Pricing Arithmetic Asian Options under Hybrid Stochastic and Local Volatility

Min-Ku Lee,Jeong-Hoon Kim,Kyu-Hwan Jang

doi:10.1155/2014/784386

Min-Ku Lee, Jeong-Hoon Kim + Show 1 more

Open Access

https://doi.org/10.1155/2014/784386

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Abstract

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.

Highlights

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
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
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 [18], where the volatility is given by the product of a multiscale stochastic process and a power of the underlying’s price

Summary

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. 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 [18], where the volatility is given by the product of a multiscale stochastic process and a power (the elasticity of variance) of the underlying’s price. The hybrid nature of this volatility enables us to capture the leverage effect produced by the constant elasticity of variance (CEV) model as well as the smile effect of implied volatility, fat-tailed and asymmetric returns distributions, the tendency of the volatility process to revert towards a long-term mean at a certain rate, and a degree of correlation between the randomness of volatility and the randomness of underlying’s price produced by a “pure” stochastic volatility (SV) model This hybrid model is called the SVCEV model.

The SVCEV Model

A PDE for Option Price

Multiscale Analysis

Approximate Option Price

Conclusion