^{1}

^{2}

^{2}

^{1}

^{2}

Recently, hybrid stochastic and local volatility models have become an industry standard for the pricing of derivatives and other problems in finance. In this study, we use a multiscale stochastic volatility model incorporated by the constant elasticity of variance to understand the price structure of continuous arithmetic average Asian options. The multiscale partial differential equation for the option price is approximated by a couple of single scale partial differential equations. In terms of the elasticity parameter governing the leverage effect, a correction to the stochastic volatility model is made for more efficient pricing and hedging of Asian options.

Since the well-known work of Black and Scholes [

This paper is concerned with one of the exotic options called an Asian option. This option is a path dependent option whose final payoff depends on the paths of its underlying asset. More precisely, the payoff is determined by the average value of underlying prices over some prescribed period of time. The name of “Asian” options is known to come from the fact that they were first priced in 1987 by David Spaughton and Mark Standish of Bankers Trust when they were working in Tokyo, Japan (cf. [

Since there is no general analytical formula for the price of Asian option, a variety of techniques have been developed to approximate the price of this option. Subsequently, there has been quite an amount of literature devoted to studying this option. For instance, Geman and Yor [

It is well known that the constant volatility assumption, on which a review of literature quoted above is based, for the underlying asset price is severely in contrast with many empirical studies which demonstrate the skew or smile effect of implied volatility, fat-tailed and asymmetric returns distributions, and the mean-reversion of volatility. Thus a number of alternative underlying models have been proposed. The constant elasticity of variance model by Cox [

So, it is desirable to study Asian options based on these alternative models. In fact, there are a number of recent studies along the lines of this type of extension. For instance, B. Peng and F. Peng [

This paper studies the pricing of an arithmetic Asian option under a hybrid stochastic and local volatility model which was introduced by Choi et al. [

This paper is structured as follows. In Section

In this section, we establish a partial differential equation for the price of Asian floating strike call option based on the SVCEV model.

As introduced by [

From the

Now, we take the process

In this section, we utilize the generalization of Vecer's dimension reduction technique given by Fouque and Han [

In this paper, a payoff function for arithmetic average Asian options is given by

First, we would like to replicate the averaged process

Next, we change the probability measure

Then we are ready to obtain the option price as the solution of a partial differential equation.

Let

By the Itô formula, we obtain the following two results:

Then we have

From (

If

Once the solution

In this section, we are interested in the solution of the multiscale PDE (

Substituting the asymptotic form of

Note that

In the following two theorems, we obtain PDEs for the leading order term

Assume that the term

From the

Next, we obtain a PDE for the first correction term

Assume that the term

Since

By applying the Fredholm alternative to the Poisson equation

Combining (

From the results in Section

If we define

In this section, we compute numerically the leading order price

Figure

The leading order term

Figure

The correction term

A frequent criticism of the stochastic volatility or local volatility models for path dependent options is that they do not produce deltas precise enough for hedging purposes. So, relevant industry experts recommend using a hybrid stochastic local volatility model of their own development for best pricing option products. See, for instance, [

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

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. The research was supported by the National Research Foundation of Korea NRF-2013R1A1A2A10006693.