Option pricing under the Bates model using the discontinuous Galerkin method

Abstract

Stochastic volatility models with jumps generalize the classical Black–Scholes framework to capture more properly the real world features of option contracts. The extension is performed by incorporating jumps and a stochastic nature of volatility of asset returns into the dynamics of underlying asset prices. In this paper, we focus on pricing of European-style options under the Bates model that combines the Merton jump-diffusion model with a stochastic volatility proposed by Heston. As a result, the pricing function is governed by a partial-integro differential equation with two spatial variables, specifically, the price of the underlying asset and its variance. Moreover, the simultaneous presence of the non-local integral term arising from jumps increases the complexity of the problem. Therefore, to improve the numerical valuation we solve the corresponding governing equation by a discontinuous Galerkin approach with a semi-implicit time stepping scheme, where the differential part is treated implicitly while the integral one explicitly by the composite trapezoidal rule. Finally, the numerical results obtained are compared within the reference benchmark.