Pricing options under stochastic volatility: a power series approach

Abstract

Options, Stochastic volatility, SDEs, PDEs, Duhamel’s principle, 60H10, 91B24, C02, G13,

Exchange option pricing under stochastic volatility: a correlation expansion

“…First, we have to introduce a further approximation to make the coefficients totally computable, since the joint distributions necessary to compute u 0 , u 1 , φ 1 and ψ 1 are usually unknown. This approximation is of the same type as the one appearing in Antonelli and Scarlatti (2009), which also contains an estimation of the relative error. …”

“…is ensured within an appropriate radius of convergence (see Antonelli and Scarlatti (2009) for the details).…”

“…(12) and we want to identify the four coefficients of the polynomial. To do so, we are going to apply the method developed in Antonelli and Scarlatti (2009) …”

“…In Antonelli and Scarlatti (2009) the setting was one dimensional, but the ease of the method made us try an application to a multidimensional situation, such as the present one. The method is based on an iterative procedure.…”

“…In this paper we tackle the pricing of an exchange or Fisher-Margrabe option (Margrabe 1978;Fisher 1978), under stochastic volatility using an analytical approximation recently introduced in a work by Antonelli and Scarlatti (2009) to evaluate plain vanilla European call options, by writing a power series expansion of the no arbitrage price with respect to the correlation parameter between asset and volatility. It resulted fairly general, working for a variety of models, including the Stein and Stein (1991), the Heston (1993) and the Hull and White (1987) ones.…”