Pricing discretely monitored barrier options: When Malliavin calculus expansions meet Hilbert transforms

Abstract

This paper proposes a novel approach that combines Malliavin calculus based expansions with Hilbert transform techniques to pricing various types of discretely monitored barrier options under jump diffusion models with multifactor stochastic volatility. Malliavin calculus based expansions are often used to deal with multidimensional models but are hard to apply directly in pricing discretely monitored barrier options, while, on the contrary, Hilbert transform techniques proposed by Feng and Linetsky (2008) (Mathematical Finance, 18(3), 337–384) have proved to be particularly useful for pricing discretely monitored barrier options but are difficult to apply directly to multidimensional models. By innovatively making these two methods complement each other, our approach takes advantage of both of them and overcomes their respective limitations. Numerical results suggest that the resulting recursive expansion pricing method is accurate and efficient under a broad range of prevalent option pricing models, including not only affine models such as the Heston-SV model and the Bates-SVJ model but also non-affine models such as the CEV model and the GARCH-SV model with/without jumps.