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It is well known that Malliavin calculus can be applied to a stochastic differential equation with Lipschitz continuous coefficients in order to clarify the existence and the smootheness of the solution. In this paper, we apply Malliavin calculus to the CEV-type Heston model whose diffusion coefficient is non-Lipschitz continuous and prove the Malliavin differentiability of the model.

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. [

In the Black-Scholes model, an underlying asset

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 [

where

However, even this model cannot grasp fluctuation of volatility accurately. In 2006 (see [

This model is called the constant elasticity of variance model (we will often shorten this model as the CEV model). Naturally, in the case

Here, consider the European call option and let

A Greek is given by

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

formula

polynomial growth. Instead, we need the Malliavin differentiability of

The solution

diffusion coefficient

continuous and then we cannot find whether the CEV-type Heston model is Malliavin differentiable or not. In [

process (1.2), that is the case where

In this paper we concentrate on the case

results in [

We give the short introduction of Malliavin calculus on the Wiener space. For further details, refer to [

We consider a Brownian motion

by

where

sequal. We define the derivative operator D, so called the Malliavin derivative operator.

Definition 2.1. (Malliavin derivative) The Malliavin derivative

We sometimes omit to write the subscript t.

Since

Lemma 2.1. We have

Lemma 2.2. For any

For any

Note that

The following result will become a very important tool.

Lemma 2.3. Suppose that a sequence

Similarly, we define the k-th Malliavin derivative of F,

Moreover we define

Lemma 2.4. For

For

Definition 2.2 (Skorohod integral). Let

where c is some constant depending on u, then u is called to belong to the domain

We can get the following results.

Lemma 2.5. Let

Lemma 2.6. Let

We give one of famous properties of

variable processes

Lemma 2.7. Let

The following result is applied to calculate Greeks. For further details, refer to ( [

Lemma 2.8. Let

where

Consider

Brownian motion on filtered probability space

where

Here

Theorem 2.1. There is a unique n-dimensional, continuous and

In the case the coefficients are Lipschitz, the solution

Theorem 2.2. Assume that coefficients are Lipschitz continuous of the stochastic differential Equation (2.10). Then the solution

Moreover the derivative

for

Let

where

Considering this as a stochastic differential equation for

The following results will also be useful to calculate Greeks later.

Lemma 2.9. Under the above conditions, we can have

Let

Lemma 2.10. Under the above conditions, we can have

Theorem 2.3. For any

For the more general case, the same result is proved as below. Let

where

Theorem 2.4. Suppose that the diffusion coefficient

process, that is,

and

The following theorem introduced in [

Theorem 2.5. Consider a stochastic process

where

for

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

Following the construction in [

with

However, in the case

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

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 ( [

We will now prove that the second claim is true. Let

that the solution

Letting

Consider the Stochastic Differential Equation

with

Lemma 3.2. Consider the solution of the (3.5). For any

Proof. At first we consider the positive moments. We define the stopping time

From the Lipschitz condition of the drift function

By Gronwall’s lemma, we can have

the negative moments. Define the stopping time as

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

Here let

Summarizing the calculation, we have

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

with

continuous functions, respectively. For all

For the functions

with

We will prove that

Lemma 3.3. The sequence

Proof. Define for all

By Gronwall’s lemma,

Next we prove that there exist square integrable processes

Lemma 3.4. For

Proof. By differentiating

Consider

Lemma 3.5. Let

with

Proof. From the definitions of

We prove the second inequality. In order to use Yamada-Watanabe’s comparison lemma, we must prove that, for

Then there is a constant

By Yamada-Watanabe’s comparison lemma, we have

Theorem 3.1. For all

Proof. From Lemma 3.5, we have

We will prove the Malliavin differentiability of both

Lemma 3.6.

for

Proof. By Theorem 2.5, we have the result.

We will now prove the Malliavin differentiability of

Lemma 3.7. For

Proof. By differentiating

By Lemma 3.7, considering the case where

We have for

so there exists a constant

Lemma 3.8. We have for all

Proof. When

Putting the scenarios together, we can prove the following.

Theorem 3.2.

for

Proof. We have proved that

convergence theorem, we can have that

Moreover we can prove the following Malliavin differentiability in more detail.

Theorem 3.3. For all

Proof. We only have to prove that

Hence we can conclude that

By the chain rule, we can conclude that

Theorem 3.4. For all

for

Proof. Consider only the case where

For all

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 [

We introduce the concept of Greeks. For example, consider a European option with payoff function

This methods are much useful since we do not require the differentiability of the payoff function

In [

where

where

where

with the initial conditions

obviously follows by Theorem 2.2. In the case

the Malliavin differentiability in [

differentiability in the case

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

Lemma 4.1. There exists a Brownian motion

Proof. From the definition of

first we prove that

Hence by the Lêvy’s theorem,

Instead of the dynamics (4.5), (4.6) and (4.7), replacing

where

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

where

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

Lemma 3.1,

It is well-known that if we can prove that

Here we have

We must prove

Integrals under P and

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

with

Here we can easily verify that

continuous. In Section 3, we have used the stochastic process

with

Lemma 4.2. We have

Proof. From the inquality

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

For the second term, since both

By the scenarios in Subsection 3.3 and Subsection 3.4, we have that for almost all

with

The following theorem implies that

Theorem 4.2.

for

Proof. Since the coefficients of stochastic differential equations for

for

Moreover we can also conclude that

for

We only consider the case

Here we have that

This converges to 0 in

This converges to 0 in

By Lemma 2.4, we have

Remark 2. For

for

From now on, we will concentrate on the underlying asset

In Subsection 4.4, we proved the Malliavin differentiability of the logarithmic price

Theorem 4.3.

for

Proof. First we consider the Malliavin derivative for

We have by Theorem 4.2

for

Hence by Theorem 4.2, we have

for

Using Theorem 2.4 and Theorem 4.4, we can calculate Greeks of

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

Hence we can directly calculate the first variation process

By Lemma 3.2, we have

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

Proof. By the definition of

and

By the above formula, we have

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.

The author declares no conflicts of interest regarding the publication of this paper.

Tsumurai, S. (2020) Malliavin Differentiability of CEV-Type Heston Model. Journal of Mathematical Finance, 10, 173-199. https://doi.org/10.4236/jmf.2020.101012