Malliavin Differentiability of CEV-Type Heston Model

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.


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 t S is assumed to follow the stochastic differential equation d d d t t t t S rS t S W σ = + , where r and σ 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 pa-rameters (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 t S and the volatility t ν follow the stochastic differential equations where t B and t W denote correlated Brownian motion s. In the Equation (1.2), κ , µ and θ 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 ( ) Here, consider the European call option and let φ 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 α 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 φ is differentiable. However, in business they often consider the payoff functions such as ( ) ( ) 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 we cannot simply prove the Malliavin differentiability in the exact same way.
In this paper we concentrate on the case 1 ,1 2 , 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.

Summary of Malliavin Calculus
We give the short introduction of Malliavin calculus on the Wiener space. For further details, refer to [8].

Malliavin Derivative
We consider a Brownian motion such that f and all its partial derivatives have polynomial growth. Let S be the space of smooth random variables expressed as The following result will become a very important tool. L Ω and the duality relation We can get the following results.  of the following n-dimensional stochastic differential equation for all

Malliavin Calculus for Stochastic Differential Equations
Here j σ 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 t  -adapted In the case the coefficients are Lipschitz, the solution i t X belongs to 1,∞ D .
Theorem 2.2. Assume that coefficients are Lipschitz continuous of the stochastic differential Equation (2.10 for r t ≤ a.e., and Let t X be the solution of the following stochastic differential equation We let t Y be the first variation of t X , that is, Considering this as a stochastic differential equation for t Y , 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 For the more general case, the same result is proved as below. Let t X denote the solution of the following n-dimensional stochastic differential equation just like as (2.10) where t W denotes m-dimensional Brownian motion. For the sake of simplification, we assume that n m = .
Theorem 2.4. Suppose that the diffusion coefficient σ is invertible and that ( ) be a random variable which does not depend on the initial condition x. Then for all measurable function φ with polynomial growth we have and k δ denotes the adjoint to the Malliavin derivative with respect to a Brownian motion k t W .
The following theorem introduced in [9] is useful. From now on, we will now denote by t ∂ the once derivative with respect to t, by x ∂ the once derivative with respect to x and by xx ∂ the second derivative with respect to x.
where t W denotes a Brownian motion and the coefficients R satisfy the linear growth condition and the Lipschitz condition. Moreover, we assume that σ is positive and bounded away from 0, and that ( ) . Then t X belongs to 1,2 D and the derivative is given by for r t ≤ and 0 r t D X = for r t > . Proof. We omit the proof. For further details, refer to (Theorem 2.1 [9]).

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 > and where µ , κ and 0 θ > . In [7], Alos and Ewald proved the Malliavin differentiability of the case 1 2 γ = of (3.1). In the case, the function 1 2 x 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.

Existence and Uniqueness
We will now prove that the solution to (3.1) not only exists uniquely but is also positive a.s.
Then we have ( ) 1 P τ =∞ = . Proof. Instead of (3.1), consider the following If we have concluded that the unique strong solution of (3. We will now prove that the second claim is true. Let In order to use ([10], Theorem 5.5.29), we verify that for a fixed number c R ∈ , Since we have known that the solution t v of (3.2) does not explode at ∞ , if we could prove that the above formula holds, we can claim that ( We can assume without restriction that 1 x < and let 1 c = . Then we have From the last inequality, there exists a constant 0 C > satisfying the following inequality and then we have as 0

L p -Integrability
Consider the Stochastic Differential Equation Proof. At first we consider the positive moments. We define the stopping time By Gronwall's lemma, we can have where both C and C′ do not depend on n. As n → ∞ , we can obtain the result. Next we consider the negative moments. Define the stopping time as Here let ( ) (

Transformation and Approximation
We consider the process transformed as σ is the solution of the stochastic differential Equation for , for , For the functions Φ and Ψ , we can easily verify that for all x ∈ R , ( ) ≤ . Define our approximations t σ  as the stochastic process following the stochastic differential equation By Gronwall's lemma, t t σ σ =  for t τ <  and by Lemma 3.1 and the fact that Next we prove that there exist square integrable processes t u and t w with . Actually, we will see that t w is t σ . Before starting with the proof, we prove the following inequality. Lemma 3.5. Let t u be the solution of the following stochastic differential equation x + ∈ R , that is, the drift coefficient of t u is smaller than one of t σ  . By Yamada-Watanabe's comparison lemma (see [10], Proposition 5.2.18) and Lemma 3.1, we have t t u σ ≤  a.s. We prove the second inequality. In order to use Yamada-Watanabe's comparison lemma, we must prove that, for x We can easily verify ( ) Then there is a constant 0 ∈ Ω .
By the dominated convergence theorem we can have the convergence.

Malliavin Differentiability
We will prove the Malliavin differentiability of both t σ and t ν . To do this, we consider our approximation sequence t σ  . The approximating stochastic differential Equation (3.14) of t σ  satisfies the assumption of Theorem 2.5, so we can prove the Malliavin differentiability of t σ  .
for r t ≤ , and Proof. By Theorem 2.5, we have the result.
We will now prove the Malliavin differentiability of t σ . To start with, we prove some useful lemmas.
We have for x <  , for all x + ∈ R . Note that ξ is independent of  . By this inequality, we have the following result.
Proof. When r t > , Putting the scenarios together, we can prove the following.
for r t ≤ , and 0 r t D σ = for r t > .
Proof. We have proved that t t σ σ →  in ( ) 2 L Ω and 1,2 t σ ∈  D . Moreover, by Lemma 3.8, we have Here t σ  converges to t σ also pointwise, we can conclude that r t D σ  converges to ( ) ( ) Hence we can conclude that 1, By the chain rule, we can conclude that t ν is also Malliavin differentiable. Theorem 3.4. For all 1 p ≥ , t ν belongs to 1,∞ D and the Malliavin derivative is given by for r t ≤ , and 0 r t D ν = for r t > . Proof. Consider only the case where r t ≤ . Similarly, we can easily prove the case where r t > . We have shown that

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 cal-Journal of Mathematical Finance culus 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.

Greeks
We introduce the concept of Greeks. For example, consider a European option with payoff function φ depending on the final value of the underlying asset T S where t S denotes a stochastic process expressing the asset and T denotes the maturity of the option. The price V is given by

CEV-Type Heston Model
In [5], Heston supposed that the stock price t S follows the stochastic differential equation where ρ is the correlation coefficient between two Brownian motion s. Moreover we assume that the dynamics following stochastic differential Equations with the initial conditions 0 S x = and 0 ν ν = . We call this model the CEV-type where t W and ˆt W are independent. Note that we assume that t S and t ν follow the dynamics (4.7) and (4.8) under the risk neutral measure.

Arbitrage
Under the real measure, the CEV-type Heston model follows the following dy- where t W and ˆt W are independent. Here u denotes the expected return of t S .
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.
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.

Malliavin Differentiability of the CEV-Type Heston Model (Logarithmic Price)
From now on, we denote by D and D two Malliavin derivatives with respect to t W and ˆt W , respectively. We now consider the logarithmic price log t t X S .
First, we will prove that t X Here we can easily verify that ( ) x φ  is bounded and continuously differentiable.
Moreover we can verify that both are Lipschitz continuous. In Section 3, we have used the stochastic process σ  with Lipschitz continuous coefficients, instead of t ν . We will now prove the Malliavin differentiability of the two stochastic processes σ  and the following approximation process X  of X with Lipschitz coefficients. Naturally, instead of t X , we consider the following stochastic differential equation We have using Cauchy-Schwarz's inequality and Itô's isometry, For the second term, since both t σ and t σ  are positive a.s. and for , 0 L Ω .
The following theorem implies that t X is Malliavin differentiable.

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.

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