A weak MLMC scheme for L\'evy-copula-driven SDEs with applications to the pricing of credit, equity and interest rate derivatives

TL;DR

This paper introduces a novel weak multilevel Monte Carlo method for Le9vy-driven SDEs, enabling efficient pricing of financial derivatives with jumps, by coupling Poisson processes for improved variance control.

Contribution

It presents a new weak MLMC scheme using a state space discretization and Poisson process coupling for Le9vy-driven SDEs, tailored for multidimensional Le9vy copulas.

Findings

Efficient pricing of credit, equity, and interest rate derivatives.

Coupling of Poisson processes simplifies simulation.

Variance bounds achieved without strong convergence.

Abstract

This paper develops a novel weak multilevel Monte-Carlo (MLMC) approximation scheme for L\'evy-driven Stochastic Differential Equations (SDEs). The scheme is based on the state space discretization (via a continuous-time Markov chain approximation) of the pure-jump component of the driving L\'evy process and is particularly suited if the multidimensional driver is given by a L\'evy copula. The multilevel version of the algorithm requires a new coupling of the approximate L\'evy 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.