1. Introduction
Malliavin calculus is the infinite-dimensional differential calculus on the Wiener space in order to give a probabilistic proof of Hölmander’s theorem. It has been developed as a tool in mathematical finance. In 1999, Founié et al. [1] gave a new method for more efficient computation of Greeks which represent sensitivities of the derivative price to changes in parameters of a model under consideration, by using the integration by parts formula related to Malliavin calculus. Following their works, more general and efficient applications to computation of Greeks have been introduced by many authors (see [2] [3] [4]). They often considered this method for tractable models typified by the Black-Scholes model.
In the Black-Scholes model, an underlying asset
is assumed to follow the stochastic differential equation
, where r and
respectively imply the risk free interest rate and the volatility. The Black-Scholes model seems standard in business. The reason is that this model has the analytic solution for famous options, so it is fast to calculate prices of derivatives and risk parameters (Greeks) and easy to evaluate a lot of deals and the whole portfolios and to manage the risk. However, the Black-Scholes model has a defect that this model assumes that volatility is a constant.
In the actual financial market, it is observed that volatility fluctuates. However, the Black-Scholes model does not suppose the prospective fluctuation of volatility, so when we use the model there is a problem that we would underestimate prices of options. Hence, more accurate models have been developed. One of the models is the stochastic volatility model. One of merits to consider this model is that even if prices of derivatives such as the European options are not given for any strike and maturity, we can grasp the volatility term structure. In particular, the Heston model, which is introduced in [5], is one of the most popular stochastic volatility models. This model assumes that the underlying asset
and the volatility
follow the stochastic differential equations
(1.1)
(1.2)
where
and
denote correlated Brownian motion s. In the Equation (1.2),
,
and
imply respectively the rate of mean reversion (percentage drift), the long-run mean (equilibrium level) and the volatility of volatility. This volatility model is called the Cox-Ingersoll-Ross model and more complicated than the Black-Scholes model. We have not got the analytic solution yet.
However, even this model cannot grasp fluctuation of volatility accurately. In 2006 (see [6]), Andersen and Piterbarg generalized the Heston model. They extended the volatility process of (1.2) to
(1.3)
This model is called the constant elasticity of variance model (we will often shorten this model as the CEV model). Naturally, in the case
, the volatility model (1.3) is more complicated than the volatility model (1.2).
Here, consider the European call option and let
is a payoff function. Then we can estimate the option price by the following formula
. However, the computation of Greeks is much important in the risk-management.
A Greek is given by
where
is one of parameters needed to compute
the price, such as the initial price, the risk free interest rate, the volatility and the maturity etc.. Most of financial institutions have calculated Greeks by using finite-difference methods but there are some demerits such that the results depend on the approximation parameters. More than anything, the methods need the assumption that the payoff function
is differentiable. However, in business they often consider the payoff functions such as
or
. Here we need Malliavin calculus. In 1999 Founié et al. in [1] gave the new methods for Greeks. To come to the point, they calculated Greeks by the following
formula
. We can calculate this even if
is
polynomial growth. Instead, we need the Malliavin differentiability of
.
The solution
satisfying the stochastic differential equation with Lipschitz continuous coefficients is known as Malliavin differentiable. Hence we can easily verify that the Black-Scholes model is Malliavin differentiable. However the
diffusion coefficient
is neither differentiable at
nor Lipschitz
continuous and then we cannot find whether the CEV-type Heston model is Malliavin differentiable or not. In [7], Alos and Ewald proved that the volatility
process (1.2), that is the case where
of (1.3), was Malliavin differentiable and gave the explicit expression for the derivative. However, in the case
, we cannot simply prove the Malliavin differentiability in the exact same way.
In this paper we concentrate on the case
, that is, we extend the
results in [7] and give the explicit expression for the derivative. Moreover we consider the CEV-type Heston model and give the formula to compute Greeks.
2. Summary of Malliavin Calculus
We give the short introduction of Malliavin calculus on the Wiener space. For further details, refer to [8].
2.1. Malliavin Derivative
We consider a Brownian motion
(in the sequel, we often denote
by
) on a complete filtered probability space
where
is the filtration generated by
, and the Hilbert space
. When fixing
, we can consider
. Then the Itô integral of
is constructed as
on
. We denote
by
the set of infinitely continuously differentiable functions
such that f and all its partial derivatives have polynomial growth. Let S be the space of smooth random variables expressed as
(2.1)
where
and
where
,
. We denote by
the set of infinitely continuously differentiable functions
such that f has compact support. Moreover we denote by
the set of infinitely continuously differentiable functions
such that ƒ and all of its partial derivatives are bounded. Denote by
and
respectively, the spaces of smooth random variables of the form (2.1) such that
and
. We can find that
and
is a linear subspace of and dense in
for all
. We use the notation
in the
sequal. We define the derivative operator D, so called the Malliavin derivative operator.
Definition 2.1. (Malliavin derivative) The Malliavin derivative
of a smooth random variable expressed as (2.1) is defined as the H-valued random variable given by
(2.2)
We sometimes omit to write the subscript t.
Since
is dense in
, we will define the Malliavin derivative of a general
by means of taking limits. We will now prove that the Malliavin derivative operator
is closable. Please refer to [8] for proves of the following results.
Lemma 2.1. We have
, for
and
.
Lemma 2.2. For any
, the Malliavin derivative operator
is closable.
For any
, we denote by
the domain of D in
and then it is the closure of
by the norm
(2.3)
Note that
is a Hilbert space with the scalar product
. Moreover, the Malliavin derivative
is regarded as a stochastic process defined almost surely with the measure
where u is a Lebesgue measure in
. Indeed, we can observe
(2.4)
The following result will become a very important tool.
Lemma 2.3. Suppose that a sequence
converges to F in
. Then F belongs to
and the sequence
converges to DF in the weak topology of
.
Similarly, we define the k-th Malliavin derivative of F,
, as a
-measurable stochastic process defined
-almost surely and the operator
is closable from
for any
and
. As with the Malliavin derivative D, from the closability of
, we can define the domain
of the operator
in
as the completion of
with the norm
(2.5)
Moreover we define
as
. We will now prove the chain rule and refer to the ( [8], Proposition 1.2.4) for details.
Lemma 2.4. For
, let
and
be a Lipschitz function with bounded partial derivatives, and then we have
and
(2.6)
2.2. Skorohod Integral
For
satisfing
, the adjoint
of the operator D which is closable and has the domain on
should be closable but with the domain contained in
. Focus on the case
. We can define the divergence operator
so called the Scorohod integral which is the adjoint of the operator D such as
(2.7)
Definition 2.2 (Skorohod integral). Let
. If for all
, we can have
(2.8)
where c is some constant depending on u, then u is called to belong to the domain
. Moreover if
, then we have that
belongs to
and the duality relation
, for all
.
We can get the following results.
Lemma 2.5. Let
and
satisfy
. And then we have that
belongs to
and
.
Lemma 2.6. Let
be an
-adapted stochastic process then
and
.
We give one of famous properties of
. The following property implies the relationship between the Malliavin derivative and the Skorohod integral. Denote by
the class of processes
such that
for almost all t and there exists a measurable version of the two
variable processes
satisfying
.
Lemma 2.7. Let
satisfy that
and that
. We have then that
belongs to
and
(2.9)
The following result is applied to calculate Greeks. For further details, refer to ( [8], Chapter 6).
Lemma 2.8. Let
. Suppose that an random variable
satisfy
a.s. and
. For any continuously differentiable function f with bounded derivatives, we have
![]()
where
.
2.3. Malliavin Calculus for Stochastic Differential Equations
Consider
and
. Let
be the m-dimensional
Brownian motion on filtered probability space
where P is the n-dimensional Wiener measure and F is the completion of the σ-field of
with P. And then
is the underlying Hilbert space. We consider the solution
of the following n-dimensional stochastic differential equation for all ![]()
(2.10)
where
and
satisfy the following : there is a positive constant
such that
(2.11)
(2.12)
Here
is the columns of the matrix
. We can have the following result related to the uniqueness and refer to ( [8], Lemma 2.2.1) for the detail.
Theorem 2.1. There is a unique n-dimensional, continuous and
-adapted stochastic process
satisfying the stochastic differential Equation (2.10) with
, for all
.
In the case the coefficients are Lipschitz, the solution
belongs to
.
Theorem 2.2. Assume that coefficients are Lipschitz continuous of the stochastic differential Equation (2.10). Then the solution
belongs to
for all
and
and satisfies
(2.13)
Moreover the derivative
satisfies the following
(2.14)
for
a.e., and
for
a.e.. Here
denotes the Malliavin derivative for
.
Let
be the solution of the following stochastic differential equation
(2.15)
where
denotes a 1-dimensional Brownian motion. Assume that
. We let
be the first variation of
, that is,
. We can easily have that
satisfies the folloing
(2.16)
Considering this as a stochastic differential equation for
, we can have the following solution
(2.17)
The following results will also be useful to calculate Greeks later.
Lemma 2.9. Under the above conditions, we can have
.
Let
be a continuous function in H such that
.
Lemma 2.10. Under the above conditions, we can have
.
Theorem 2.3. For any
of polynomial growth, we have
where
.
For the more general case, the same result is proved as below. Let
denote the solution of the following n-dimensional stochastic differential equation just like as (2.10)
(2.18)
where
denotes m-dimensional Brownian motion. For the sake of simplification, we assume that
.
Theorem 2.4. Suppose that the diffusion coefficient
is invertible and that
, for some
, where Y denotes the first variation
process, that is,
. Let
be a random variable which does not depend on the initial condition x. Then for all measurable function
with polynomial growth we have
, where
is an
-adapted process satisfying
,
(2.19)
and
denotes the adjoint to the Malliavin derivative with respect to a Brownian motion
.
The following theorem introduced in [9] is useful. From now on, we will now denote by
the once derivative with respect to t, by
the once derivative with respect to x and by
the second derivative with respect to x.
Theorem 2.5. Consider a stochastic process
satisfying the 1-dimensional stochastic differential equation
(2.20)
where
denotes a Brownian motion and the coefficients
and
satisfy the linear growth condition and the Lipschitz condition. Moreover, we assume that
is positive and bounded away from 0, and that
and
are bounded for all
. Then
belongs to
and the derivative is given by
(2.21)
for
and
for
.
Proof. We omit the proof. For further details, refer to (Theorem 2.1 [9]).
3. Mean-Reverting CEV Model
Following the construction in [7], we will now prove that the mean-reverting constant elasticity of variance model is Malliavin differentiable. The mean-reverting CEV model follows the stochastic differential equation
(3.1)
with
and where
,
and
. In [7], Alos and Ewald proved the Malliavin differentiability of the case
of (3.1). In the case, the function
is neither continuously differentiable in 0 nor Lipschitz continuous so they circumvented various problems by some transforming and approximating.
However, in the case
, there are more complex problems. Following [7], we will extend their results.
3.1. Existence and Uniqueness
We will now prove that the solution to (3.1) not only exists uniquely but is also positive a.s.
Lemma 3.1. There exists a unique strong solution to (3.1) which satisfies
. Moreover, let
with
. Then we have
.
Proof. Instead of (3.1), consider the following
(3.2)
If we have concluded that the unique strong solution of (3.2) is positive a.s., then (3.2) coincides with (3.1). The existence of non-explosive weak solution for (3.2) follows from the continuity and the sub-linear growth condition of drift and diffusion coefficients. Moreover, from ( [10], Proposition 5.3.20, Corollary 5.3.23), we have the pathwise uniqueness. From ( [10], Proposition 5.2.13), we can verify that the pathwise uniqueness holds for (3.2).
We will now prove that the second claim is true. Let
with
. In order to use ( [10], Theorem 5.5.29), we verify that for a fixed number
,
where
is defined as
. Since we have known
that the solution
of (3.2) does not explode at
, if we could prove that the above formula holds, we can claim that
, that is,
. We can assume without restriction that
and let
. Then we have
(3.3)
Letting
, we can calculate
. From the last inequality, there exists a constant
satisfying the following inequality and then we have as
,
(3.4)
3.2. Lp-Integrability
Consider the Stochastic Differential Equation
(3.5)
with
, where b is such that
and satisfies the Lipschitz condition,
and
. The following lemma ensures the existence of its moments of any order.
Lemma 3.2. Consider the solution of the (3.5). For any
, we have
and
.
Proof. At first we consider the positive moments. We define the stopping time
with
. By Itô’s formula,
(3.6)
From the Lipschitz condition of the drift function
, there exists a positive constant K which satisfies
. By the above inequality and Young’s inequality, we have
(3.7)
By Gronwall’s lemma, we can have
, where both C and
do not depend on n. As
, we can obtain the result. Next we consider
the negative moments. Define the stopping time as
, with
. By Itô’s formula, we have
(3.8)
Taking the expectation and using the Fubini’s theorem, we have
(3.9)
Here let
, then we can easily evaluate the boundedness for any ![]()
(3.10)
Summarizing the calculation, we have
, and from Gronwall’s lemma we finally have
. Taking the limit
, then
so we have
. Hence we can deduce the result.
Remark 1. Since the CEV model satisfies the assumptions of Lemma 3.2, so the result holds for the CEV model.
3.3. Transformation and Approximation
We consider the process transformed as
. By Itô’s formula, we have
(3.11)
with
. If
is the solution of the stochastic differential Equation (3.11), then we can prove that
is also the solution of the stochastic differential Equation (3.1) satisfying the initial condition
. By this transformation, we can replace (3.1) by (3.11) with the constant volatility term. In order to use Theorem 2.5, we must approximate
and
by the Lipschitz
continuous functions, respectively. For all
, define the continuously differentiable functions
and
as
(3.12)
(3.13)
For the functions
and
, we can easily verify that for all
,
and
and then we have that for all
,
and
. Moreover, note that for all
,
and
. Define our approximations
as the stochastic process following the stochastic differential equation
(3.14)
with
for all
. The coefficients of the Equation (3.14) are Lipschitz continuous because we can have for all
,
(3.15)
We will prove that
converges to
in
. First we prove that
converges to
pointwise.
Lemma 3.3. The sequence
converges to
a.s., for all
.
Proof. Define for all
the stopping time as
with
. By the definition of
,
, and
, we have
(3.16)
By Gronwall’s lemma,
for
and by Lemma 3.1 and the fact that
for
, we have
a.s. so
for all
.
Next we prove that there exist square integrable processes
and
with
for all
. Actually, we will see that
is
. Before starting with the proof, we prove the following inequality.
Lemma 3.4. For
and
, let
. We have, for
,
(3.17)
Proof. By differentiating
, we can easily have the result.
Consider
and
in the above inequality, then we can have the below result.
Lemma 3.5. Let
be the solution of the following stochastic differential equation
(3.18)
with
, where
. Then
a.s. for all
.
Proof. From the definitions of
and
,
for all
, that is, the drift coefficient of
is smaller than one of
. By Yamada-Watanabe’s comparison lemma (see [10], Proposition 5.2.18) and Lemma 3.1, we have
a.s.
We prove the second inequality. In order to use Yamada-Watanabe’s comparison lemma, we must prove that, for
,
. Let
. We can easily verify
, for
and
for
. For all
, we have
(3.19)
(3.20)
Then there is a constant
with
for all
and
for all
. For
,
is decreasing for all
. Then
and
imply for all
,
, that is, for ![]()
(3.21)
By Yamada-Watanabe’s comparison lemma, we have
a.s.
Theorem 3.1. For all
, the sequence
converges to
in
.
Proof. From Lemma 3.5, we have
. Lemma 3.2 implies
. Moreover, the Ornstein-Uhlenbeck process
. By the dominated convergence theorem we can have the convergence.
3.4. Malliavin Differentiability
We will prove the Malliavin differentiability of both
and
. To do this, we consider our approximation sequence
. The approximating stochastic differential Equation (3.14) of
satisfies the assumption of Theorem 2.5, so we can prove the Malliavin differentiability of
.
Lemma 3.6.
belongs to
and we have
(3.22)
for
, and
for
.
Proof. By Theorem 2.5, we have the result.
We will now prove the Malliavin differentiability of
. To start with, we prove some useful lemmas.
Lemma 3.7. For
and
, let
, then for
we have
(3.23)
Proof. By differentiating
we can easily have the result.
By Lemma 3.7, considering the case where
and
, we have for
,
(3.24)
We have for
,
(3.25)
so there exists a constant
such that for all
,
. Hence, for
, we have
, for all
. Note that
is independent of
. By this inequality, we have the following result.
Lemma 3.8. We have for all
and
,
(3.26)
Proof. When
,
so the result follows. Moreover when
, putting above results together, we obtain the result.
Putting the scenarios together, we can prove the following.
Theorem 3.2.
belongs to
and we have
(3.27)
for
, and
for
.
Proof. We have proved that
in
and
. Moreover, by Lemma 3.8, we have
. Here
converges to
also pointwise, we can conclude that
converges to
. Using the bounded
convergence theorem, we can have that
converges to G in
. Hence by Lemma 2.4, we can conclude that
and
.
Moreover we can prove the following Malliavin differentiability in more detail.
Theorem 3.3. For all
,
belongs to
, that is,
belongs to
.
Proof. We only have to prove that
. We have
(3.28)
Hence we can conclude that
.
By the chain rule, we can conclude that
is also Malliavin differentiable.
Theorem 3.4. For all
,
belongs to
and the Malliavin derivative is given by
(3.29)
for
, and
for
.
Proof. Consider only the case where
. Similarly, we can easily prove the case where
. We have shown that
and
. By Lemma 2.5, we have
(3.30)
For all
, using Young’s inequality and the fact
and
, we can prove that
belongs to
. Indeed, we have
(3.31)
4. CEV-Type Heston Model and Greeks
We will now consider the CEV-type Heston model and Greeks. Fournié et al. introduced new numerical methods for calculating Greeks using Malliavin calculus for the first time in 1999 (see [1]). We call this methods Malliavin Monte-Carlo methods. They focused on models with Lipschitz continuous coefficients, and then a lot of researchers have considered Malliavin Monte-Carlo methods to compute Greeks. However, lately, there is need to focus on models with non-Lipschitz coefficients such as stochastic volatility models. In 2008, Alos and Ewald proved that the Cox-Ingersoll-Ross model was Malliavin differentiable (see [7]). We apply Malliavin calculus for calculating Greeks of the CEV-type Heston model which is one of the important in business but mathematically complex models. Basically, we consider the European option but we can easily extend this result to other options.
4.1. Greeks
We introduce the concept of Greeks. For example, consider a European option with payoff function
depending on the final value of the underlying asset
where
denotes a stochastic process expressing the asset and T denotes the maturity of the option. The price V is given by
where r is the risk-free rate. We can estimate this by Monte-Carlo simulations. Greeks are derivatives of the option price V with respect to the parameters of the model. Greeks are the useful measure for the portfolio risk management by traders in financial institutions. Most of financial institutions estimate Greeks by finite difference methods. However, there are some demerits. For examples, the numerical results depend on the approximation parameters and, in the case where
is not differentiable, this methods do not work well. In [1], Founié et al. gave the new methods to circumvent these problems. The idea is that we calculate Greeks by multiplying the weight, so-called Malliavin weight, as following
(4.1)
This methods are much useful since we do not require the differentiability of the payoff function
. Instead, there is need to assume that the underlying assert
is Malliavin differentiable. From Theorem 2.2, we find that the solution of the stochastic differential equation with Lipschitz continuous coefficients are Malliavin differentiable. However, if a model under consideration becomes more complex just like the CEV-type Heston model, we could not apply this Malliavin methods. Through Section 4, we consider the Malliavin differentiability of the CEV-type Heston model in order to give formulas for Greeks, in particular, Delta and Rho. Here, Delta
and Rho
respectively measure the sensitivity of the option price with respect to the initial price and the risk-free rate. In particular,
is one of the most important Greeks which also describes the replicating portfolio.
4.2. CEV-Type Heston Model
In [5], Heston supposed that the stock price
follows the stochastic differential equation
(4.2)
where
, r and
respectively mean a Brownian motion , the risk-free rate and the volatility. Moreover Heston assumed that the volatility process
becomes a mean-reverting stochastic process of the form
(4.3)
where
,
,
and
respetively mean a Brownian motion , the long-run mean, the rate of mean reversion and the volatility of volatility. This model is called the Cox-Ingersoll-Ross model. Here
and
are two correlated Brownian motion s with
(4.4)
where
is the correlation coefficient between two Brownian motion s. Moreover we assume that the dynamics following stochastic differential Equations (4.1), (4.2), and (4.3) are satisfied under the risk neutral measure. However even the Heston model cannot grasp the fluctuation of the volatility accurately. In [6], Andersen and Piterbarg extended the Heston model to the model of which dynamics follow
(4.5)
(4.6)
(4.7)
with the initial conditions
and
. We call this model the CEV-type Heston model. For the Equation (4.5) with
, the Malliavin differentiability
obviously follows by Theorem 2.2. In the case
, Alos and Ewald proved
the Malliavin differentiability in [7]. In Section 3, we have proved the Malliavin
differentiability in the case
. Fron now on, we concentrate on
.
In order to give the formulas for the CEV-type Heston model, we will now prove
the Malliavin differentiability of the model. Before considering the Malliavin differentiability, we now prove that there is a following Brownian motion
which will become useful later.
Lemma 4.1. There exists a Brownian motion
independent of
with
.
Proof. From the definition of
, we have
. At
first we prove that
is independent of
. Since we easily have
, so
is independent of
. Using Lêby’s theorem, we conclude
is a Brownian motion. We can easily verify that
is also martingale. Consider the quadratic variation
of
. Then we have
(4.8)
Hence by the Lêvy’s theorem,
is a Brownian motion.
Instead of the dynamics (4.5), (4.6) and (4.7), replacing
by
, then we can consider the following
(4.9)
(4.10)
where
and
are independent. Note that we assume that
and
follow the dynamics (4.7) and (4.8) under the risk neutral measure.
4.3. Arbitrage
Under the real measure, the CEV-type Heston model follows the following dynamics
(4.11)
(4.12)
where
and
are independent. Here u denotes the expected return of
. In business, u is assumed to equal to the risk free rate. In order to do this, we will change the real measure P to the measure Q called the risk-neutral measure. We consider the arbitrage but this problem is complicated, since the volatility is not tractable. However, we obtain the following theorem.
Theorem 4.1. The CEV-type Heston model following (4.9) and (4.10) is free of arbitrage and there is a risk-neutral measure Q
(4.13)
(4.14)
Proof. We consider the interval
. First we solve the equation
. In order to solve this, we put
. From
Lemma 3.1,
is positive a.s. so we have
. Here
is obviously progressively measurable. Moreover, we can easily see that
is locally bounded and in
. Let
where
.
It is well-known that if we can prove that
is a martingale, then the market is free of arbitrage and under the risk neutral measure Q with
. Note that
is replaced by
which is a Brownian motion under Q. Here we must prove that for all
,
. Fix
and let
with
. Here
is bounded, so we have
is bounded. From Novikov’s criteria, we have that
is a uniformly integrable martingale for any
. Moreover, from the continuity of
and Lemma 3.1,
increases to infinity. Since
is positive a.s.,
converges to
as
, and then by using the monotone convergence theorem
(4.15)
Here we have
, so letting
be the measure satisfying
, and then we have
(4.16)
We must prove
. First we prove
. From Girsanov’s theorem, the processes
and
are
-Brownian motion s under the measure
. Note that
is an
-adapted Brownian motion under
for all n. We have known that under the measure P,
follows the equation
(4.17)
Integrals under P and
are the same, so
also satisfies the above stochastic differential equation under
. From Lemma 3.1, the solution
is unique. Hence the distribution of
under the measure
must be the same as the distribution of
under the measure P, and then we can conclude that the distribution
is the same under P and
, that is,
. Since
tends to
a.s.,
. Hence we can conclude
and
is a martingale. Then the market is free of arbitrage.
This theorem implies that the dynamics for the volatility process is preserved, and the drift term of the underlying asset is changed from u to r. In the sequel, we will consider the CEV-type Heston model under the risk-neutral measure denoted by P not by Q.
4.4. Malliavin Differentiability of the CEV-Type Heston Model (Logarithmic Price)
From now on, we denote by D and
two Malliavin derivatives with respect to
and
, respectively. We now consider the logarithmic price
. First, we will prove that
is Malliavin differentiable. By Itô’s formula, we have
(4.18)
with
. Here
is neither differentiable at
in 0 nor Lipschitz continuous. Hence we will now approximate this stochastic differential equation by one with Lipschitz continuous coefficients and prove the Malliavin differentiability of
. Let
(4.19)
Here we can easily verify that
is bounded and continuously differentiable. Moreover we can verify that both
and
are Lipschitz
continuous. In Section 3, we have used the stochastic process
with Lipschitz continuous coefficients, instead of
. We will now prove the Malliavin differentiability of the two stochastic processes
and the following approximation process
of X with Lipschitz coefficients. Naturally, instead of
, we consider the following stochastic differential equation
(4.20)
with
.
Lemma 4.2. We have
in
.
Proof. From the inquality
, we have
(4.21)
We have using Cauchy-Schwarz’s inequality and Itô’s isometry,
(4.22)
For the second term, since both
and
are positive a.s. and for
,
, we have
(4.23)
By the scenarios in Subsection 3.3 and Subsection 3.4, we have that for almost all
there exists a positive constant
such that for all
,
. For such
, let
(4.24)
with
, then we have
for
. Hence we can have
, for
. And then we can have
as
,
for all
. Since
and
,
. Here
is
-integrable for all
so we can conclude that for all
,
in
. We have from Fubini’s theorem,
in
.
The following theorem implies that
is Malliavin differentiable.
Theorem 4.2.
belongs to
and the Malliavin derivatives are given by
(4.25)
(4.26)
for
, and
for
.
Proof. Since the coefficients of stochastic differential equations for
and
are Lipschitz continuous, we can use Theorem 2.2. At first, we can conclude that
and the derivatives are given by
(4.27)
(4.28)
for
and
for
.
Moreover we can also conclude that
and the derivatives are given by the following
(4.29)
(4.30)
for
, and
for
.
We only consider the case
. First we consider the Malliavin derivative
. By Lemma 4.2 and the proof, we have
in
and
in
. Moreover,
is bounded, so we can use Lemma 2.4. Hence we can conclude
. We consider the Malliavin derivative
. For the first term, we need prove
(4.31)
Here we have that
(4.32)
This converges to 0 in
by the proof of Lemma 4.2, Lemma 3.8, Theorem 3.2, and Lemma 3.1. Hence we can conclude
in
. For the second term, as well as the case for
, we can prove that
in
. For the third term, we will prove
in
. We have from Itô’s isometry,
(4.33)
This converges to 0 in
as well as the first term, so we can conclude that
(4.34)
By Lemma 2.4, we have
(4.35)
Remark 2. For
, as well as Theorem 4.1, we can more easily prove
(4.36)
(4.37)
for
, and
for
.
4.5. Malliavin Differentiability of the CEV-Type Heston Model (Actual Price)
From now on, we will concentrate on the underlying asset
and the volatility
.
In Subsection 4.4, we proved the Malliavin differentiability of the logarithmic price
and the transformed volatility
. Here we can prove that both of the underlying asset
and the volatility
are Malliavin differentiabile by the chain rule.
Theorem 4.3.
and
belong to
and we have
(4.38)
(4.39)
(4.40)
(4.41)
for
, and
for
.
Proof. First we consider the Malliavin derivative for
. By Lemma 2.5, we have
(4.42)
(4.43)
We have by Theorem 4.2
(4.44)
(4.45)
for
, and
for
. Next, we consider the Malliavin derivative for
. By Lemma 2.5, we have
(4.46)
(4.47)
Hence by Theorem 4.2, we have
(4.48)
(4.49)
for
and
for
.
4.6. Delta and Rho
Using Theorem 2.4 and Theorem 4.4, we can calculate Greeks of
. We now consider the following stochastic differential equations
(4.50)
(4.51)
Rewrite the stochastic differential Equations (4.15) and (4.16) as the integral form, and then we have
(4.52)
We now give the formula for Delta of this model.
Theorem 4.4. Consider the CEV-type Heston model following the dynamics (4.15) and (4.16). We have for any funtion with polynomial growth ![]()
(4.53)
Proof. Let
be the diffusion matrix
, then we can have the inverse
. We can have from the Itô’s formula
(4.54)
Hence we can directly calculate the first variation process
of
as
. Then we can have
(4.55)
By Lemma 3.2, we have
. As with Theorem 4.3, let
be the column with the form
. Since
and
are Malliavin differentiable we have from Theorem 2.4
(4.56)
Moreover we can calculate a Greek, Rho
.
Theorem 4.5. Consider the CEV-type Heston model following the dynamics (4.15) and (4.16). Then for any
of polynomial growth, we have
(4.57)
Proof. By the definition of
, we have
(4.58)
and
as
. Here we have
(4.59)
By the above formula, we have
(4.60)
5. Conclusions
From Sections 3 and 4, it is proved by using unique transformation and approximation that we can apply Malliavin calculus to the CEV model and the CEV-type Heston model both of which have non-Lipschitz coefficients in their processes. Then we can provide the formulas to calculate important Greeks as Delta and Rho of these models and contribute to finance, in particular for traders in financial institutions to measure market risks and hedge their portfolios in terms of Delta Hedge.
In the future, it will be required how to calculate the Vega, one of the most important Greeks, for general stochastic volatility models including the CEV-type Heston model. Vega is the sensitivity for volatility but it is difficult to measure Vega for the stochastic volatility models since the volatility is also stochastic process. After the financial crisis, the necessity to grasp the behavior of volatility is increasing. We believe that we can calculate the vega of some important stochastic volatility models such as the Heston model or the CEV-type Heston model by using our results in Sections 3 and 4.