Å t Y +t/ p+t, Y +t /, Ö+t // Å B1 +t/
q+t, Y +t /, Ö+t // Å t w+t, Y +t /, Ö+t // , U+t, Y +t/,
Ö t Å B1 t + //
+ /
1 U t, Y t , Ö t
r 2 + + / + //
Å B2 t
(0.1)
+ /0
où B1 et B2 représentent des mouvements browniens indépendants, Á est le taux d’appréciation du titre, ª1 est le dividende, p 0 est la volatilité, et v est un facteur scalaire arbitraire (par exemple, la volatilité : p+t, Y, Ö/ Ö ) à direction q, à diffusion w 0 et où 1 U 1 est le coefficient de corrélation entre prix et facteur. Soit également un marché à options constitué d’une option unique sur le titre ci-dessus, de gain fixe X+Y / à la date de maturité T , et au prix V +t, Y +t /, Ö+t // à la date t T , et où V +t, Y, Ö/ est une fonction inconnue a priori. Soit X ! 0, la richesse. L’utilité associée à la richesse est mesurée par une fonction d’utilité de type HARA : \J + X / X 1J s +1 J/ pour J ± +0, /, J 1, et \1 + X / log+ X / .
Considérons une stratégie de couverture du portefeuille par autofinancement 3+t, X , Y, Ö/ 31 , 32 , où 31 est la valeur monétaire de l’investissement dans l’action et 32 est la valeur de l’investissement dans l’option. Etant donné (0.1), et pour une stratégie fixe 3 , il existe une EDS qui caractérise l’évolution de la richesse X +t/ X 3 +t / . La stratégie de portefeuille J -optimale (le problème de Merton), notée 3ÚJ 3ÚJ 1 , 3ÚJ 2 est telle que :
2
Ú
Et, X,Y,Ö \J , X 3J +T /0.
sup Et, X,Y,Ö \J + X 3 +T // 3
(0.2)
DEFINITION 1. Pour tout J ± +0, / , la fonction du prix de l’option VJ +t, Y, Ö/ est définie comme la solution d’une équation Black-Scholes (abstraite) 3ÚJ 2 0. THEOREME 1. Pour tout J ± +0, /, J 1, le prix de l’option correspondant VJ +t, Y, de l’EDP Black-Scholes généralisée VJ +1,0,0/
1 2
cccc Y 2 VJ 0,2,0 +
/
p2 Y w U VJ +0,1,1/ p
fJ 0,0,1 Ár wU q cccccccccccccccccccccccccccccc w2 1 U2 cccccccccccccccc ccccc +
L M M
+
/
+
p
N
/
de condition terminale VJ +T, Y, Ö/
Á rcccc2cc r J cccccccccccccccc 2 2 L + M M
/
p
N
\ ] ] ^
XY +
qJ fJ 2 - cccccccccccccc c J1
/
fJ
\ ] ]
/
^
cccc w2 VJ 0,0,2 Y r ª1 1 2
+
/
+
/
+
//
+
/
0
, et où fJ est une solution appropriée de l’EDP de type Monge-Ampère
w Ár U cccccccccccccccccccccccccccccc +
/
p
1
w2 J cccccccccccccccc cccccc c f 0,0,2 fJ 2 J1 J p2 Y 2 J fJ cccccccccccccccccccccc cc fJ 0,2,0 fJ 2 J1
fJ +0,0,1/ fJ
/
+
+
de condition terminale fJ +T, Y, Ö/
+
est (formellement) solution
(0.3)
VJ +0,0,1/
+
+
/
/
/
+
+
+
/
VJ +0,1,0/ r VJ
Y Á r J r ª1 p wY J U fJ 0,1,0 fJ ccccccccccccccccccccccccccc fJ 0,1,1 cccccccccccccccccccccccccccccccc cccccccccccccccc cccccccc c c c J1 J1 1 1 J cccccccccccc1cc c fJ 1,0,0 fJ cc2cc w2 U2 fJ 0,0,1 2 cc2cc p2 Y 2 fJ 0,1,0 2 p w Y U fJ 0,0,1 J +
Ö
/
/
+
+
/
/
(0.4)
/
+
/
/
fJ +0,1,0/
0
1.
Pour J 1, le prix de l’option correspondant V1 V1 +t, Y, Ö/ est solution de l’EDP Black-Scholes généralisée (0.3) où f1 1, soit pour f1 +0,0,1/ s f1 0 ((0.4) ne sont pas nécessaires dans ce cas). The author wishes to thank very much to Dr. Christelle Viauroux for the French translation.
1 Statement of problem and it's solution Consider a "stochastic volatility stock market," consisting of a single security with price Y +t/ at time t , obeying Itô SDE system Y +t / +Á+t, Y +t /, Ö+t // ª1 +t, Y +t /, Ö+t /// Å t Y +t/ p+t, Y +t /, Ö+t // Å B1 +t/
ÅY t ÅÖ t
+ /
q+t, Y +t /, Ö+t // Å t w+t, Y +t /, Ö+t // , U+t, Y +t/,
(1.1) Ö t Å B1 t 1 U t, Y t , Ö t 2 Å B2 t where B1 and B2 are independent Brownian motions, and where Á is the appreciation rate, ª1 is the dividend rate, p is the volatility, and Ö is an arbitrary scalar factor (for example, volatility: p t, Y, Ö Ö ), with a drift q, diffusion w, and where U is the price/factor correlation. Of course, it is not essential that Ö is scalar. + /
+ //
+ /
r + + / + //
+
+ /0
/
Consider also an associated option market, consisting of a single option on the above underlying, with a fixed payoff X+Y / (for example, Xcall+Y / Max#0, Y k ' , for some strike price k ) at a fixed expiration time T , with a price V +t, Y +t/, Ö+t // at time t T , where function V +t, Y, Ö/ is a priori unknown. Assuming (1.1), it is not difficult to write down an Itô SDE characterizing the evolution of V +t, Y +t /, Ö+t // . The problem is to characterize, and be able to compute, "a fair option price function" V +t, Y,
Ö. /
Let X ! 0 denotes the wealth. We measure utility of wealth via HARA class of utility functions \J + X / X 1J s +1 J/ , for J ± +0, /, J 1, and \1 + X / log+ X / . Parameter J is called risk-aversion.
Consider a self-financing portfolio hedging strategy 3+t, X , Y, Ö/ 31 +t, X , Y, Ö/, 32 +t, X , Y, Ö/ , where 31 is the cash value of the investment into the underlying stock, and 32 is the cash value of the investment into the option. Assuming (1.1), and for a fixed strategy 3 , it is not difficult to write down an Itô SDE characterizing corresponding evolution of the wealth X +t / X 3 +t / (see (2.3) below). The J -optimal portfolio strategy (Merton's problem) is a strategy 3ÚJ 3ÚJ 1 , 3ÚJ 2 , such that sup Et, X,Y,Ö \J + X 3 +T // 3
Ú
Et, X,Y,Ö \J , X 3J +T /0.
(1.2)
3
DEFINITION 1. For every J ± +0, / , the fair option price function VJ +t, Y, Ö/ is defined as a solution of the (abstract) Black–Scholes equation 3ÚJ 2 0. Intuitively, "a fair option price" is such a price for which it is not rational to speculate by investing, long or short, into options. Variants of this definition have been present in the literature already (see, e.g., [11] and references given there). We shall use alternative notations for partial derivatives: for example, 2 V +t, Y, Ö/ s Ö2 V +0,0,2/ +t, Y, Ö/ .
THEOREM 1. For every J ± +0, /, J 1 , the corresponding fair option price VJ (formally) as a solution of a generalized Black–Scholes PDE VJ +1,0,0/
1 2
cccc Y 2 VJ 0,2,0 +
/
p2 Y w U VJ +0,1,1/ p
fJ 0,0,1 Ár wU q cccccccccccccccccccccccccccccc w2 1 U2 cccccccccccccccc ccccc +
L M M
+
/
+
p
N
/
/
fJ
\ ] ] ^
cccc w2 VJ 0,0,2 Y r ª1 1 2
+
/
+
/
VJ +t, Y,
Ö
/
is characterized
VJ +0,1,0/ r VJ (1.3)
VJ +0,0,1/
0
with the terminal condition VJ +T, Y, and where fJ
Ö XY fJ t, Y, Ö /
+
+
Á rcccc2cc r J cccccccccccccccc 2 2 L + M M
/
p
N
(1.4)
/ /
\ ] ] ^
is an appropriate solution of a Monge–Ampère type PDE qJ fJ 2 - cccccccccccccc c J1
w Ár U cccccccccccccccccccccccccccccc +
/
p
1
w2 J cccccccccccccccc cccccc c f 0,0,2 fJ 2 J1 J p2 Y 2 J fJ cccccccccccccccccccccc cc fJ 0,2,0 fJ 2 J1
fJ +0,0,1/ fJ
+
+
/
/
Y Á r J r ª1 p wY J U cccccccccccccccccccccccccccccccc ccc f 0,1,1 cccccccccccccccccccccccccc c fJ 0,1,0 fJ cccccccccccccccccccccccc J1 J1 J 1 1 J cccccccccccc1cc c fJ 1,0,0 fJ cc2cc w2 U2 fJ 0,0,1 2 cc2cc p2 Y 2 fJ 0,1,0 2 p w Y U fJ 0,0,1 J +
+
//
+
/
+
/
+
+
+
/
+
+
/
/
+
+
/
/
+
/
(1.5)
/
/
fJ +0,1,0/
0
with the terminal condition fJ +T, Y, Ö/
1.
(1.6)
For J 1, the corresponding fair option price V1 V1 +t, Y, Ö/ is characterized as a solution of the generalized Black–Scholes PDE (1.3) with f1 1, i.e., with f1 +0,0,1/ s f1 0 ((1.5)–(1.6) is not needed in that case). REMARK 1. Comparing (1.3) with the literature (see, e.g., equation (15) in [6]) we can see that Theorem 1 imples that, for any J ± +0, / , the so-called "market price of volatility risk," which we denote ³J , is given by
³J
w
1 U2 fJ +0,0,1/ s fJ
r
(1.7)
where fJ is an appropriate solution of (1.5)–(1.6). The exact expression for the market price of volatility risk was not known so far. In particular, ³1 0. Analysing (1.3), we can see that fair option price is unique, i.e., it is same for all J ± +0, / , iff ³J s J 0, and therefore iff ³J 0, and assuming furthermore w2 +1 U2 / ! 0, iff fJ +0,0,1/ 0, for all J ± +0, / . It was claimed recently in the literature (see [3, 4]) that for uniqueness of fair option prices it suffices that the Sharpe ratio +Á r/ s p is constant with respect to Ö . This does not seem to be correct, since although the Sharpe ratio may have a dominant effect on the solution of (1.5), and therefore indeed the "uniqueness" of prices under the above condition may hold approximatelly, ++Á r/2 s +2 p2 // fJ 2 is not the only term in (1.5) that may cause fJ +0,0,1/ 0. Indeed, suppose that fJ +0,0,1/ 0; terms p2 Y 2 J fJ +0,2,0/ fJ s +2 +J 1// and p2 Y 2 + fJ +0,1,0/ /2 s 2 remain in (1.5), and unless, which is trivial, the volatility p p+t, Y, Ö/ does not depend on the factor Ö , we cannot expect that fJ +0,0,1/ 0. Numerical computations confirm my findings. REMARK 2. Multiplying equation (1.5) by +J 1/ , sending J 1, and dividing by f1 , one arrives (at least formally) at 1 1 ccccc f1 +0,0,2/ w2 p Y U f1 +0,1,1/ w q f1 +0,0,1/ Y Á f1 +0,1,0/ Y ª1 f1 +0,1,0/ ccccc 2
2
p2 Y 2 f1 +0,2,0/ f1 +1,0,0/
0
(1.8)
which (is a linear equation, and) together with (1.6), is solved, obviously, by f1 1. This argument is not necessary to derive the above result in the case J 1—it follows from the portfolio theory below. REMARK 3. System (1.3)–(1.6) is uncoupled, which is very useful. Furthermore, fJ does not depend on a particular option payoff X .
4
REMARK 4. If w 0, or if U 1, then ³J 0 , and the same fair option price V VJ VJ +t, Y, Ö/ holds for all J ± +0, / . For example, if w 0, the price is characterized as a solution of a (hypoelliptic (see [7], and also [15, 16])) Black–Scholes PDE 1 + / p2 Y +r ª1 / V +0,1,0/ r VJ q VJ +0,0,1/ 0 2 with the terminal condition (1.4). If also q 0, then (1.9) simplifies further to the (usual) Black–Scholes PDE. VJ +1,0,0/
cccc Y 2 VJ 0,2,0
(1.9)
2 Preliminary results in general portfolio theory Consider, quite generally, a set of m factors A+t /
ÅAt
A1 +t /, ..., Am +t / obeying the Itô SDE dynamics
b+t, A+t // Å t c+t, A+t //.Å B+t /
+ /
(2.1)
where B+t / B1 +t /, ..., Bn +t/ is the n-dimensional Brownian motion (so, b+t, A/ , is an m-vector valued function, and c+t, A/ is an m n-matrix valued function). In addition to the factors, consider a set of k tradable assets with prices S +t / S1 +t /, ..., Sk +t / , obeying (non-linear in A ) Itô SDE dynamics S +t / +as +t, A+t // ª+t, A+t /// Å t S +t / Vs +t, A+t //.Å B+t /
ÅS t
+ /
(2.2)
where the vector-valued function as +t, A/ is the k -vector of appreciation rates, ª+t, A/ is the k -vector of dividend rates of the corresponding assets, Vs +t, A/ is the (volatility) k n-matrix valued function. If factor is tradable, then it can be represented also as one of the equations in system (2.2), and vice-versa. We assume Vs .Vs T ! 0. It can be shown that
ÅX
t
+ /
+
X +t /, A+t //.+as +t, A+t// r/ r X +t //Å t 3+t, X +t /, A+t //.Vs +t, A+t //.Å B+t /
3 t, +
(2.3)
and (2.1) and (2.3) form a closed SDE system, to be controlled. In particular, ª is eliminated, while components of ª may still be hidden in (2.1) and (2.3), and more specifically in as and b (see (3.1)). We shall refer to functions as ,
Vs , b, and c as the market coefficients.
The objective of the investor is to, for a given utility function \ (not necessarily HARA), maximize the expected value of the utility of the final total wealth, i.e., to find an optimal hedging strategy 3Ú +t, X , A/ such that:
M t, +
Ú
sup Et, X, A \+ X 3 +T //
X , A/
Et, X, A \, X 3 +T /0.
3
(2.4)
The standard formalism for solving the stochastic control problem (2.4) is to attempt to solve the associated Hamilton–Jacobi–Bellman (HJB) PDE characterizing the value function M M+t, X , A/ :
M
1 M 2 M 3. as r r X ccccccccccc b.´ A M cccc 3.Vs .Vs T .3 cccccccccccc2cc 2 X X 1 M 3.Vs .cT .´ A ccccccccccc cccc Tr c.cT .´ A ´ A M 0 2 X with the terminal condition M T, X , A \ X . After some simplifications, one Max$ ccccccccc 3 t
+
+
/
/
-
1
#
(2.5)
'(
can see that solving the HJB PDE + / + / (2.5) is equivalent to finding an appropriate solution of the Monge–Ampère type PDE:
2 M M 1 M 2 2 M M 2 M cccccccccccc2ccc cccccccccc ccccc cccccccccc c as r . Vs .Vs T 1 . as r r X cccccccccccc2ccc cccccccccc c b.´ A M cccccccccccc2ccc X t 2 X X X X M M M M 1 T T 1 T T 1 cccccccccc c as r . Vs .Vs .Vs .c .´A cccccccccc c ccccc ´ A cccccccccc c .c.Vs . Vs .Vs .Vs .cT .´ A cccccccccc c X X X X 2 1 2 M ccccc cccccccccccc2ccc Tr c.cT .´ A ´ A M 0 2 X -
+
/ +
1
+
/
/ +
/
-
+
1
/
-
1
+
/
-
1
(2.6)
'
#
with the same terminal condition, while the optimal hedging strategy is given by
3Ú t, +
X , A/
ccccMXcc c cccccccc cccccc 2 ccccXccccM2cc
as r/.+Vs .Vs T /
+
1
1 M cccccccc cccccc ´ A ccccccccccc 2 M X cccc Xcccc2cc -
1
.c.Vs T .+Vs .Vs T /1 .
(2.7)
5
As in Merton's case (see also [14, 15]), in the case of HARA utility \J , we seek the solution of (2.6), in the form X , A/ X 1J fJ +t, A/ s +1 J/ , if J ± +0, /, J 1, and in the form M+t, X , A/ f1 +t, A/ log+ X / if J 1. In + the first case f fJ solves
M t,
J cccccccccccccc c 1J
f 1 cccccccccc cccc t 2
f
f 2 ++as r/.+Vs .Vs T /1 .+as r/ 2 r J/
as r/.+Vs .Vs T /1 .Vs .cT
f
-+
1 2
cccc ´ A f .c.Vs T . Vs .Vs T +
1
/
Jb cccccccccccccc c .´ A f 1J
1 Jf cccc cccccccccccccc c Tr c.cT .´ A ´ A f 2 1J
.Vs .cT .´ A f
#
together with the terminal condition f +T, A/
3ÚJ t, +
X , A/
X MM+as r/ L N
(2.8)
1
0
'
1, while the optimal hedging strategy is now given by
´ f ccccccccAccccccJc .c.VsT . J Vs .Vs T \ ] + ]
fJ
/
1
X PJÚ +t, A/.
^
(2.9)
In the case J 1, f1 solves instead a simple linear PDE, which is even irrelevant, since the optimal hedging strategy 3Ú does not depend on f : 1
3Ú1 t, +
X , A/
X +as r/.+Vs .Vs T /1
X PÚ1 +t, A/.
(2.10)
We refer to equation (2.8) as the reduced Monge–Ampère PDE of optimal portfolio hedging. REMARK 5. Of course, (2.6), (2.7), (2.8), (2.9), and (2.10) have much broader significance then being a tool for proving Theorem 1.
3 A sketch of the proof of Theorem 1
We shall consider only the case J ± +0, /, J 1. We apply the general portfolio theory of Section 2 in the case of a "stochastic volatility market" of Section 1, and in particular (2.8) and (2.9). To that end we identify factors A+t/ Y +t /, Ö+t / , tradable assets S +t / Y +t /, V +t, Y +t /, Ö+t // , and the market coefficients as
Vs
Á, ccccV1cc cc12ccc Y 2 V +0,2,0/ p2 Y w U V +0,1,1/ p q V +0,0,1/ cc12ccc w2 V +0,0,2/ Y Á ª1 V +0,1,0/ V +1,0,0/
L M M M M M M N
-
p
0
w U V +0,0,1/ Y p V +0,1,0/
w 1U2 V +0,0,1/
cccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccc cccccccccccccc c V V
\ ] ] ] ] ] ]
Y +Á ª1 /, q , c
,b
V +1,0,0/
L M M M M N
L M M M
N
^
So, we extract the "abstract Black–Scholes" equation 3ÚJ 2 Ú PJ 2
/
+
Yp
1!
(3.1)
0
wU w
\ ] ] r ] ^
.
1 U2
0. Indeed, we compute, using (2.9),
ccccc Y 2 V +0,2,0/ p2 Y w U V +0,1,1/ p ccccc w2 V +0,0,2/ 1 2
1 2
fJ +0,0,1/ r Á wU V Y r ª1 V +0,1,0/ r V q cccccccccccccccccccccccccccccccccc w2 1 U2 cccccccccccccccc cccccc c V +0,0,1/ cccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccccc 2 p fJ J w 1 U2 V +0,0,1/ 2 +
L M M M M
/
+
/
+
/
N
\ ] ] ] ]
\ ] ] ] ]
^
^
+
(3.2)
/
and therefore (1.3) follows. Equation (1.3) is not closed, since fJ f is not known and in fact may depend on V . Equation (1.3) is therefore coupled with the equation (2.8), which now reads as
J w2 cccccccccccccccccc c f 2J2 2 Y J p2 cccccccccccccccccccccc f 0,2,0 2J2
a1 +V / f 2 a2 +V / f +0,0,1/ f Y J pwU ccccccccccccccccccccccccccc J1
f +0,1,1/ f
+
J Y r ª1 r Á Y 0,1,0 cccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccccc f J1 1 J cccccccccccccc c f 1,0,0 f cccc w2 f 0,0,1 2 2 J1
+0,0,2/
/
f
/
+
f
+
/
+
/
+
+
/
1 f +0,1,0/ 2 Y p w U f +0,0,1/ f +0,1,0/ 0 2 where the remaining coefficients a1 +V / and a2 +V / depend on V , and are given by
cccc Y 2 p2
/
f (3.3)
6
a1 +V /
1 2
ccccc 2 r J -
Y 2 V +0,2,0/ p2 2 Y w U V +0,1,1/ p 2 r V 2 q V +0,0,1/ w2 V +0,0,2/
++
2 Y Á V +0,1,0/ 2 Y ª1 V +0,1,0/ 2 V +1,0,0/ Y 2 V +0,2,0/ p3 2 Y w U V +0,1,1/ p2 V +0,0,2/ w2 2 r V 2 q V +0,0,1/ 2 r Y V +0,1,0/ 2 Y ª1 V +0,1,0/ 2 V +1,0,0/ p 2 w r Á U V +0,0,1/ 4 p w2 U2 1 V +0,0,1/ 2 Á r r Á w2 V +0,0,1/ 2 /+
/
+
+
/
// t ,
+
0
/
-+
/ -+
/,
2 Y p w U V +0,1,0/ V +0,0,1/ Y 2 p2 V +0,1,0/ 2 ccccc
1 p +w U V +0,0,1/ Y p V +0,1,0// 2 p 2 r V 2 q V +0,0,1/ w2 V +0,0,2/ 2 Y Á V +0,1,0/ 0
Y 2 V +0,2,0/ p2 2 Y w U V +0,1,1/
+
2 Y ª1 V +0,1,0/ 2 V +1,0,0/
/11 u ,
p2 w2 + U2 1/ V +0,0,1/
2
01
and a2 + V /
1
cccccccccccccccccccccccccccccccc ccccccccccccc 2 r J 1 V 2 q V 0,0,1 J 1 Y 2 V 0,2,0 p2 2 J 1 V 0,0,1 2 Y w U V 0,1,1 p w2 V 0,0,2 2 Y r ª1 V 0,1,0 2 V 1,0,0 /
+
+
+
/
+
+
+
/
/
+
/
/
+
+
/ +
+
/
+
/
/
+
/
(3.5) //
making (1.3) and (3.3) a coupled system. Moreover the coupling (3.4) and (3.5) appears quite challenging (high derivatives of V are involved, and V +0,0,1/ is in the denominator). Now, using (1.3), we express V +1,0,0/
1
cc2cc Y 2 V 0,2,0 +
/
Ár wU f 0,0,1 q cccccccccccccccccccccccccccccc w2 1 U2 cccccccccccccccccc c L M M
+
+
/
+
p
N
cc2cc w2 V 0,0,2 Y r ª1 1
p2 Y w U V +0,1,1/ p /
/
f
\ ] ]
+
/
+
/
V +0,1,0/ r V (3.6)
V +0,0,1/
^
and using (3.6) in (3.4) and (3.5), we get (uncoupled) a1 + V /
a1
1 Á rcccc2cc cccccccccccccccccccccccccccccccc w2 U2 1 f 0,0,1 2 cccc cccccccccccccccc cccccccccccccccccccccc 2 r J 2 2 2
L M M N
+
/
+
p
/
+
/
f
\ ] ]
(3.7)
^
and a2 + V /
a2
+
f
+
J pq J1 +
/
w +r Á/ U/ +J 1/ p w2 + U2 1/ f +0,0,1/ / s ++J 1/ f p/.
(3.8)
Plugging (3.7) and (3.8) into (3.3), and simplifying, we get (1.5).
4 Computational Example
Let Á ,3 Ö 0.10 Ö , ª1 0, p Ö , q 16 +0.12 Ö/ , U 1 s 2, w Ö , r r (Monte–Carlo generated, see, e.g., [15]) price-Y /volatility– Ö trajectories looks like: r
r
r
r
• !!! v
Y 0.5 0.45 0.4 0.35 0.3 0.25
52.5 50 47.5 45 42.5 0.1 0.2 0.3 0.4 0.5 For J1
0.1 0.2 0.3 0.4 0.5
1 s 10, and some time t T , fJ and corresponding fJ
+0,0,1/
s
fJ look like:
.025, T
.5 . One of the
7
t=0.0166667, g=0.1
3 2 1
500 450 400 350 300 0.1
40 0.2 v 0.3
while for J2
0.1
60 80 Y
60 0.2 80 Y v 0.3 100
100, and same time t T , fJ and corresponding fJ
+0,0,1/
s
40
fJ look like
t=0.0166667, g=100
0.24 0.235 0.23 0.1 0.2 v 0.3
60 80 Y
40
-0.1 -0.15 -0.2 0.1
60 0.2 v 0.3 80 Y 100
40
The spread V1s10 +t, Y, Ö/ V100 +t, Y, Ö/ between two corresponding price-functions, for a call-option with strike price k 60 (and expiration T ), as well as the computed option prices V1s10 +t, Y, Ö/ and V100 +t, Y, Ö/ , look like: t=0.0166667, v=0.2
t=0.0166667 0.2 0.15 0.1 0.05 0.1 0.2 v 0.3
60 80 Y
40
8 6 4 2 58 60 62 64
Y
REMARK 6. Once the pricing problem is solved, i.e., V is computed, the hedging problem can be settled quickly either via (constrained) portfolio rule 3Ú w U V +0,0,1/ s p Y V +0,1,0/, V , which is an appropriate analogue of the Black–Scholes hedging, or via (constrained) portfolio rule 3Ú1 +Á r/ s p2 w U V +0,0,1/ s p Y V +0,1,0/ , V , which is the "log-utility" (constrained) portfolio rule. A full scale of analogous (constrained) portfolio rules corresponding to J ± +0, /, J 1, are possible as well, but they are beyond the scope of this note, and will be discuss elsewhere.
5 References [1] [2] [3] [4] [5] [6] [7] [8]
Fleming W. H. and Sheu S. J., Risk-sensitive control and an optimal investment model, Mathematical Finance, 10(2), (2000), 197-213. Gutiérrez C. E., The Monge–Ampère Equations, Birkhäuser, Boston, 2001. Henderson V., Analytical comparisons of option prices in stochastic volatility models, preprint. Henderson V., Hobson D., Howison S., and Kluge T., A comparison of option prices under different pricing measures in a stochastic volatility model with correlation, preprint. Heston S. L., A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, Vol. 6, No. 2, (1993), 327–343. Hobson D. G., Stochastic volatility, D. Hand and S. Jacka (eds.), Applications of Statistics Series, Arnold, London, 1998. Hörmander L., Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. Hull J. and White A., The pricing of options on assets with stochastic volatility, The Journal of Finance, Vol. 42, No. 2 (1987), 281–300.
8
Jonsson M. and Sircar R., Partial hedging in a stochastic volatility environment, Mathematical Finance, Vol. 12, No. 4, (2002), 375–409. Jonsson M. and Sircar R., Optimal investment problems and volatility homogenization approximations, A. Bourlioux and M.J. [10] Gander (eds.), Modern Methods in Scientific Computing and Applications, 255–281, Kluwer, Dordrecht, 2002. Kallsen J., Utility-based derivative pricing in incomplete markets, Mathematical Finance—Bachelier Congress 2000, Geman, [11] H., Madan, D., Pliska, S. R., Vorst, T. (Eds.), Springer, Berlin, 2002. [12] Merton R. C., Optimum consumption and portfolio rules in a continuous-time model, J. Economic Theory, 3, (1971), 373–413.
[9]
[13]
Merton R. C., Continuous-Time Finance, Blackwell, Cambridge, 1992.
[14]
Stein E. M. and Stein J. C., Stock price distributions with stochastic volatility: an analytic approach, The Review of Financial Studies, Vol. 4, No. 4, (1991), 727–752. Stojanovic S., Computational Financial Mathematics using Mathematica®: optimal trading in stocks and options, Birkhäuser, Boston, 2003. Stojanovic S., Optimal momentum hedging via hypoelliptic reduced Monge–Ampère PDEs, to appear in SIAM Journal on Control & Optimization.
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