We show that the Heston volatility or equivalently the Cox-Ingersoll-Ross process is Malliavin differentiable and give an explicit expression for the derivative. This result assures the applicability of Malliavin calculus in the framework of the Heston stochastic volatility model and the Cox-Ingersoll-Ross model for interest rates.Malliavin calculus, stochastic volatility models, Heston model, Cox-Ingersoll-Ross process

Christian-Olivier Ewald

Elisa Alòs

English

Research Papers in Economics

A Note on the Malliavin diﬀerentiability
of the Heston Volatility
Elisa Alos1, Christian-Oliver Ewald2
1 Department of Economics, University of Pompeu Fabra, c/Ramon Trias
fargas, 25-27, 08005 Barcelona, Spain ( e-mail : elisa.alos@upf.edu )
2 School of Mathematics, Woodhouse Lane, University of Leeds, LS2 9JT,
Leeds, UK ( e-mail : ewald@maths.leeds.ac.uk )
Abstract
We show that the Heston volatility or equivalently the Cox-Ingersoll-
Ross process satisfying
dvt = κ(θ − vt)dt + ν
√
vtdWt
is Malliavin diﬀerentiable and give an explicit expression for the deriv-
ative. This result assures the applicability of Malliavin calculus in
the framework of the Heston stochastic volatility model and the Cox-
Ingersoll-Ross model for interest rates.
Keywords: Malliavin calculus, stochastic volatility models, Heston model,
Cox-Ingersoll-Ross process
AMS Subject Classiﬁcation: 60H07,65H10, 65H30, 91B28, 91B70
JEL Subject Classiﬁcation: G12, G19, C19, E431 Introduction
In recent years, Malliavin calculus has appeared as a major tool in both
theoretical and computational mathematical ﬁnance. This fact is document
by the large amount of published work in this area, see [1], [3], [11], [16], [19]
for contributions on the theoretical aspects and [2],[4], [5],[6], [8],[10], [12],
[13], [14] for computational aspects. In all these publications, the assumption
on the possibly multidimensional diﬀusion process (Xt) which determines the
factors of the model, are very strict. In fact, the classical results assume that
the coeﬃcient functions β and σ in
dXt = β(Xt,t)dt + σ(Xt,t)dWt
have bounded derivatives of all orders and have linear growth at inﬁnity. In
some publications the assumption are even more strict. These assumptions
work ﬁne with the standard Black-Scholes model. Problems occur however
when one uses more advanced models, like the Heston stochastic volatility
model. In this model the stock price is given by the equation
dSt = St(bdt +
√
vtdBt) (1)
where (Bt) denotes a Brownian motion but in contrast to the standard Black-
Scholes model the volatility vt is itself a diﬀusion process, satisfying the
stochastic diﬀerential equation
dvt = κ(θ − vt)dt + ν
√
vtdWt (2)
where Wt denotes a possibly correlated second Brownian motion. As one
can see on ﬁrst glance, the coeﬃcient functions of this model do not satisfy
the standard assumptions. The square root function does not have bounded
derivatives. However the Heston stochastic volatility model is one of the
most popular among the stochastic volatility models, both from the theo-
retical and empirical point of view. Malliavin calculus in the framework of
the Heston stochastic volatility model has been considered in [10], but in this
article a direct application of the Malliavin derivative operator on the Heston
volatility has been avoided. Among the experts there was widely speculation
about whether the Heston volatility is Malliavin diﬀerentiable. In this article
we give an answer to this longstanding and signiﬁcant question. We give a
direct proof of the Malliavin diﬀerentiability of the Heston volatility and its
square root and give explicit expressions for their derivatives. This is a key
result, as it justiﬁes the applicability of Malliavin calculus in the framework
of the Heston stochastic volatility model and the Cox-Ingersoll-Ross interest
rate model. Our result turns out to be a powerful tool, many application can
1be found in the literature mentioned in the beginning of this introduction.
Given our result, these application can now be directly adapted to the Hes-
ton stochastic volatility model. This is the reason why we omit any explicit
application in this article. The same process as in equation (2) but classically
denoted with rt rather than vt has been used in interest rate theory in the
framework of the famous Cox-Ingersoll-Ross model [7]. Our results therefore
directly apply to this model too. However, here we leave this by side. In
our proof we use a result by Detemple et al. [9] which allows to represent
the Malliavin derivative of a diﬀusion in an elegant and computational ad-
vantageous way. Though Detemple’s result also uses strict assumptions such
as boundedness of all derivatives of the coeﬃcient functions, we are able to
generalize Detemple’s formula for the case of the Heston volatility.
2 The Heston volatility model and an ap-
proximating sequence
As mentioned in the introduction, the Heston stochastic volatility model
consists of a bond, which we do not specify in this article, a stock (St)
and the volatility process (vt) with dynamics speciﬁed in (1) and (2) in the
introduction. It is assumed that κ,θ and ν are positive constants. This
model was ﬁrst used by Heston in [15]. Heston computed explicit formulas
for the value of European calls in this model. In the following we consider
one ﬁxed probability space (Ω,F,P) on which there is deﬁned a Brownian
motion (Wt) and which is ﬁltered by the augmented and completed Brownian
ﬁltration. We also ﬁx an interval [0,T]. A standard assumption, when using
the Heston model is
2κθ ≥ ν
2. (3)
This is often called the Novikov condition. Given that v0 > 0 this condition
guarantees that the volatility process is always positive, i.e.
P({vt > 0 ∀ t > 0}) = 1. (4)
We assume that v0 > 0 and that the Novikov condition holds. It is then
possible to consider the square root process σt :=
√
vt. It follows from the
Itˆ o formula that this process satisﬁes
dσt =

κθ
2
−
ν2
8

1
σt
−
κ
2
σt +
ν
2
dWt. (5)
2We note that the Novikov condition implies in particular that the factor 
κθ
2 − ν2
8

appearing in the drift term of σt is positive. This will play a role
later. We will need the following Lemma to compare processes and to apply
the dominated convergence theorem in our approximation of the process σt.
The Lemma is a special case of the Comparison Lemma as it can be found
in [17], Chapter 5, Proposition 2.18.
Lemma 2.1. Consider the two Itˆ o processes
X
1
t = X0 +
Z t
0
β1(X
1
s)ds + kWt, (6)
X
2
t = X0 +
Z t
0
β2(X
2
s)ds + kWt, (7)
where β1,β2 : R → R are two measurable functions and k is a positive con-
stant. Assume that (6) and (7) have a strong solution and suppose that β1 ≥
β2and at least one of them satisﬁes a Lipschitz condition. Then X1
t ≥ X2
t
for all t, a.s.
In order to show in section 4, that σt is Malliavin diﬀerentiable we now
deﬁne an approximating sequence. This sequence shows to be useful in other
aspects of the Heston model as well. Let ε > 0 and Φε (x) be a continuously
diﬀerentiable function satisfying
Φε (x) =



1 if x ≥ 2ε
0 if x < ε
as well as Φε (x) ≤ 1 for all x ∈ R. We note that
Φ
0
ε (x) =



0 if x ≥ 2ε
0 if x < ε
Furthermore we deﬁne the function
Λε(x) = Φε(x)
1
x
with Λε(0) = 0. The function Λε(x) is bounded and continuously diﬀeren-
tiable with
Λ
0
ε (x) = Φ
0
ε (x)
1
x
− Φε (x)
1
x2 =



− 1
x2 if x ≥ 2ε
0 if x < ε
3Let us now deﬁne our approximations σε
t as the solutions of the stochastic
diﬀerential equations
dσ
ε
t =

κθ
2
−
ν2
8

Λε (σ
ε
t) −
κ
2
σ
ε
t +
ν
2
dWt, (8)
with σε
0 = σ0 for all ε > 0.
Proposition 2.1. For each t ∈ [0,T] the sequence σε
t converges to σt in
L2(Ω).
Proof. We use the dominated convergence theorem in order to obtain this
result. Let us ﬁrst prove that σε
t converges to σt point wise. This follows
from a standard localization argument. For each ε > 0 deﬁne a stopping
time τε via
τε(ω) := inf{t|σt(ω) ≤ ε}. (9)
Letting ε go to zero, the sequence of stopping times (τε) deﬁnes an increasing
sequence of stopping times, and it follows from equation (4) that limε→0 τε =
∞ a.s. Denoting with στε the process obtained from σ by stopping at τε,
then it follows from the choice of the function Λε(x) and equations (5) and
(8), that
σ
τ2ε
t = σ
ε
t ∀ t ≤ τ
ε.
Now, for ﬁxed t ∈ [0,T] letting ε go to zero one obtains that
lim
ε→0σ
ε
t = σt a.s.
Let us now prove that for each t ∈ [0,T] σε
t converges to σt in L2 (Ω). For
this let us consider the Ornstein-Uhlenbeck process ut satisfying u0 = σ0 and
dut = −
κ
2
ut +
ν
2
dWt.
The Novikov condition as well as 0 ≤ Φ(x) ≤ 1 for all choices of ε and x
imply that
−
κ
2
≤ −
κ
2
+

κθ
2
−
ν2
8

Λε (x) ≤ −
κ
2
+

κθ
2
−
ν2
8

1
x
.
It therefore follows from Lemma 2.1 that
ut ≤ σ
ε
t ≤ σt
4and hence |σε
t| ≤ |ut|+|σt|. The dominated convergence theorem implies the
desired convergence.
Corollary 2.1. For each t ∈ [0,T] the sequence vε
t = (σε
t)2 converges to the
Heston volatility vt in L1(Ω).
3 A short review on Malliavin calculus
Let us consider the cylindrical functional F : Ω → R given by
F = f (Wt1,...,Wtl)
where f ∈ C∞
b
 
Rl
is a smooth function with bounded derivatives of all
orders. Given h ∈ L2([0,T],R) we have that
R ·
0 h(s)ds ∈ C0([0,T],R) where
the dot indicates that the upper bound of the integral is taken as a variable.
The directional derivative of F in direction
R ·
0 h(s)ds at ω is given by
DhF(ω) :=
d
d

  
=0
f

Wt1(ω) +
Z t1
0
h(s)ds,...,Wtl(ω) +
Z tl
0
h(s)ds

=
m X
i=1
∂
∂xi
f (Wt1(ω),...,Wtl(ω)) ·
Z ti
0
h(s)ds.
Now for ﬁxed ω consider the linear bounded functional on L2([0,T]) given
by
h 7→ DhF(ω).
By the Riesz-representation theorem there is an element DF(ω) in
L2([0,T],R) such that
DhF(ω) =
Z T
0
DF(ω)(s)h(s)ds , ∀h ∈ L
2([0,T]).
In the following we denote DF(ω)(s) with DsF(ω). Let us now consider ω
as a variable. The assumption that f has bounded derivatives of all orders
ensures that for all p ≥ 1 we have DF ∈ Lp (Ω,L2([0,T])) when considered
as an L2([0,T]) valued functional in ω
Assume now that the functional F is not necessarily cylindrical but there
5exists a sequence of cylindrical functionals Fi such that (Fi) converges to F
in Lp(Ω) and (DFi) converges to G in Lp(Ω,L2([0,T])). Then we deﬁne
DF := G = lim
i→∞
DFi.
The Cameron-Martin Theorem shows that this is indeed well deﬁned and
that the operator
D : L
p(Ω) → L
p(Ω,L
2([0,T]))
deﬁned on the cylindrical functionals is closable. The closure is called the
Malliavin derivative operator and is denoted by D. The domain of D is the
closure of the cylindrical functionals under the norm
kFk1,p := kFkLp(Ω) + kDFkLp(Ω,(L2[0,T])).
It is denoted with D1,p. In this article we concentrate on the case p = 2. We
will later make use of the following chain-rule ( see [19], Lemma 2.1 ) :
Proposition 3.1. Let φ : R → R be a continuously diﬀerentiable function
and F ∈ D1,2. Then φ(F) ∈ D1,2 if and only if φ(F) ∈ L2(Ω) and φ0(F)DF ∈
L2(Ω × [0,T]) and in this case
Dtφ(F) = φ
0(F) · DtF.
The following formula for the Malliavin derivative of a diﬀusion is due to
Detemple et al. [9]
Proposition 3.2. Consider a diﬀusion process (Xt) satisfying the SDE
dXt = µ(Xt) + σ(Xt)dWt
and assume that the coeﬃcient functions satisfy µ ∈ C1(R) and σ ∈ C2(R)
as well as a linear growth condition and Lipschitz condition. Furthermore
assume that σ is positive and bounded away from zero. Then Xt ∈ D1,2 and
for θ ≤ t one has
DθXt = σ(Xt)exp
Z t
θ
µ
0(Xs) −
µ(Xs)σ0(Xs)
σ(Xs)
−
1
2
σ
00(Xs)σ(Xs)ds

.
64 The Malliavin derivative of the Heston volatil-
ity
In this section we will show that both the Heston volatility vt as well as it’s
square root σt belong to D1,2. To do this we use our approximating sequence
σε
t from section 2. We have the following lemma :
Lemma 4.1. σε
t ∈ D1,2 and Dθσε
t = ν
2 exp
nR t
θ
h
−κ
2 +

κm
2 − ν2
8

Λ0
ε (σε
s)
i
ds
o
Proof. This follows directly from Proposition 3.2.
We are now ready to proof our main result.
Theorem 4.1. We have σ ∈ D1,2 and Dθσt = ν
2 exp
nR t
θ
h
−κ
2 −

κm
2 − ν2
8

1
σ2
t
i
ds
o
.
Proof. We know from Proposition 2.1 that for each t ∈ [0,T] the sequence
σε
t converges to σt in L2(Ω). Since this convergence is also point wise, we
conclude by using the properties of the function Λε(x) that
Dθσ
ε
t =
ν
2
exp
Z t
θ

−
κ
2
+

κθ
2
−
ν2
8

Λ
0
ε (σ
ε
t)

ds

converges point wise to
G :=
ν
2
exp
Z t
θ

−
κ
2
−

κθ
2
−
ν2
8

1
σ2
t

ds

.
It follows from the Novikov condition, that the exponent in Dθσε
t is negative
for all choices of ε and therefore that |Dθσε
t| ≤ ν
2 for all ε. It then follows
from the bounded convergence theorem that Dθσε
t converges to G in L2(Ω).
Finally it follows from Lemma 1.2.3 in [18] that σt ∈ D1,2 and Dθσt = G.
Corollary 4.1. The Malliavin derivative of σt is bounded and satisﬁes |Dθσε
t| ≤
ν
2. Furthermore σt ∈ L1,2.
Proof. This result follows from the proof above. It directly implies that
σt ∈ L1,2 ( see Deﬁnition 1.3.2 ) in [18].
Corollary 4.2. The Heston volatility process vt is in L1,2 and
Dθvt = ν exp
Z t
θ

−
κ
2
−

κθ
2
−
ν2
8

1
vt

ds

√
vt
7Proof. For ﬁxed t ∈ [0,T] we have vt ∈ L2(Ω) and
ν exp
Z t
θ

−
κ
2
−

κθ
2
−
ν2
8

1
vt

ds

√
vt ∈ L
2 (Ω).
It then follows from Lemma 2.1 in [19] that vt ∈ D1,2. As in Corollary 4.1
one concludes from the explicit expression, that (vt) ∈ L1,2.
Remark 4.1. Note that due to the Novikov condition, the expressions
R t
θ
1
vtds
(and then
R t
θ
h
−κ
2 −

κm
2 − c2
8

1
vt
i
ds) are well deﬁned a.s.
5 Conclusion
We have proved that the Heston stochastic volatility or alternatively the Cox-
Ingersoll-Ross process and their square roots are Malliavin diﬀerentiable and
have given compact formulas for their derivatives. This is a key result in
so far as that it opens the door for applications of Malliavin calculus in the
framework of the Heston stochastic volatility model. The results underlines
the role of the Heston stochastic volatility model, both under a theoretical
and computational aspects.
References
[1] Alos, E. A Generalization of Hull and White Formula and Applica-
tions to Option Pricing Approximation (February 2004)
[2] Benhamou, E. Optimal Malliavin Weighting Function for the Com-
putation of the Greeks. Mathematical Finance , Vol 13, Issue 1 (2003)
[3] Benth, F.-E.; Di Nunno, G.; Løkka, A.; Øksendal, B.; Proske, F.
Explicit Representation of the Minimal Variance Portfolio in Markets
Driven by L´ evy Processes. Mathematical Finance , Vol 13, Issue 1
(2003)
[4] Bermin, H.-P. Hedging Options: The Malliavin Calculus Approach
versus the ∆-Hedging Approach. Mathematical Finance, Vol 13, Issue
1 (2003)
[5] Bouchard, B.; Ekeland I.; Touzi N. On the Malliavin approach to
Monte Carlo approximation of conditional expectations. Finance and
Stochastics 8 (2004)
8[6] Bouleau, N. Error Calculus and Path Sensitivity in Financial Models.
Mathematical Finance , Vol 13, Issue 1 (2003)
[7] Cox, J.; Ingersoll, J. and Ross, S. A theory of the term structure of
interest rates. Econometrica, 53: 385-408, (1985)
[8] Cvitanic,J.; Ma, J.; Zhang, J. Eﬃcient Computation of Hedging Port-
folios for Options with Discontinuous Payoﬀs. Mathematical Finance,
Vol 13, Issue 1. (2003)
[9] Detemple, J.; Garcia, R.; Rindisbacher, M. Representation formulas
for Malliavin derivatives of diﬀusion processes. Finance and Stochas-
tics Volume 9, Number 3 (2005)
[10] Ewald, C.-O. Local Volatility in the Heston Model : A Malliavin
Calculus Approach. Journal of Applied Mathematics and Stochastic
Analysis, Vol. 2005, Issue 3, (2005)
[11] Ewald, C.-O. Optimal logarithmic Utility and optimal Portfolios for
an Insider in a Stochastic Volatility Model. International Journal of
Theoretical and Applied Finance, Vol. 8 No. 3 (2005)
[12] Ewald, C.-O.; Zhang, A. A new Technique for Calibrating Stochastic
Volatility Models : The Malliavin Gradient Method. Submitted to
Journal of Quantitative Finance (2005)
[13] Fourni´ e, E.; Lasry, J-M.; Lebuchous, J.; Lions, P.-L., Touzi, N. Ap-
plications of Malliavin calculus to Monte Carlo methods in ﬁnance I.
Finance Stoch. 3, No.4, 391-412 (1999).
[14] Fourni´ e, E.; Lasry, J.-M.; Lebuchous, J.; Lions, P.-L. Applications
of Malliavin calculus to Monte Carlo methods in ﬁnance II. Finance
Stoch. 5, No.2, 201-236 (2001).
[15] Heston, S.L.; A Closed-Form Solution for Options with Stochastic
Volatility with Applications to Bond and Currency Options; Review
of Financial Studies, Vol.6, Issue 2 (1993)
[16] Imkeller, P. Malliavin’s Calculus in Insider Models: Additional Utility
and Free Lunches. Mathematical Finance, Vol 13, Issue 1 (2003)
[17] Karatzas, I.; Shreve, S.-E. Brownian motion and stochastic calculus.
Graduate Texts in Mathematics, 113. New York etc. Springer-Verlag
(1988)
9[18] Nualart, David The Malliavin calculus and related topics. Probability
and Its Applications. Springer-Verlag (1995).
[19] Leon J.A., Navarro R., Nualart D.: An Anticipating Calculus Ap-
proach to the Utility Maximization of an Insider; Mathematical Fi-
nance, Vol 13, Issue 1 (2003)
10

A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options;

A Generalization of Hull and White Formula and Applications to Option Pricing Approximation

A new Technique for Calibrating Stochastic Volatility Models : The Malliavin Gradient Method. Submitted to Journal of Quantitative Finance

A theory of the term structure of interest rates.

An Anticipating Calculus Approach to the Utility Maximization of an Insider;

Applications of Malliavin calculus to Monte Carlo methods in ﬁnance I.

Applications of Malliavin calculus to Monte Carlo methods in ﬁnance II.

Brownian motion and stochastic calculus. Graduate Texts in Mathematics, 113. New York etc.

Eﬃcient Computation of Hedging Portfolios for Options with Discontinuous Payoﬀs.

Hedging Options: The Malliavin Calculus Approach versus the ∆-Hedging Approach.

Local Volatility in the Heston Model : A Malliavin Calculus Approach.

Malliavin’s Calculus in Insider Models: Additional Utility and Free Lunches.

On the Malliavin approach to Monte Carlo approximation of conditional expectations.

Optimal logarithmic Utility and optimal Portfolios for an Insider in a Stochastic Volatility Model.

Optimal Malliavin Weighting Function for the Computation of the Greeks.

Representation formulas for Malliavin derivatives of diﬀusion processes.

A note on the Malliavin differentiability of the Heston volatility

http://www.econ.upf.edu/docs/papers/downloads/880.pdf

A note on the Malliavin differentiability of the Heston volatility

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics