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We obtain a decomposition of the call option price for a very general stochastic volatility diffusion model, extending a previous decomposition formula for the Heston model. We realize that a new term arises when the stock price does not follow an exponential model. The techniques used for this purpose are nonanticipative. In particular, we also see that equivalent results can be obtained by using Functional Itô Calculus. Using the same generalizing ideas, we also extend to nonexponential models the alternative call option price decomposition formula written in terms of the Malliavin derivative of the volatility process. Finally, we give a general expression for the derivative of the implied volatility under both the anticipative and the nonanticipative cases.

Stochastic volatility models are a natural extension of the Black-Scholes model in order to manage the skew and the smile observed in real data. It is well known that in these models the average of future volatilities is a relevant quantity. See, for example, [

During last years, different developments for finding approximations to the closed-form option pricing formulas have been published. Malliavin techniques are naturally used to solve this problem in [

In the present paper, we generalize [

The main ideas developed in this paper are the following:

A generic call option price decomposition is found without having to specify the volatility structure.

A new term emerges when the stock option prices do not follow an exponential model, as, for example, in the SABR case.

The Feynman-Kac formula is a key element in the decomposition. It allows expressing the new terms that emerge under the new framework (i.e., stochastic volatility) as corrections of the Black-Scholes formula.

The decomposition found using Functional Itô calculus appears to be the same as the decomposition obtained through our techniques.

A general expression of the derivative of the implied volatility, both for nonanticipative and anticipative cases, is given.

Let

Black-Scholes model:

CEV model:

Heston model:

SABR model:

For existence and unicity of the solution in the Heston case, see, for example, [

The following notation will be used in all the paper:

We will denote by

We use in all the paper the notation

In our setting, the call option price is given by

Recalling that from the Feynman-Kac formula, the operator

We will also use the following definitions for

In this section, following the ideas in [

It is well known that if the stochastic volatility process is independent from the price process, then the pricing formula of a plain vanilla European call is given by

The idea used in [

For all

Notice that

As the derivatives of

So, applying Itô formula, using the fact that

Taking conditional expectation and multiplying by

In [

We observe the following:

We have extended the decomposition formula in [

Note that when

Indeed, we show that, due to the use of Feynman-Kac formula,

In this section, we give the insights of the Functional Itô Calculus developed in [

Let

We define

Under this framework, we have the next definitions of derivative.

The horizontal derivative of a functional

The vertical derivative of a functional

We also have the following Itô formula that works for nonanticipative functionals:

For any nonanticipative functional

See [

In this section, we apply Functional Itô Calculus to the problem of finding a decomposition for the call option price. The decomposition problem is an anticipative path-dependent problem. Using a smart choice of the volatility process into the Black-Scholes formula, we can convert it into a nonanticipative one. It is natural to wonder whether the Functional Itô Calculus brings some new insides into the problem.

We consider the functional

Under this framework, we calculate the derivatives using Functional Itô Calculus with respect to the variance. Then we write them in terms of the classical Black-Scholes derivatives. We must realize that, for simplicity, the new derivatives are calculated with respect to the variance instead of the volatility of the process.

If

Alternative Vega:

Alternative Vanna:

Alternative Vomma:

Alternative Theta:

For all

Notice that

We deduce that

Note that

as

as

So, we have

Note that Functional Itô formula proved in [

Note that Theorem

Realize that Theorem

In the next section, we present a brief introduction to the basic facts of Malliavin calculus. For more information, see [

Let us consider a Brownian motion

We will use the next Itô formula for anticipative processes.

Let us consider the processes

See [

The next proposition is useful when we want to calculate the Malliavin derivative.

Let

See [

In this section, we use the Malliavin calculus to extend the call option price decomposition in an anticipative framework. This time, the decomposition formula has one term less than in the Itô formula’s setup.

We recall the definition of the future average volatility as

For all

Notice that

Taking conditional expectation and multiplying by

As it is expected, a new term emerges when it is considered (

In particular, when

Note that when

When

In this section, we give a general expression for the derivative of the implied volatility under the framework of Itô Calculus and Malliavin calculus. A previous calculation of this derivative in the case of exponential models by using Malliavin calculus is given in [

Let

Under (

Taking partial derivatives with respect to

Now, we derive the implied volatility using Malliavin calculus. This is done in [

Under (

See [

Note that this is generalization of the formula proved in [

In this section, we provide some applications of the decomposition formula to well-known models in Finance.

We consider that the stock price follows the Heston Model (

Using Theorem

We consider that the stock price follows the SABR model (

Using Theorem

In this paper, we notice that the idea used in [

The authors declare that there is no conflict of interests regarding the publication of this paper.