An IMEX-Scheme for Pricing Options under Stochastic Volatility Models with Jumps

Abstract

Partial integro-differential equation (PIDE) formulations are often preferable for pricing options under models with stochastic volatility and jumps, especially for American-style option contracts. We consider the pricing of options under such models, namely the Bates model and the so-called stochastic volatility with contemporaneous jumps (SVCJ) model. The nonlocality of the jump terms in these models leads to matrices with full matrix blocks. Standard discretization methods are not viable directly since they would require the inversion of such a matrix. Instead, we adopt a two-step implicit-explicit (IMEX) time discretization scheme, the IMEX-CNAB scheme, where the jump term is treated explicitly using the second-order Adams–Bashforth (AB) method, while the rest is treated implicitly using the Crank–Nicolson (CN) method. The resulting linear systems can then be solved directly by employing LU decomposition. Alternatively, the systems can be iterated under a scalable algebraic multigrid (AMG) method. For pricing American options, LU decomposition is employed with an operator splitting method for the early exercise constraint or, alternatively, a projected AMG method can be used to solve the resulting linear complementarity problems. We price European and American options in numerical experiments, which demonstrate the good efficiency of the proposed methods.