Abstract

We obtain a Hull and White type option price decomposition for a general local volatility model. We apply the obtained formula to CEV model. As an application we give an approximated closed formula for the call option price under a CEV model and an approximated short term implied volatility surface. These approximated formulas are used to estimate model parameters. Numerical comparison is performed for our new method with exact and approximated formulas existing in the literature.

1. Introduction

In [1], a decomposition of the price of a plain vanilla call under the Heston model is obtained using Itô calculus. Recently, in [2], the decomposition obtained in [1] has been used to infer a closed form approximation formula for a plain vanilla call price in the Heston case, and on the basis of this approximated price, a method to calibrate model parameters has been developed and successfully applied. In this paper, we use the ideas presented in [1] to obtain a closed form approximation to plain vanilla call option price under a spot-dependent volatility model.

The model presented here assumes the volatility is a deterministic function of the underlying stock price, and therefore, there is only one source of randomness in the model. These models are sometimes called local volatility models in the industry and GARCH-type volatility models in financial econometrics. Recall that these models are different from the so-called stochastic volatility models, like Heston model, where the volatility process is driven by an additional source of randomness, not perfectly correlated with the stock price innovations.

As an application, for the particular case of CEV model, we obtain an approximation of the at-the-money (ATM) implied volatility curve as a function of time and an approximation of the implied volatility smile as a function of the log-moneyness, close to the expiry date. We use these approximations to calibrate the CEV model parameters.

2. Preliminaries and Notations

Let be a positive price process under a market chosen risk neutral probability that follows the modelwhere is a standard Brownian motion, is the constant interest rate, and is a function of such that is a square integrable random variable that satisfies enough conditions to ensure the existence and uniqueness of a solution of (1).

The following notation will be used in all the paper:(i)We define the Black-Scholes function as a function of and such that where denotes the cumulative probability function of the standard normal law, and are strictly positive constants, and  Note that the price of a plain vanilla European call under the classical Black-Scholes theory is where is the price of the underlying process at , is the constant volatility, is the strike price, and is the expiry date.(ii)We will denote frequently by the time to maturity.(iii)We use in all the paper the notation , where is the completed natural filtration of (iv)In our setting, the call option price is given by (v)Recall that from the Feynman-Kac formula, the operator satisfies .(vi)We define the operators , , and In particular, we have that

Lemma 1. Then, for any , and for any positive quantities , , , and , one haswhere is a constant that depends on , , and

Proof. For any we have where is a polynomial of order and the exponential decreasing on of the Gaussian kernel compensates the possible increasing of and

3. A General Decomposition Formula

Here we obtain a general abstract decomposition formula for a certain family of functionals of that will be the basis of all later computations.

Assume we have a functional of the form where is a function of and is a function of

Then we have the following lemma.

Lemma 2 (generic decomposition formula). For all , one has

Proof. Applying the Itô formula to process we obtainNow, applying Feynman-Kac formula for , multiplying by , and taking conditional expectations, we obtainOn the other hand, using Itô calculus rules, it is easy to see thatFinally, substituting this expression in (12) we finish the proof.

For the Black-Scholes function previous lemma reduces to the following corollary.

Corollary 3 (BS decomposition formula). For all , one has

Proof. Applying Lemma 2 to and , and using equalities the corollary follows straightforward. Note that to apply Itô formula to Black-Scholes function, because the derivatives of this function are not bounded, we have to use an approximation to the identity and the dominated convergence theorem as it is done, for example, in [3]. For simplicity we skip this mollifying argument across the paper.

Remark 4. For clarity, in the following we will refer to terms of the previous decomposition as

Remark 5. In [4], an alternative formula that can be used for local volatility models is proved. The formula presented in [4] uses, as a base function, function , but this formula is numerically worse than the new formula presented here that uses as a base function . This happens because in the formula presented in [4] the volatility is put into the approximated term, instead of keeping it on the Black-Scholes term as we do here. It is precisely because volatility is a deterministic function of the underlying asset price that we can do that.

4. Approximation Formula

In this section we obtain an approximation formula to plain vanilla call price by approximating terms (I)–(IV). The main idea is to use again Lemma 2 to estimate the errors.

Theorem 6 (BS decomposition formula with error term). For all , one haswhere is an error. Terms of are written in Appendix A.

Proof. We apply Lemma 2 to terms (I)–(IV). Concretely, functions and in every case are
(I)(II)(III)(IV)

5. CEV Model

The constant elasticity of variance (CEV) model is a diffusion process that solves the stochastic differential equationNote that, writing , CEV model can be seen as a local volatility model. This model, introduced in [5], is one of the first alternatives to Black-Scholes point of view that appeared in the literature. The parameter is called the elasticity of the volatility and is a scale parameter. Note that for , the model reduces to Osborne-Samuelson model, for , the model reduces to Bachelier model, and for , the model reduces to Cox-Ingersoll-Ross model. Parameter controls the steepness of the skew exhibited by the implied volatility.

There exists a closed form formula for call options; see [5, 6]. An approximated formula is given in [7].

5.1. Approximation of the CEV Model

Applying Corollary 3 to CEV model, we obtain the following.

Corollary 7 (CEV exact formula). For all , one has

We will write

The exact formula can be difficult to use in practice, so we will use the following approximation.

Corollary 8 (CEV approximation formula). For all , one haswhere is an error. Terms of are written in Appendix B. We have that and is an increasing function on every parameter.