On the boundary conditions adopted in stochastic volatility option pricing models

Song-Ping Zhu and Chun-Yang Liu

Abstract

In quantitative finance, stochastic volatility models have gradually become a dominant trend since the publication of Heston’s seminal 1993 paper, supported by some very convincing empirical evidence (eg, that obtained in 2009 by Christoffersen et al). Although imposing the right boundary condition is an important aspect of adopting the Heston model in pricing derivative contracts (such as options), particularly from a computational point of view, the boundary conditions that should be adopted in an option pricing problem have never been clearly discussed; despite the huge number of published papers with various forms of boundary conditions, some provide only limited explanations, while others give no explanations. This paper aims to fill the gap in the literature by presenting appropriate boundary conditions that should be adopted for pricing European- and American-style options under the Heston model after conducting a comprehensive review of the literature on various boundary conditions used in the past.