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Malliavin Monte Carlo Greeks for jump diffusions
Mark H.A. DavisDepartment of Mathematics, Imperial College
London SW7 [email protected]
Martin P. JohanssonImperial College and Citigroup∗
Citigroup Centre, 33 Canada Square, 6th Floor, Canary WharfLondon E14 5LB
July 3, 2005
Abstract
In recent years efficient methods have been developed for calculat-ing derivative price sensitivities using Monte Carlo simulation. Malliavincalculus has been used to transform the simulation problem in the casewhere the underlying follows a Markov diffusion process. In this work,recent developments in the area of Malliavin calculus for Levy processesare applied and slightly extended. This allows for derivation of similarstochastic weights as in the continuous case for a certain class of jump-diffusion processes.
Keywords: jump process, Levy process, Monte Carlo estimation,mathematical finance
1 Introduction
For many financial applications Monte Carlo simulation is the favoured pricingtool because of its flexibility. Pricing is, however, only the first step in man-aging a trade. As the contract start to run it needs to be protected againstunfavourable price moves and its risk needs to be properly managed. The pricesensitivities with respect to the model parameters—the Greeks—are vital in-puts in this context. For calculating the Greeks Monte Carlo simulation leavesmuch to be desired. The slow convergence, especially for discontinuous payofffunctions, is well known, and for this reason less flexible pricing tools are oftenused.
∗The analysis and conclusions set forth are those of the authors. Citigroup is not respon-sible for any statement or conclusion herein, and opinions or theories presented herein do notnecessarily reflect the position of the institution.
1
The Greeks are calculated as differentials of the derivative price, which can beexpressed as an expectation (in risk–neutral measure) of the discounted payoff:
v(x) = E [φ(XT ) |X0 = x] .
When this expectation is estimated using Monte Carlo simulation the deriva-tives can be found by finite difference approximations. If the terms in the finitedifference approximation are estimated using independent random number se-quences the convergence rate is at best n−1/3, but by using the same randomnumbers for all terms the convergence rate can be improved to n−1/2, which isthe best that can be achieved using Monte Carlo methods.1
The theoretical convergence rates for finite difference approximations arenot met for discontinuous payoff functions. Fournie et al. (1999) propose amethod with faster convergence for the case when the underlying price processis a Markov diffusion in Rn,
dXt = µ(Xt)dt + σ(Xt)dWt,
where Wt, t ≥ 0 is an n–dimensional Brownian motion. They use the Malli-avin integration by parts formula to transform the problem of calculating deriva-tives by finite difference approximations to calculating expectations of the form
E [φ(XT )π |X0 = x] ,
where π is a random variable.The objective of this work is to derive stochastic weights for calculating
the Greeks in a jump diffusion setting where the jump amplitude is deter-ministic. In particular we consider jump diffusions, Xt, which can be writtenXt = f(Xc
t , Xdt ), where Xc
t is a Markov diffusion for which drift and diffussioncoefficients are assumed to have bounded continuous derivatives, whereas Xd
t isdriven by the Poisson process and not dependent on the the initial value X0.This includes the stochastic volatility model
dX(1)t = µ1
(X
(2)t , . . . , X
(n)t
)X
(1)t− dt + σ1
(X
(2)t , . . . , X
(n)t
)X
(1)t− dWt
+m∑
k=1
(αk − 1)X(1)t− (dN
(k)t − λkdt), X
(1)0 = x1
dX(i)t = µi
(X
(2)t , . . . , X
(n)t
)dt + σi
(X
(2)t , . . . , X
(n)t
)dWt,
X(i)0 = xi, i = 2, . . . , n,
and the jump diffusion version of the Vasicek model for interest rates
drt = a(b− r)dt + σdWt +m∑
k=1
αk(dNt − λkdt),
where Wt, t ≥ 0 is a Brownian motion in Rn, N (k), t ≥ 0, k = 1, . . . ,m areindependent Poisson processes with intensities λk and αk are positive determin-istic constants.
1See Glasserman and Yao (1992).
2
Malliavin-style stochastic calculus for jump processes has been studied sincethe 1980s. Early works such as Bismut (1983) or Bichteler et al. (1987) ex-clude the case of fixed jump size, while Carlen and Pardoux (1990) study thepure Poisson case (no Brownian motion component). More recently, Nualartand Schoutens (2000) developed a theory of chaos expansions for functionals ofLevy processes when the Levy measure satisfies an exponential moment condi-tion. Leon et al. (2002) have used this as the basis for a Malliavin calculus. Adrawback of this approach is that there is no general chain rule. We developtheir approach to include a ‘Skorokhod integral’ and show that a restricted formof chain rule is enough to give formulas for Monte Carlo Greeks which are validfor important applications.
Since the first version of this paper was written, at least two other treat-ments of similar problems have appeared, both dealing with stochastic differen-tial equations driven by Brownian motion and compound Poisson components.Bavouzet and Massaoud (2005) and Bally et al. (2005) reduce the problem toa setting in which only ‘finite-dimensional’ Malliavin calculus is required. El-Khatib and Privault (2004) consider a market driven by jumps alone. Theirsetup allows for random jump sizes, and by imposing a regularity condition onthe payoff they use Malliavin calculus on Poisson space to derive weights forAsian options. Forster et al. (2005) take a dynamical systems approach whichtreats the Poisson component as a quasi-deterministic parameter, permittingthem to bypass jump process Malliavin calculus altogether. While each of theseapproaches has advantages for specific applications, no one of them uniformlydominates the others.
In Section 2 some theoretical results are stated and the existing theory ex-tended for later use. In Section 3 the random weights for calculating the Greeksare derived and in Section 4 some numerical examples are presented. Somederivations are left to the Appendix.
2 Malliavin calculus for simple Levy processes
In this section some of the theory of Malliavin calculus for simple Levy processesas developed in Leon et al. (2002) will be revisited. The theory will also beextended with some results we will need in the following sections.
On a filtered probability space (Ω,F , (Ft), P ), define a simple Levy processas given by
Xt = σWt + α1N(1)t + · · ·+ αmN
(m)t , t ≥ 0,
where Wt, t ≥ 0 is a standard Brownian motion and N (j)t , t ≥ 0, j =
1, . . . ,m, are independent Poisson processes with intensities λ1, . . . , λm, whichare also independent of the Brownian motion. The jump amplitudes αj , j =1, . . . ,m are different non-null constants. Also, let the filtration Ft be theone generated by the simple Levy process Xt, which is the same filtration asσW., N
(j). j = 1, . . . ,m.
The idea in Leon et al. (2002) is to represent random variables on the proba-bility space (Ω,F , P ) by iterated integrals, and from there, develop a Malliavincalculus as in the Gaussian case. See Nualart (1995) for a comprehensive treat-ment of the Malliavin calculus in the Gaussian setting.
3
We use the notation
G0(t) = Wt
Gj(t) = N(j)t − λjt, j = 1, . . . ,m,
and define L(i1,...,in)n (f) to be the iterated integral of some deterministic square
integrable function f with respect to Gi1 , . . . , Gin :
L(i1,...,in)n (f) =
∫ ∞
0
(∫ t−n
0
. . .
(∫ t−2
0
f(t1, . . . , tn)dGi1(t1)
). . .
dGin−1(tn−1)
)dGin
(tn).
It will also be useful to introduce the notation
Σn =(t1, . . . , tn) ∈ Rn
+ : 0 < t1 < t2 < . . . < tn
,
and
Σ(k)n (t) = (t1, . . . , tk−1, tk+1, . . . , tn) ∈ Σn−1 :
0 < t1 < . . . < tk−1 < t < tk+1 < . . . < tn .
Leon et al. (2002) formulate the following chaotic representation property,which is a modification of the more general result of Nualart and Schoutens(2000).
Theorem 2.1 (Leon, Sole, Utzet and Vives) Let F ∈ L2(Ω,F , P ), then Fhas a representation of the form
F = E[F ] +∞∑
n=1
∑0≤i1,...,in≤m
L(i1,...,in)n (fi1,...,in),
where fi1,...,in∈ L2(Σn).
The reason for introducing the chaotic representation is because we want todefine a Malliavin derivative similar to the Gaussian analogue. In this discon-tinuous case the expressions in the definition of the iterated integral and thechaotic representation property are a bit more involved since we are integratingwith respect to different martingales instead of only Brownian motion. There-fore, we cannot restrict our attention to symmetric functions.
For the definition of the Malliavin derivative we must first define the spacesof random variables which are differentiable in the l–th direction (l = 0, . . . ,m).We use the same notation as in Leon et al. (2002).
Definition 2.2 (Differentiable in the l–th direction) We say that F is dif-ferentiable in the l–th direction (l = 0, . . . ,m) if F ∈ D(l) where
D(l) = F ∈ L2(Ω), F = E[F ] +∞∑
n=1
∑0≤i1,...,in≤m
L(i1,...,in)n (fi1,...,in
) :
∞∑n=1
∑0≤i1,...,in≤m
n∑k=1
1ik=lλi1 . . . λik−1λik+1 . . . λin∫ ∞
0
‖fi1,...,in(. . . , t, . . . )1Σ(k)
n (t)‖2L2([0,∞)n−1)dt < ∞,
4
and λ0 = 1.
This definition ensures that the Malliavin derivative defined below is in L2(Ω×R+). In fact, the spaces D(l) are all dense in L2(Ω) and it can also be shownthat if we remove the Poisson processes the space D(0) collapses to the classicalGaussian space D 1,2 discussed e.g. in Nualart (1995).
Definition 2.3 (Malliavin derivative) For F ∈ D(l) such that
F = E[F ] +∞∑
n=1
∑0≤i1,...,in≤m
L(i1,...,in)n (fi1,...,in),
we define the derivative of F in the l–th direction as the element of L2(Ω×R+)given by
D(l)t F =
∞∑n=1
∑0≤i1,...,in≤m
n∑k=1
1ik=l
L(i1,...,ik−1,ik+1,...,in)n−1
(fi1,...,in
(. . . , t, . . . )1Σ(k)n (t)
).
We can now state the following Clark–Ocone formula and the chain rule.
Theorem 2.4 (Leon, Sole, Utzet and Vives) If F ∈⋂m
j=0 D(j) then
F = E[F ] +∫ ∞
0
p(D(0)t F )dWt +
m∑j=1
∫ ∞
0
p(D(j)t F )d(N (j)
t − λjt) (2.1)
Theorem 2.5 (Leon, Sole, Utzet and Vives) Let F = f(Z,Z ′) ∈ L2(Ω),where Z only depends on the Brownian motion W , and Z ′ only depends onthe Poisson processes N (1), . . . , N (m). Assume that f(x, y) is a continouslydifferentiable function with bounded partial derivatives in the variable x, andthat Z ∈ D(0). Then F ∈ D(0) and
D(0)F =∂f
∂x(Z,Z ′)D(0)Z
where D(0)Z is the usual Gaussian Malliavin derivative.
The chain rule is one of the main results in Leon et al. (2002) and it will play acentral role in the next section. It allows us to calculate the Malliavin derivativein W direction using the classical rules. This, together with the integration byparts formula Theorem 2.6, which will be proved later in this section, is whatallows us to derive the stochastic weights for calculating the Greeks using MonteCarlo simulation.
For the theory to work in a multidimensional setting we add Brownian mo-tions as
Gj(t) = W(j)t , j = 1, . . . , d
Gj(t) = (N (j−d)t − λj−dt), j = d + 1, . . . , d + m.
Definition 2.2 with l = 1, . . . , d defines spaces of random variables differentiablewith respect to the respective Brownian motion and we group the d Brownian
5
motions together in one d-dimensional column vector denoted simply by Wt.For a random variable F ∈
⋂di=1 D(i) we write D
(0)t F—the Malliavin derivative
with respect to Brownian motion—as a row vector where each component is theMalliavin derivative as defined in Definition 2.3.
For functions F ∈ D(l+d), l = 1, . . . ,m we will denote by D(l)t F the Malliavin
derivative defined as
D(l)t F=
∞∑n=1
∑0≤i1,...,in≤m
n∑k=1
1ik=l+d
L(i1,...,ik−1,ik+1,...,in)n−1
(fi1,...,in
(. . . , t, . . . )1Σ(k)n (t)
).
Note that with these new definitions D(0)t F will be a d-dimensional row vector,
whereas D(i)t F , i = 1, . . . ,m will be scalars denoting the Malliavin derivative
with respect to the ith Poisson process. Now the above theory still holds truewith d-dimensional Brownian noise and applying the results we can prove thefollowing theorem.
Theorem 2.6 (Integration by parts formula) Let F ∈⋂m+d
j=1 D(j), u be aprevisible process in L2(Ω×R+; Rd) and v be a previsible process in L2(Ω×R+).Then
E
[F
∫ ∞
0
u∗t dWt
]= E
[∫ ∞
0
(D(0)t F )utdt
]
E
[F
∫ ∞
0
vt(dN(l)t − λldt)
]= E
[∫ ∞
0
(D(l)t F )vtλldt
],
where l = 1, . . . ,m and ∗ denotes transpose.
Proof. This follows from the Clark-Ocone representation (2.1) for F .
Remark. When ut is a matrix–process the Brownian part of Theorem 2.6 be-comes
E
[(F
∫ ∞
0
u∗t dWt
)∗ ]= E
[∫ ∞
0
(D(0)t F )utdt
],
where the integration is done component–wise.
We now continue the development of a Malliavin calculus for simple Levy pro-cesses by defining a Skorohod integral as the adjoint of the derivative operator.We can do this since D is a densely defined operator.
Definition 2.7 (Skorohod integral) Let ut be a stochastic process in L2(Ω×R+), not necessarily adapted, such that∣∣∣∣E [∫ ∞
0
(D(l)t F )utλldt
]∣∣∣∣ < c‖F‖L2(Ω),
for some constant c depending on u and any F ∈ D(l). We say that u is Skorohodintegrable in the l–th direction or u ∈ Dom δ(l). We define the Skorohod integral
6
in the l–th direction, δ(l)(u), as the operator mapping L2(Ω×R+) to L2(Ω) forwhich
E
[∫ ∞
0
(D(l)t F )utλldt
]= E
[Fδ(l)(u)
],
for any F ∈ D(l).
In the case of multidimensional Brownian motion we will denote by δ(0)(u)the Skorohod integral of a vector process u. Analogous to the Ito integral theSkorohod integral is here the sum of the Skorohod integral of the componentsof u integrated with respect to its respective Brownian motion.
In both the continuous and the discontinuous case the integral is a linearoperator due to the linearity of the Malliavin derivative. It is likely that theSkorohod integral in Definition 2.7 has other properties in common with itsGaussian analogue. Another such similarity that will be discussed here is theresult presented in Proposition 2.10, which provides a good tool for calculatingthe Skorohod integral with respect to Brownian motion. For the proof we willneed the following Lemmas, which are also interesting in their own right.
Lemma 2.8 Let g(t) ∈ L2(R+) and H ∈ L2(Ω) with finite chaos expansion
H = E[H] +N∑
n=1
∑0≤i1,...,in≤m
L(i1,...,in)n (hi1,...,in).
Then Hg ∈ Dom δ(l) for l = 1, . . . ,m.
Proof. It is enough to consider the case when H is a single iterated integral:
H = L(j1,...,jq)n (hj1,...,jq
).
The general result follows from linearity. For any F ∈ D(l) we have
E
[∫ ∞
0
(D(l)t F )Hg(t)λldt
]
= λl
∫ ∞
0
E
∞∑n=1
∑0≤i1,...,in≤m
n∑k=1
1ik=l
L(i1,...,ik,...,in)n−1
(fi1,...,in
(. . . , t, . . . )1Σ(k)n (t)
)×(L(j1,...,jq)
q (hj1,...,jq ))
g(t)]dt.
We have (by isometry as shown in Leon et al. (2002)) that
E[L(i1,...,in)n (f)L(j1,...,jm)
m (h)] =
λi1 . . . λin
∫Σn
fh dt1 . . . dtn,
if multi–indices identical,
0, otherwise,
7
so we get (note how the infinite sum for F simplifies due to the above fact)
E
[∫ ∞
0
(D(l)t F )Hg(t)λldt
]=
λl
∫ ∞
0
( ∑0≤i1,...,iq+1≤m
q+1∑k=1
1ik=l1(i1,...,ik−1,ik+1...,iq+1)=(j1,...,jq)
λi1 . . . λik−1λik+1 . . . λiq+1
∫ ∞
0
. . .
∫ tik+1
0
∫ tik−1
0
. . .
∫ t−2
0
fi1,...,iq+1(. . . , t, . . . )1Σ(k)q+1(t)
hj1,...,jq(t1, . . . , tk−1, tk+1, . . . , tq+1)g(t)dt1 . . . dtk−1dtk+1 . . . dtq+1
)dt.
We can ‘bring the dt integration in to the kth position’ and transform back intostochastic integrals:
E
[∫ ∞
0
(D(l)t F )Hgλldt
]= E
[(E[F ] +
∞∑n=1
∑0≤i1,...,in≤m
∫ ∞
0
∫ tn−
0
. . .
∫ t2−
0
fi1,...,in(t1, . . . , tn)dGi1(t1) . . . dGin−1(tn−1)dGin
(tn)
)
×
(q+1∑k=1
∫ ∞
0
. . .
∫ tk+2−
0
∫ tk+1−
0
∫ tk−
0
. . .
∫ t2−
0
hj1,...,jq(t1, . . . , tk−1, tk+1, . . . , tq+1)
g(tk)dGj1(t1) . . . dGjk−1(tk−1)dGl(tk)dGjk(tk+1) . . . dGjq
(tq+1)
)].
An application of Schwarz inequality concludes the proof.
Corollary 2.9 The Skorohod integral is a densely defined operator in L2(Ω ×R+).
Proof. Processes of the form Hg(t) where H ∈ L2(Ω) and g ∈ L2(R+) are densein L2(Ω× R+).
Remark. The Skorohod integral of Hg as in the above Lemma is given by
δ(l)(Hg) =∫ ∞
0
E[H]g(t1)dGl(t1) +N∑
n=1
∑0≤i1,...,in≤m
n+1∑k=1
∫ ∞
0
. . .
∫ tk+2−
0
∫ tk+1−
0
∫ tk−
0
. . .
∫ t2−
0
hi1,...,in(t1, . . . , tk−1, tk+1, . . . , tn+1)
g(tk)dGi1(t1) . . . dGik−1(tk−1)dGl(tk)dGik(tk+1) . . . dGin
(tn+1). (2.2)
We see that Skorohod integration increases the order of the chaos expansionwith a dGl integral mixed in at every possible position. This is in line withSkorohod integration in the Gaussian case.
8
Proposition 2.10 For Fu in L2(Ω × R+), where F ∈ D(0) and ut previsible,we have
δ(0)(Fu) = F
∫ ∞
0
utdWt −∫ ∞
0
(D(0)t F )utdt. (2.3)
in the sense that Fu ∈ Dom δ(0) if and only if the right hand side of (2.3) is inL2(Ω).
Proof. It is not very surprising that when ut is stochastic but previsible, theexpression for the Skorohod integral of Fu is very similar to the expression (2.2).In Appendix A we show that
δ(l)(Fu) =∫ ∞
0
E[F ]ut1dG(l)(t1) +∞∑
n=1
∑0≤i1,...,in≤m
n+1∑k=1
∫ ∞
0
. . .
∫ tk−1−
0
∫ tk−
0
∫ tk+1−
0
. . .
∫ t2−
0
fi1,...,in(t1, . . . , tk−1, tk+1, . . . , tn+1)dGi1(t1)
. . . dGik−1(tk−1)utkdGl(tk)dGik
(tk+1) . . . dGin(tn+1).
We will now focus on the expression F∫∞0
utdG0(t), where 0 denotes the Brow-nian motion. By the chaos expansion of F we see that the product will be givenby terms of the form∫ ∞
0
∫ t−n
0
. . .
∫ t−2
0
fi1,...,in(t1, . . . , tn)dGi1(t1) . . . dGin−1(tn−1)dGin
(tn)∫ ∞
0
utdG0(t),
and the strategy is to prove that∫ ∞
0
∫ t−n
0
. . .
∫ t−2
0
fi1,...,in(t1, . . . , tn)dGi1(t1) . . . dGin−1(tn−1)dGin
(tn)∫ ∞
0
utdG0(t)
=n+1∑k=1
∫ ∞
0
. . .
∫ tk+2−
0
∫ tk+1−
0
∫ tk−
0
. . .
∫ t2−
0
fi1,...,in(t1, . . . , tk−1, tk+1, . . . , tn+1)
dGi1(t1) . . . dGik−1(tk−1)utkdG0(tk)dGik
(tk+1) . . . dGin(tn+1)
+n∑
k=1
∫ ∞
0
. . .
∫ t−k
0
. . .
∫ t−2
0
fi1,...,in(t1, . . . , tn)dGi1(t1) . . . utk
d[Gik, G0]tk
. . . dGin(tn).
(2.4)
The first sum of the right hand side of (2.4) is the contribution from the chaosexpansion of δ(l)(Fu) and the second sum is the contribution from the chaosexpansion of
∫∞0
(D(0)t F )utdt, establishing (2.3). Equation (2.4) can be shown
by induction using Ito’s formula.
Remark. In the case when F is a d-dimensional random column–vector and ut
is a (d× d) matrix–process Proposition 2.10 translates to
δ(0)(uF ) = F ∗∫ ∞
0
utdWt −∫ ∞
0
Tr((D(0)
t F )ut
)dt,
with the convention that the Ito integral for a matrix process is a column–vector.
9
3 Monte Carlo Greeks
In this section we will apply the results from the previous section to calculatethe Greeks for a certain class of price processes. We focus our attention on jumpdiffusion processes in Rd for which we make a particular assumption.
Assumption 3.1 (Separability) Assume that b, σ and α are continuouslydifferentiable functions with bounded derivatives and consider Markov jump dif-fusions, Xt ∈
⋂d+mj=1 D(j), of the form
dXt = b(Xt−)dt + σ(Xt−)dWt +m∑
k=1
αk(Xt−)(dN(k)t − λkdt), X0 = x,
for which the we have a continuously differentiable function f with boundedderivative in the first argument such that
Xt = f(Xct , Xd
t ), Xc0 = x.
Here Xct satisfies a stochastic differential equation
dXct = bc(Xc
t )dt + σc(Xct )dWt, Xc
0 = x,
with smooth coefficients bc and σc while Xdt is adapted to the natural filtra-
tion FNt of the Poisson processes (N (1)
t , . . . , N(m)t ). In particular, Xd
t does notdepend on x. We say that the jump diffusion process is separable.
An important property of the class of jump diffusions that satisfies Assumption3.1 is that they are differentiable with respect to Brownian motion and theMalliavin derivative can be calculated using the chain rule given in Theorem2.5. This, together with the integration by parts formula (Theorem 2.6), allowsus to derive stochastic weights for calculating the Greeks using Monte Carlosimulation by more or less mimicking the work of Fournie et al. (1999).
Examples of separable pricing processes are the stochastic volatility model
dX(1)t = b1
(X
(2)t , . . . , X
(d)t
)X
(1)t− dt + σ1
(X
(2)t , . . . , X
(d)t
)X
(1)t− dWt
+m∑
k=1
(αk − 1)X(1)t− (dN
(k)t − λkdt), X
(1)0 = x1
dX(i)t = bi
(X
(2)t , . . . , X
(d)t
)dt + σi
(X
(2)t , . . . , X
(d)t
)dWt,
X(i)0 = xi, i = 2, . . . , d,
and the jump–diffusion version of the Vasicek model for interest rates
drt = a(b− rt−)dt + σdWt +m∑
k=1
αk(dNt − λkdt),
where αk, k = 1, . . . ,m are deterministic constants. The claim that the jump–diffusion version version of the Vasicek model is separable can be proved by
10
solving the short rate SDE analytically, and the claim regarding the stochasticvolatility model will be proved in section 4.
An important role for the derivation of the stochastic weights will be playedby the first variation process for Xc
t defined by
dYt = b′c(Xct )Ytdt +
d∑i=1
σ′ci(Xct )YtdW
(i)t , Y0 = I, (3.1)
where I is the identity matrix, σci is the i-th column vector of σc and primedenotes derivatives. It is true that
Yt = ∇xXct ,
and it is also known that, since bc and σc are continuously differentiable functionswith bounded derivatives and since Xc
t is continuous, the Malliavin derivativeof Xc
t can be written as2
D(0)s Xc
t = YtY−1s σc(Xc
s)1s≤t. (3.2)
We define the payoff function
φ = φ(Xt1 , . . . , Xtn), (3.3)
to be a square integrable function discounted from maturity T and evaluated atthe times t1, . . . , tn. The price of a contingent claim is then expressed as
v(x) = E [φ(Xt1 , . . . , Xtn)] .
The objective is to differentiate v with respect to the model parameters and forthe proofs we will need to assume the diffusion matrix to be uniformly elliptic,that is
∃η > 0, ξ∗σ∗(x)σ(x)ξ ≥ η|ξ|2 for any ξ, x ∈ Rn.
We will also need the following technical lemma:
Lemma 3.2 Let g : R → R be a continuously differentiable function withbounded derivative and let G be a separable random variable in
⋂d+mj=1 D(j). Then
g(G) ∈d+m⋂j=1
D(j).
Proof. That g(G) ∈⋂d
j=1 D(j) is given by the chain rule (Theorem 2.5). Fork ∈ [d + 1, d + m] we can write the Malliavin derivative as3
D(k)t G = G(ω + θ
(k)t )−G(ω),
where ω + θ(k)t means “an extra jump in the kth Poisson process at time t”. We
have,
E
[∫ ∞
0
(D
(k)t g(G)
)2
dt
]= E
[∫ ∞
0
(g(G(ω + θ
(k)t ))− g(G(ω))
)2
dt
]≤ E
[M2
∫ ∞
0
(G(ω + θ
(k)t )−G(ω)
)2
dt
]< ∞,
where M is a bound on the derivative of g. 2See Nualart (1995) section 2.3.1.3See Leon et al. (2002) Proposition 2.4(b).
11
3.1 Variations in the drift coefficient
In order to assess the sensitivity of the price of the contingent claim v to changesin the drift coefficient we introduce the perturbed process
dXεt =
(b(Xε
t−)+ εγ(Xεt−))dt + σ(Xε
t−)dWt +m∑
k=1
αk(Xεt−)(dN
(k)t − λkdt)
Xε0 = x,
where ε is a scalar and γ is a bounded function. The following Proposition tellsus how sensitive the price of a claim on the perturbed process is to ε in thepoint ε = 0.
Proposition 3.3 Suppose that the diffusion matrix σ is uniformly elliptic. Forvε(x) defined as
vε(x) = E[φ(Xε
t1 , . . . , Xεtn
)],
we have
∂
∂εvε(x)
∣∣∣∣ε=0
= E
[φ(Xt1 , . . . , Xtn)
∫ T
0
(σ−1(Xt−)γ(Xt−))∗dWt
]Proof. The proof builds on an application of the Girsanov Theorem which holdstrue even in the presence of Poisson jumps. See e.g. the proof of Theorem E2in Karatzas and Shreve (1998).
Remark. The result holds for more general path-dependent claims φ = φ(X(.))and also for processes which are not separable as in Assumption 3.1.
3.2 Variations in the initial condition
This is where we for the first time will make full use of the theory developedin Section 2. We rely heavily on the chain rule and the integration by partsformula, which allow us to repeat the steps in Fournie et al. (1999).
First define the set Γ, of square integrable functions in R, as
Γ =
a ∈ L2([0, T )) :∫ ti
0
a(t)dt = 1, ∀i = 1, . . . , n
,
where ti are as defined in (3.3).
Proposition 3.4 Assume that the diffusion matrix σc is uniformly elliptic.Then for any a(t) ∈ Γ
(∇v(x))∗ = E
[φ(Xt1 , . . . , Xtn)
∫ T
0
a(t)(σ−1
c (Xct )Yt
)∗dWt
].
Proof. First assume that φ is continuously differentiable with bounded gradient.In this case it is possible to “differentiate inside the expectation”.4 We will
4The proof of this claim is in Fournie et al. (1999) built on the fact that almost surelyconvergence together with uniform integrability implies convergence in L1 norm. The almostsurely convergence holds true in the present treatment exactly as stated in the continuous
case. The uniform integrability follows from the the boundedness of ∂XT∂Xc
Tand the fact that
the map x 7→ Xt is a.s. continuous even in the presence of jumps.
12
denote by ∇iφ(Xt1 , . . . , Xtn) the gradient of φ with respect to Xti
, and by ∂Xti
∂xthe (d×d) matrix of derivatives of the d–dimensional random variable Xti
withrespect to its initial condition.
Remember that from the separability assumption (Assumption 3.1) XdT does
not depend on x so that
∇v(x) = E
[n∑
i=1
∇iφ(Xt1 , . . . , Xtn)∂Xti
∂x
]
= E
[n∑
i=1
∇iφ(Xt1 , . . . , Xtn))
∂Xti
∂Xcti
Yti
]. (3.4)
We want to rewrite Ytiin terms of the Malliavin derivative for Xti
using (3.2),but to do that we must also use the chain rule (Theorem 2.5). We have
D(0)t Xti
=∂Xti
∂Xcti
D(0)t Xc
ti=
∂Xti
∂Xcti
YtiY −1
t σc(Xct ) 1t≤ti.
For any a(t) ∈ Γ we can express ∂Xti
∂Xcti
Yti as
∂Xti
∂Xcti
Yti =∫ T
0
a(t)(D(0)t Xti)σ
−1c (Xc
t )Ytdt.
Inserting in (3.4) yields
∇v(x) = E
[∫ T
0
n∑i=1
∇iφ(Xt1 , . . . , Xtn)(D(0)
t Xti)a(t)σ−1
c (Xct )Ytdt
],
We know from Lemma 3.2 that φ(Xt1 , . . . , Xtn) ∈⋂d+m
j=1 D(j) since φ is contin-
uously differentiable with bounded gradient and Xt ∈⋂d+m
j=1 D(j) is separable.We can thus use the chain rule again to get
∇v(x) = E
[∫ T
0
(D
(0)t φ(Xt1 , . . . , Xtn
))
a(t)σ−1c (Xc
t )Ytdt
].
Since the diffusion matrix σc is uniformly elliptic by assumption we can deducethat a(t)σ−1
c (Xct )Yt ∈ L2(Ω × [0, T ]). We can therefore use the integration by
parts formula (Theorem 2.6) to get
(∇v(x))∗ = E
[φ(Xt1 , . . . , Xtn
)∫ T
0
a(t)(σ−1
c (Xct )Yt
)∗dWt
].
The fact that the class of continuously differentiable functions with boundedgradient is dense in L2 can be used, exactly as in Fournie et al. (1999), to provethe general result for φ ∈ L2.
Remark. The above argument can easily be repeated to get derivatives of higherorder.
13
Remark. Proposition 3.4 is known as the Bismut–Elworthy formula. See Elwor-thy and Li (1994) for an alternative proof in the continuous diffusion case.
We see that the discontinuities do not appear in the stochastic weight. How-ever, the fact that the payoff function is evaluated for the full price processensures that the sensitivity commonly known as delta does depend on the jumpparameters.
3.3 Variations in the diffusion coefficient
Calculating the stochastic weight for what is commonly known as vega is a bitmore involved than the previous Greeks. It is here the need for a Skorohodintegral arises and we use Proposition 2.10 to interpret the result.
As in Section 3.1 we need to define the perturbed process with respect tothe property under investigation; in this case the diffusion coefficient:
dXεt = b(Xε
t−)dt +(σ(Xε
t−) + εγ(Xεt−))dWt +
m∑k=1
αk(Xεt−)(dN
(k)t − λkdt)
Xε0 = x,
where again ε is a scalar and γ is a continuously differentiable function withbounded derivative. We will also need to introduce the variation process withrespect to the parameter ε
dZεt = b′(Xε
t−)Zεt−dt +
d∑i=1
(σ′i(X
εt−) + εγ′i(X
εt−))Zε
t−dW(i)t + γ(Xε
t−)dWt
+m∑
k=1
α′k(Xεt−)Zε
t−(dN(k)t − λkdt), Zε
0 = 0n,
so that ∂Xεt
∂ε = Zεt . Further, we define the set Γn, of square integrable functions
in R, as
Γn =
a ∈ L2([0, T )) :
∫ ti
ti−1
a(t)dt = 1, ∀i = 1, . . . , n
.
In this setting Proposition 3.5 is the analog of Proposition 3.3.
Proposition 3.5 Assume that the diffusion matrix σc is uniformly elliptic and
that for βti=(
∂Xti
∂Xcti
Yti
)−1
Zti, i = 1, . . . , n we have σ−1
c (Xc)Y β ∈ Dom δ(0). For
vε(x) defined asvε(x) = E
[φ(Xε
t1 , . . . , Xεtn
)],
we have for any a ∈ Γn
∂
∂εvε(x)
∣∣∣∣ε=0
= E[φ(Xt1 , . . . , Xtn
)δ(0)(σ−1
c (Xc· )Y·β·
)],
where
βt =n∑
i=1
a(t)(βti − βti−1
)1t∈[ti−1,ti),
14
for t0 = 0. Furthermore, if β ∈ D(0) the Skorohod integral can be calculatedaccording to
δ(0)(σ−1
c (Xc· )Y·β·
)=
n∑i=1
β∗ti
∫ ti
ti−1
a(t)(σ−1
c (Xct )Yt
)∗dWt
−∫ ti
ti−1
a(t)Tr((D(0)
t βti)σ−1
c (Xct )Yt
)dt
−∫ ti
ti−1
a(t)(σ−1
c (Xct )Ytβti−1
)∗dWt
Proof. As in the proof of Proposition 3.4 we may assume φ to be continuouslydifferentiable with bounded gradient, and we can take the differential under theexpectation to get
∂
∂εvε(x) = E
[n∑
i=1
∇iφ(Xεt1 , . . . , X
εtn
)Zεti
]. (3.5)
Next we define βt =(
∂Xt
∂XctYt
)−1
Zt =(
∂Xt
∂XctYt
)−1
Zε=0t , and proceed as in the proof
of Proposition 3.4. Because of (3.2) and since a ∈ Γn we can write∫ T
0
∂Xti
∂Xcti
(D(0)t Xc
ti)σ−1
c (Xct )Ytβtdt =
∫ ti
0
∂Xti
∂Xcti
Ytiβtdt
=∂Xti
∂Xcti
Yti
i∑j=1
∫ tj
tj−1
a(t)(βtj
− βtj−1
)dt
=∂Xti
∂Xcti
Ytiβti = Zti .
Inserting into (3.5) we get, by using the chain rule and Lemma 3.2,
∂
∂εvε(x)
∣∣∣∣ε=0
= E
[∫ T
0
n∑i=1
∇iφ(Xεt1 , . . . , X
εtn
)∂Xti
∂Xcti
(D(0)t Xc
ti)σ−1
c (Xct )Ytβtdt
]
= E
[∫ T
0
(D(0)t φ(Xε
t1 , . . . , Xεtn
))σ−1c (Xc
t )Ytβtdt
].
By assumption and the linearity of the Skorohod integral σ−1c (Xc)Y β ∈ Dom δ(0).
So,∂
∂εvε(x)
∣∣∣∣ε=0
= E[φ(Xε
t1 , . . . , Xεtn
) δ(0)(σ−1
c (Xc· )Y·β·
)],
and Proposition 2.10 can be used to calculate the Skorohod integral.
Remark. The assumption of σ−1c (Xc)Y β ∈ Dom δ(0) might seem restrictive at
first, but it can be shown that important examples do satisfy the assumption.See the Appendix for the explicit calculations in the case of the Heston stochasticvolatility model.
15
3.4 Variations in the jump intensity
The stochastic weight for variations in jump intensity is derived using the sametechnique as for variations in drift coefficient and it is given by the followingresult.
Proposition 3.6 Suppose that the diffusion matrix σ is uniformly elliptic. Forj = 1, . . . ,m we have
∂v
∂λj= E
[φ(Xt1 , . . . , Xtn
)
(N
(j)T
λj− T −
∫ T
0
(σ−1(Xt−)αj(Xt−))∗dWt
)].
Proof. The argument will be carried out for the first of the m Poisson processesto simplify the notation. Consider the perturbed process
dXεt = b(Xε
t−)dt + σ(Xεt−)dWt + α1(Xε
t−)(dN εt − (λ1 + ε)dt)
+m∑
k=2
αk(Xεt−)(dN
(k)t − λkdt), Xε
0 = x,
where ε is a positive deterministic parameter and N εt is a Poisson process with
intensity λ1 + ε. As in the proof of Proposition 3.3 we note that by changingmeasure we can write
E[φ(Xε
t1 , . . . , Xεtn
)]
= E [M εT φ(Xt1 , . . . , Xtn
)] ,
where M εt = MW
t MNt is the Radon-Nikodym derivative for which the altered
drift due to the perturbation is controlled for by MWt and the increased jump
intensity is governed by MNt . Explicitly,
MWt = exp
(−ε
∫ T
0
(σ−1(Xt−)α1(Xt−))∗dWt −ε2
2
∫ T
0
‖σ−1(Xt−)α1(Xt−)‖2ds
)
MNt =
(1 +
ε
λ1
)N (1)t
exp(−ε t).
Note that we can write
M εT = 1− ε
∫ T
0
MWt (σ−1(Xt−)α1(Xt−))∗dWt +
ε
λ1
∫ T
0
MNt (dN
(1)t − λ1dt),
which implies that
∂v
∂λj= lim
ε→0E
[φ(Xε
t1 , . . . , Xεtn
)− φ(Xt1 , . . . , Xtn)
ε
]= lim
ε→0E
[φ(Xt1 , . . . , Xtn)
(M ε
T − 1ε
)]= E
[φ(Xt1 , . . . , Xtn
)
(∫ T
0
1λ1
(dN(1)t − λ1dt)−
∫ T
0
(σ−1(Xt−)α1(Xt−))∗dWt
)].
Remark. As was the case for variations in the drift coefficient the above resultholds true even for more general path–dependent claims and non–separable pricedynamics.
16
3.5 Variations in the jump amplitude
To derive a stochastic weight for the sensitivity to the amplitude parameter αwe adopt the same technique as in the proof of Proposition 3.5—the diffusioncoefficient result. To that end, consider the perturbed process
dXεt = b(Xε
t−)dt + σ(Xεt−)dWt + (α1(Xε
t−) + εγ(Xεt−))(dN
(1)t − λ1dt)
+m∑
k=2
αk(Xεt−)(dN
(k)t − λkdt), Xε
0 = x,
where ε is a deterministic parameter and γ is a continuously differentiable func-tion with bounded derivative. Again, for notational simplicity, we considervariations in the first of the m jump amplitudes. The variation process withrespect to the parameter ε becomes
dZεt = b′(Xε
t−)Zεt−dt +
d∑i=1
σ′i(Xεt−)Zε
t−dW(i)t
+(α′1(Xεt−) + εγ′(Xε
t−))Zεt−(dN
(1)t − λ1dt)
+m∑
k=2
α′k(Xεt−)Zε
t−(dN(k)t − λkdt) + γ(Xε
t−)(dN(1)t − λ1dt), Zε
0 = 0.
The stochastic weight in this context is obtained similarly to the vega weight.The statement of the following proposition is therefore almost identical to Propo-sition 3.5.
Proposition 3.7 Assume that the diffusion matrix σc is uniformly elliptic and
that for βti =(
∂Xti
∂Xcti
Yti
)−1
Zti , i = 1, . . . , n we have σ−1c (Xc)Y β ∈ Dom δ(0). For
vε(x) defined asvε(x) = E
[φ(Xε
t1 , . . . , Xεtn
)],
we have for any a ∈ Γn
∂
∂εvε(x)
∣∣∣∣ε=0
= E[φ(Xt1 , . . . , Xtn)δ(0)
(σ−1
c (Xc· )Y·β·
)],
where
βt =n∑
i=1
a(t)(βti
− βti−1
)1t∈[ti−1,ti),
for t0 = 0. Furthermore, if β ∈ D(0) the Skorohod integral can be calculatedaccording to
δ(0)(σ−1
c (Xc· )Y·β·
)=
n∑i=1
β∗ti
∫ ti
ti−1
a(t)(σ−1
c (Xct )Yt
)∗dWt
−∫ ti
ti−1
a(t)Tr((D(0)
t βti)σ−1c (Xc
t )Yt
)dt
−∫ ti
ti−1
a(t)(σ−1
c (Xct )Ytβti−1
)∗dWt
17
Proof. Exactly the same proof as for Proposition 3.5 with redefined βt and Zt.
Remark. As long as αk 6= 0 for k = 1, . . . ,m we have βti ∈⋂d+m
j=1 D(j) for thestochastic volatility and interest rate models as presented earlier. See the Ap-pendix for the explicit calculations in the case of the Heston stochastic volatilitymodel.
4 Examples
In this section we will explicitly derive examples of stochastic weights for acouple of different equity models. In particular we will look at jump–diffusionversions of the Black–Scholes model and the Heston model.
We start by proving that the separability assumption (Assumption 3.1) holdsfor the stochastic volatility model
dX(1)t = b1
(X
(2)t , . . . , X
(d)t
)X
(1)t− dt + σ1
(X
(2)t , . . . , X
(d)t
)X
(1)t− dWt
+∑m
k=1(αk − 1)X(1)t− (dN
(k)t − λkdt), X
(1)0 = x1
dX(i)t = bi
(X
(2)t , . . . , X
(d)t
)dt + σi
(X
(2)t , . . . , X
(d)t
)dWt,
X(i)0 = xi, i = 2, . . . , d,
(4.1)
where bi and σi have bounded continuous derivatives for i = 1, . . . , d, andαk, k = 1, . . . ,m are deterministic constants. We introduce the d-dimensionalcontinuous process Xc
t defined by
dXc (1)t =
(b1
(X
c (2)t , . . . , X
c (d)t
)+∑m
k=1 λk(1− αk))
Xc (1)t dt
+σ1
(X
c (2)t , . . . , X
c (d)t
)X
c (1)t− dWt X
c (1)0 = x1
dXc (i)t = bi
(X
c (2)t , . . . , X
c (d)t
)dt + σi
(X
c (2)t , . . . , X
c (d)t
)dWt,
Xc (i)0 = xi, i = 2, . . . , d,
(4.2)
and the d-dimensional discontinuous process
X(1)t = α
N(1)t
1 . . . αN
(m)t
m Xc (1)t
X(i)t = X
c (i)t , i = 2, . . . , d.
(4.3)
It is clear that (X(2), . . . , X(d)) and (X(2), . . . , X(d)) are indistinguishable dueto pathwise uniqueness of the solutions to the SDEs, and by applying the Itoformula to X
(1)t we find that X
(1)t = X
(1)t satisfies (4.2).
We summarise the result in the following Lemma.
Lemma 4.1 Let Xt, Xct and Xt be as defined in (4.1),(4.2) and (4.3)respectively.
Then Xt = Xt a.s.
Next we consider a jump–diffusion version of the Black–Scholes model. Let theprice process follow
dXt = µXt−dt + σXt−dWt + (α− 1)Xt−(dNt − λdt), X0 = x,
18
where µ, σ and α are constants and the intensity of the Poisson process is λ.The price process is clearly separable since it is a special case of the stochasticvolatility model defined in (4.1). In particular we see that the continuous priceprocess contribution Xc
t is identical to the Black–Scholes price process withmodified drift. By straight forward derivations we conclude that for a contingentclaim
v = E[e−rT φ(XT )
], (4.4)
we have
RhoBS = ∂v∂r = E
[e−rT φ(XT )WT
σ
]− E
[Te−rT φ(XT )
]DeltaBS = ∂v
∂x = E[e−rT φ(XT ) WT
xσT
]GammaBS = ∂2v
∂x2 = E[e−rT φ(XT ) 1
x2σT
(W 2
T
σT −WT − 1σ
)]VegaBS = ∂v
∂σ = E[e−rT φ(XT )
(W 2
T
σT −WT − 1σ
)]LambdaBS = ∂v
∂λ = E[e−rT φ(XT )
(NT−λT
λ − (α−1)WT
σ
)]AlphaBS = ∂v
∂α = E[e−rT φ(XT )
(NT
α − λT)
WT
σT
]Rho might need some further explanation. Rho is defined as the sensitivity ofthe claim (4.4) with respect to interest rate, and here the interest rate appearsboth in the discount factor and in the drift. Analogous to section 3.1 we definethe Rho as the sensitivity of the claim with respect to the factor ε, as in theperturbed interest rate r+εγ, in the point ε = 0. Two examples involving Deltaare shown in Figure 1 and Figure 2.
The jump–diffusion version of the Heston model
dX(1)t = rX
(1)t− dt +
√X
(2)t X
(1)t− dW
(1)t + (α− 1)X(1)
t− (dNt − λdt), X(1)0 = x1
dX(2)t = κ(θ −X
(2)t )dt + σ
√X(2)dW
(2)t , X
(2)0 = x2,
with E[dW(1)t dW
(2)t ] = ρdt is also a special case of (4.1) and the separability
assumption therefore holds. X(1)t is the process for the security on which the
contingent claim is written, and X(2)t is a process governing the volatility. The
price of the claim in this setting is
v = E[e−rT φ(X(1)
T )],
and the continuous process Xct is identical to the price process in the origi-
nal Heston model, but again with modified drift. The stochastic weights forcalculating the Greeks for the jump–diffusion version are found to be:
19
Figure 1: Monte Carlo simulation of DeltaBS for a call option using finite differenceapproximation and Malliavin weighting. The model parameters are x = 100, Strike =100, r = 0.05, σ = 0.3, α = 0.5 and λ = 0.1.
RhoH = E
[e−rT φ(X(1)
T )(∫ T
0dW
(1)t√
X(2)t
− ρ√1−ρ2
∫ T
0dW
(2)t√
X(2)t
)]−E[Te−rT φ(X(1)
T )]
DeltaH = E
[e−rT φ(X(1)
T ) 1x1T
(∫ T
0dW
(1)t√
X(2)t
− ρ√1−ρ2
∫ T
0dW
(2)t√
X(2)t
)]
GammaH = E
[e−rT φ(X(1)
T ) 1x21T 2
((∫ T
0dW
(1)t√
X(2)t
− ρ√1−ρ2
∫ T
0dW
(2)t√
X(2)t
)2
− 11−ρ2
∫ T
0dt
X(2)t
− T
(∫ T
0dW
(1)t√
X(2)t
− ρ√1−ρ2
∫ T
0dW
(2)t√
X(2)t
))]
Vegax2H = E
[e−rT φ(X(1)
T ) 1T
(∫ T
0Y
(1,2)t
Xc(1)t
√X
(2)t
dW(1)t
+∫ T
0
1σ Y
(2,2)t −ρX
c(1)t Y
(1,2)t√
1−ρ2Xc(1)t
√X
(2)t
dW(2)t
)]
20
Figure 2: Monte Carlo simulation of DeltaBS for a digital option using finite differenceapproximation and Malliavin weighting. The model parameters are x = 100, Strike =100, r = 0.05, σ = 0.3, α = 0.5 and λ = 0.1.
VegaPH = E
[e−rT φ(X(1)
T ) 1T
((W
(1)T −
∫ T0
√X
(2)t dt
)(∫ T
0dW
(1)t√
X(2)t
− ρ√1−ρ2
∫ T
0dW
(2)t√
X(2)t
)−∫ T
0dt√X
(2)t
)]
LambdaH = E
[e−rT φ(X(1)
T )(
NT
λ − T
−(α− 1)(∫ T
0dW
(1)t√
X(2)t
− ρ√1−ρ2
∫ T
0dW
(2)t√
X(2)t
))]
AlphaH = E
[e−rT φ(X(1)
T )((
NT
αT − λ)(∫ T
0dW
(1)t√
X(2)t
− ρ√1−ρ2
∫ T
0dW
(2)t√
X(2)t
))]The derivations are placed in the Appendix and examples are shown in Figure 3and Figure 4, where the stochastic Euler scheme has been used in the MonteCarlo simulation.
The two vegas are different in the sense that Vegax2H is the sensitivity to
changes in the initial value of the volatility process and VegaPH is the sensitivity
to a perturbation as in section 3.3. Since the initial value of the volatility processis just another parameter coming from the calibration Vegax2
H is perhaps not asinteresting as VegaP
H which is the analogue of VegaBS.Looking at the convergence plots it is evident that the finite difference ap-
proximation performs better for the call option, but the Malliavin weight ap-proach performs better for the digital. The reason for this is that the stochasticweight adds some randomness to the expression to be simulated and for the call
21
Figure 3: Monte Carlo simulation of DeltaH for a call option using finite differenceapproximation and Malliavin weighting. The model parameters are x1 = 100, Strike =100, x2 = 0.04, r = 0.05, κ = 1, θ = 0.04, σ = 0.04, ρ = −0.8, α = 0.5 and λ = 0.1.
Figure 4: Monte Carlo simulation of DeltaH for a digital option using finite differenceapproximation and Malliavin weighting. The model parameters are x1 = 100, Strike =100, x2 = 0.04, r = 0.05, κ = 1, θ = 0.04, σ = 0.04, ρ = −0.8, α = 0.5 and λ = 0.1.
22
option this effect worsens the convergence. However, for the discontinuous pay-off function of the digital option, the finite difference approximation produceslarge errors since the contributions to be averaged are either zero or one. Inthis case the Malliavin weight and payoff is still smooth and the effect of addedrandomness through the stochastic weight is not prominent.
A Skorohod integral for Fut when ut previsible
Claim: For Fut in L2(Ω× R+) where
F = E[F ] +∞∑
n=1
∑0≤i1,...,in≤m
L(i1,...,in)n (fi1,...,in
),
and ut previsible, we have
δ(l)(Fu) =∫ ∞
0
E[F ]ut1dGl(t1) +∞∑
n=1
∑0≤i1,...,in≤m
n+1∑k=1
∫ ∞
0
. . .
∫ tk−1−
0
∫ tk−
0
∫ tk+1−
0
. . .
∫ t2−
0
fi1,...,in(t1, . . . , tk−1, tk+1, . . . , tn+1)dGi1(t1)
. . . dGik−1(tk−1)utkdGl(tk)dGik
(tk+1) . . . dGin(tn+1),
in the sense that Hu ∈ Dom δ(l) if the right hand side is in L2(Ω× R+).
Proof: By linearity of the Skorohod integral it is enough to prove the claim forthe case when F is a single iterated integral. Take any H ∈ D(l).
E
[∫ ∞
0
(D(l)t H)Futdt
]
=∫ ∞
0
E
[( ∞∑n=1
∑0≤i1,...,in≤m
n∑k=1
1ik=l1Σ(k)n (t)
∫ ∞
0
. . .
∫ tk+2−
0
∫ tk−
0
. . .
∫ t2−
0
hi1,...,in(. . . , t, . . . )dGi1(t1) . . . dGik−1(tk−1)dGik+1(tk+1) . . . dGin(tn)
)
×
(∫ ∞
0
∫ tq−
0
. . .
∫ t2−
0
fj1,...,jq(. . . )dGj1(t1) . . . dGjq−1(tq−1)dGjq
(tq)
)ut
]dt
= E
[ ∞∑n=1
∑0≤i1,...,in≤m
n∑k=1
1ik=l
(∫ ∞
0
. . .
∫ tk+2−
0
∫ tk+1−
0
∫ t−
0
. . .
∫ t2−
0
hi1,...,in(. . . , t, . . . )dGi1(t1)
. . . dGik−1(tk−1)utdt dGik+1(tk+1) . . . dGin(tn)
)
×
(∫ ∞
0
∫ tq−
0
. . .
∫ t2−
0
fj1,...,jq(. . . )dGj1(t1) . . . dGjq−1(tq−1)dGjq
(tq)
)].
23
For the D(l)t H contribution, the integrals outside the dt integral can be matched
up with the outer integrals in the F–contribution with an isometry argument:
E
[∫ ∞
0
(D(l)t H)Futdt
]
= E
[ ∞∑n=1
∑0≤i1,...,in≤m
n∑k=1
1ik=l1(ik+1,...,in)=(jq−n+k+1,...,jq)
λik+1 . . . λin
∫ ∞
0
∫ tn−
0
. . .
∫ tk+1−
0
(∫ tk−
0
. . .
∫ t2−
0
hi1,...,in(. . . )dGi1(t1) . . . dGik−1(tk−1)
)
×
(∫ tq−n+k+1−
0
. . .
∫ t2−
0
fj1,...,jq (. . . )dGj1(t1) . . . dGjq−n+k(tq−n+k)
)
utkdtk . . . dtn−1dtn
].
By the same isometry argument we can now transform back into stochasticintegrals.
E
[∫ ∞
0
(D(l)t H)Futdt
]
= E
[ ∞∑n=1
∑0≤i1,...,in≤m
n∑k=1(∫ ∞
0
. . .
∫ tk+1−
0
. . .
∫ t2−
0
hi1,...,in(. . . )dGi1(t1) . . . dGl(tk) . . . dGin
(tn)
)
×
(∫ ∞
0
. . .
∫ tq−n+k+2−
0
. . .
∫ t2−
0
fj1,...,jq(. . . )dGj1(t1)
. . . utq−n+k+1dGl(tq−n+k+1) . . . dGjq (tq+1)
)]
= E
[ ∞∑n=1
∑0≤i1,...,in≤m
(∫ ∞
0
∫ tn−
0
. . .
∫ t2−
0
hi1,...,in(. . . )dGi1(t1)
. . . dGin−1(tn−1)dGin(tn)
q+1∑k=1
∫ ∞
0
. . .
∫ tq+1−
0
. . .
∫ t2−
0
fj1,...,jq(. . . )dGj1(t1) . . . utk
dGl(tk) . . . dGjq(tq+1)
)].
B Stochastic weights in the Heston model
We write the pricing process in matrix form as
dXt = b(Xt−) dt + a(Xt−) dWt + c(Xt−) (dNt − λdt)
24
where
b(Xt−) =
rX(1)t−
κ(θ −X(2)t )
, c(Xt−) =
((α− 1)X(1)
t−
0
)and
a(Xt−) =
√
X(2)t X
(1)t− 0
σ
√X
(2)t ρ σ
√X
(2)t
√1− ρ2
,
with the notation used in section 4. The inverse of the diffusion matrix becomes
a(Xt−)−1 =1
σX(1)t− X
(2)t
√1− ρ2
σ
√X
(2)t
√1− ρ2 0
−σ
√X
(2)t ρ
√X
(2)t X
(1)t−
.
Stochastic weight for Rho in the Heston model
We perturb the original drift with γ(x) = (x1, 0)∗ to get the perturbed process
dXεt = (b(Xε
t−) + εγ(Xεt−)) dt + a(Xε
t−)) dWt + c(Xεt−) (dNt − λdt).
Now it is clear that the row–vector that should be integrated with respect tothe Brownian motion is 1√
X(2)t
, − ρ√X
(2)t
√1− ρ2
,
and we get the expression in section 4
Stochastic weight for Delta in the Heston model
The first variation process is in this case a matrix process
dYt = b′(Xct )Ytdt + a′1(X
ct )YtdW
(1)t + a′2(X
ct )YtdW (2), Y0 = I,
with
b′(Xct−) =
(r + λ(1− α) 0
0 −κ
)
a′1(Xct−) =
√
Xc(2)t
12
Xc(1)t√X
c(2)t
0 12
σρ√X
(c2)t
a′2(Xct−) =
0 0
0 12
σ√
1−ρ2√X
c(2)t
.
25
For the Delta we are interested in the first row of the matrix (a−1(Xct )Yt)∗
integrated over the 2-dimensional Brownian motion. From the above we candeduce that the row vector of interest is 1
Xc(1)t
√X
c(2)t
Y(1,1)t , − ρ√
1− ρ2
1
Xc(1)t
√X
c(2)t
Y(1,1)t
.
The first–row first–column element in the first variation process can be seen tobe
Y(1,1)t =
1x
Xc(1)t ,
since
dY(1,1)t =(r + λ(1− α))Y (1,1)
t dt +√
Xc(2)t Y
(1,1)t dW
(1)t +
12
Xc(1)t√X
c(2)t
Y(2,1)t dW
(1)t
Y(1,1)0 = 1,
and Y(2,1)t = 0 due to the fact that X
c(2)t does not depend explicitly on X
c(1)t
and in particular not its initial value. This leaves us with the expression forDeltaH presented in section 4.
Stochastic weight for Vegax2 in the Heston model
The interpretation of Vega as the sensitivity with respect to initial volatilitylevel follows directly from the discussion above. In this case the row–vector tobe integrated over the Brownian motion is 1
Xc(1)t
√X
c(2)t
Y(1,2)t ,
1√1− ρ2
1
Xc(1)t
√X
c(2)t
[1σ
Y(2,2)t − ρX
c(1)t Y
(1,2)t
] .
Stochastic weight for VegaP in the Heston model
We perturb the original diffusion matrix with γ to get the perturbed process
dXεt = b(Xε
t−) dt + (a(Xεt−) + εγ(Xε
t−)) dWt + c(Xεt−) (dNt − λdt)
where
γ(x) =(
x1 00 0
).
Again using the fact that X(2)t does not depend explicitly on X
(1)t we can de-
duce that the variation process with respect to ε, Zεt , has a vanishing second
26
component. We write Z(2)t = 0. Furthermore,
βt = Y −1t
∂Xct
∂XtZt
=1
Y(1,1)t Y
(2,2)t
(Y
(2,2)t −Y
(1,2)t
0 Y(1,1)t
) ∂Xc(1)t
∂X(1)t
∂Xc(1)t
∂X(2)t
∂Xc(2)t
∂X(1)t
∂Xc(2)t
∂X(2)t
(Z(1)t
0
)
=
(Z
(1)t
∂Xc(1)t
∂X(1)t
/ Y(1,1)t
0
).
Note that by the Ito formula
Z(1)t = X
(1)t
(W
(1)t −
∫ t
o
√X
(2)s ds
),
so that
β(1)t = x1
(W
(1)t −
∫ t
0
√X
(2)s ds
).
Except for expressions already derived, the only thing we need in order to cal-culate the weight for VegaP is the Malliavin derivative of βt in the directionof the Brownian motion. Using the chain rule (Theorem 2.5) on a sequence ofcontinuously differentiable functions with bounded derivatives approximating√
X(2)s , together with (3.2) we get
D(0)t β
(1)T = x1
((1, 0)− 1
2
∫ T
0
1√X
(2)s
D(0)t X(2)
s ds
)
= x1
((1, 0)− 1
2
∫ T
t
σ
√X
(2)t√
X(2)s
Y(2,2)s
Y(2,2)t
(ρ,√
1− ρ2)ds
).
To arrive at the expression presented in section 4 we note that
Tr((D(0)
t βT )a−1Yt
)=
1√X
(2)t
.
Stochastic weight for Lambda in the Heston model
The quantity we need to expand in this case is a−1(Xt−)c(Xt−). From what wasstated in the beginning of this section it is clear that the vector to be integratedover Brownian motion is
(α− 1)
1√X
(2)t
, − ρ√X
(2)t
√1− ρ2
.
Stochastic weight for Alpha in the Heston model
27
The perturbed process is chosen as
dXεt = b(Xε
t−) dt + a(Xεt−) dWt + (c(Xε
t−) + εγ(Xεt−)) (dNt − λdt),
with
γ(x) =(
x1
0
).
As was the case for VegaP X(2)t does not depend explicitly on X
(1)t so we get
βt = Y −1t
∂Xct
∂XtZt
=1
Y(1,1)t Y
(2,2)t
(Y
(2,2)t −Y
(1,2)t
0 Y(1,1)t
) ∂Xc(1)t
∂X(1)t
∂Xc(1)t
∂X(2)t
∂Xc(2)t
∂X(1)t
∂Xc(2)t
∂X(2)t
(Z(1)t
0
)
=
(Z
(1)t
∂Xc(1)t
∂X(1)t
/ Y(1,1)t
0
).
Note that by the Ito formula
Z(1)t = X
(1)t
(Nt
α− λt
),
so that
β(1)t = x1
(Nt
α− λt
).
The Skorohod integral expansion in Proposition 2.10 is easy to compute inthis case since D
(0)t βt = 0. Adding up the results we arrive at the formula in
Section 4.
References
V. Bally, M.-P. Bavouzet, and M. Massaoud. Integration by part formula forlocally smooth laws and applications to sensitivity computations. Report5567, INRIA, Rocquencourt, France, 2005.
M.-P. Bavouzet and M. Massaoud. Computation of Greeks using Malliavin’scalculus in jump-type market models. Report 5482, INRIA, Rocquencourt,France, 2005.
K. Bichteler, J.B. Graveraux, and J. Jacod. Malliavin calculus for processeswith jumps. Gordon and Breach, 1987.
J.-M. Bismut. Calcul des variations stochastique et processus de saut. Z.Wahrscheinlichkeitstheorie ver. Geb., 63:147–235, 1983.
E.A. Carlen and E. Pardoux. Differential calculus and integration by parts onPoisson space. In S. Albeverio et al., editor, Stochastics, Algebra and Analysisin Classical and Quantum Dynamics. Kluwer Academic Publishers, 1990.
Y. El-Khatib and N. Privault. Computations of greeks in a market with jumpsvia the malliavin calculus. Finance and Stochastics, 8(2):161–179, 2004.
28
K.D. Elworthy and X.-M. Li. Formulae for the derivatives of heat semigroups.Journal of Functional Analysis, 125:252–286, 1994.
B. Forster, E. Lutkebohmert, and J. Teichmann. Calculation of the greeks forjump-diffusions. Financial and Actuarial Mathematics, Technical Universityof Vienna, 2005.
E. Fournie, J.-M. Lasry, J. Lebuchoux, P.-L. Lions, and N. Touzi. Applica-tions of Malliavin calculus to Monte Carlo methods in finance. Finance andStochastics, 3(4):391–412, 1999.
P. Glasserman and D.D. Yao. Some guidelines and guarantees for commonrandom numbers. Management Science, 38(6):884–908, 1992.
I. Karatzas and S.E. Shreve. Methods of Mathematical Finance. Springer–Verlag, 1998.
J. A. Leon, J. L. Sole, F. Utzet, and J. Vives. On Levy processes, Malliavincalculus and market models with jumps. Finance and Stochastics, 6(2):197–225, 2002.
D. Nualart. The Malliavin Calculus and Related Topics. Springer–Verlag, 1995.
D. Nualart and W. Schoutens. Chaotic and predictable representations for Levyprocesses. Stochastic Processes and their Applications, 90(1):109–122, 2000.
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