A Weak MLMC Scheme for Lévy-Copula-Driven SDEs with Applications to the Pricing of Credit, Equity and Interest Rate Derivatives

Abstract

This paper develops a novel weak multilevel Monte-Carlo (MLMC) approximation scheme for Lévy-driven Stochastic Differential Equations (SDEs). The scheme is based on the state space discretization (via a continuous-time Markov chain approximation [Mijatović, Aleksandar, Matija Vidmar, and Saul Jacka. 2012. “Markov Chain Approximations for Transition Densities of Lévy Processes.” Electronic Journal of Probability 19]) of the pure-jump component of the driving Lévy process and is particularly suited if the multidimensional driver is given by a Lévy copula. The multilevel version of the algorithm requires a new coupling of the approximate Lévy drivers in the consecutive levels of the scheme, which is defined via a coupling of the corresponding Poisson point processes. The multilevel scheme is weak in the sense that the bound on the level variances is based on the coupling alone without requiring strong convergence. Moreover, the coupling is natural for the proposed discretization of jumps and is easy to simulate. The approximation scheme and its multilevel analogous are applied to examples taken from mathematical finance, including the pricing of credit, equity and interest rate derivatives.