Introduction

Abstract

Since more than twenty-five years the markets for financial derivatives have developed exponentially. The Black-Scholes (1973) model provides us with a basic tool for the valuation of financial instruments traded in the markets. Pricing formulae for almost all options, both plain vanilla and exotic style, have been derived in the Black-Scholes framework and form a valuation formula class with identical assumptions and a standardized structure: constant interest rate, constant volatility and no discontinuous component in the asset price process. Obviously, in many circumstances, this framework is too restrictive to meet the new emerging challenges and empirical findings in more involved financial markets. Nowadays, option pricing theory is undergoing a significant technical innovation: Fourier analysis and characteristic functions, which are successfully applied to stochastic volatility models. This class of models is suggested to capture the leptokurtic property of empirical distributions of stock returns, a property that is not consistent with the normal distribution in the Black-Scholes model. Wiggins (1987), Hull and White (1987), Scott (1987) as well as Stein and Stein (S&S) (1991) undertook their pioneer works in modelling stochastic volatility. The first closed-form solution for stochastic volatility whose square is specified as a mean-reverting square root process was given by Heston (1993). His influential paper introduces characteristic functions to express a tractable closed-form solution for options.