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

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

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

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

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

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

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

Definition 2.2 (Skorohod integral). Let. If for all, we can have

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

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.

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

where and satisfy the following : there is a positive constant such that

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

Moreover the derivative satisfies the following

for a.e., and for a.e.. Here denotes the Malliavin derivative for.

Let be the solution of the following stochastic differential equation

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

Considering this as a stochastic differential equation for, we can have the following solution

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)

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,

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

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

for and for.

Proof. We omit the proof. For further details, refer to (Theorem 2.1 [9]).

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

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

Letting, we can calculate. From the last inequality, there exists a constant satisfying the following inequality and then we have as,

Consider the Stochastic Differential Equation

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,

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

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

Taking the expectation and using the Fubini’s theorem, we have

Here let, then we can easily evaluate the boundedness for any

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.

We consider the process transformed as. By Itô’s formula, we have

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

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

with for all. The coefficients of the Equation (3.14) are Lipschitz continuous because we can have for all,

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

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,

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

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

Then there is a constant with for all and for all. For, is decreasing for all. Then and imply for all, , that is, for

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.

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

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

Proof. By differentiating we can easily have the result.

By Lemma 3.7, considering the case where and, we have for,

We have for,

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,

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

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

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

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

For all, using Young’s inequality and the fact and, we can prove that belongs to. Indeed, we have

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

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.

In [5], Heston supposed that the stock price follows the stochastic differential equation

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

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

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

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

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

where and are independent. Note that we assume that and follow the dynamics (4.7) and (4.8) under the risk neutral measure.

Under the real measure, the CEV-type Heston model follows the following dynamics

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

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

Here we have, so letting be the measure satisfying, and then we have

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

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.

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

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

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

with.

Lemma 4.2. We have in.

Proof. From the inquality, we have

We have using Cauchy-Schwarz’s inequality and Itô’s isometry,

For the second term, since both and are positive a.s. and for, , we have

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

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

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

for and for.

Moreover we can also conclude that and the derivatives are given by the following

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

Here we have that

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,

This converges to 0 in as well as the first term, so we can conclude that

By Lemma 2.4, we have

Remark 2. For, as well as Theorem 4.1, we can more easily prove

for, and for.

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

for, and for.

Proof. First we consider the Malliavin derivative for. By Lemma 2.5, we have

We have by Theorem 4.2

for, and for. Next, we consider the Malliavin derivative for. By Lemma 2.5, we have

Hence by Theorem 4.2, we have

for and for.

Using Theorem 2.4 and Theorem 4.4, we can calculate Greeks of. We now consider the following stochastic differential equations

Rewrite the stochastic differential Equations (4.15) and (4.16) as the integral form, and then we have

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

Proof. Let be the diffusion matrix, then we can have the inverse. We can have from the Itô’s formula

Hence we can directly calculate the first variation process of as. Then we can have

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

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

Proof. By the definition of, we have

and as. Here we have

By the above formula, we have