%0 Journal Article
%T Pricing Arithmetic Asian Options under Hybrid Stochastic and Local Volatility
%A Min-Ku Lee
%A Jeong-Hoon Kim
%A Kyu-Hwan Jang
%J Journal of Applied Mathematics
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/784386
%X 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. 1. Introduction Since the well-known work of Black and Scholes [1] on the classical vanilla European option, there has been concern about the pricing of more complicated exotic options. An exotic option is a derivative which has a payoff structure more complex than commonly traded vanilla options. They are usually traded in over-the-counter market or embedded in structured products. Also, the pricing of them tends to require more complex methods than the classical Black-Scholes approach. 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. [2]). The main motivation of creating these options is that their averaging feature could reduce the risk of market manipulation of the underlying risky asset at maturity. Since Asian options reduce the volatility inherent in the option, the price of these options is usually lower than the price of classical European vanilla options. Note that there are two types of Asian options depending on the style of averaging: one is the arithmetic average Asian option and the other is the geometric average Asian option. 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 [3] computed the Laplace transform of the price of continuously sampled Asian options. However, there is a problem of slow convergence for low
%U http://www.hindawi.com/journals/jam/2014/784386/