Multilevel Richardson-Romberg and Importance Sampling in Derivative Pricing

TL;DR

This paper introduces a combined multilevel Richardson-Romberg and importance sampling method for derivative pricing, improving computational efficiency and accuracy through advanced Monte Carlo estimators and adaptive parameter optimization.

Contribution

It develops a novel Monte Carlo estimator integrating importance sampling with multilevel Richardson-Romberg, utilizing Robbins-Monro for parameter optimization and higher-order schemes for SDE simulation.

Findings

Reduced computational time for derivative pricing.

Achieved desired accuracy with fewer simulations.

Numerical results confirm improved efficiency.

Abstract

In this paper, we propose and analyze a novel combination of multilevel Richardson-Romberg (ML2R) and importance sampling algorithm, with the aim of reducing the overall computational time, while achieving desired root-mean-squared error while pricing. We develop an idea to construct the Monte-Carlo estimator that deals with the parametric change of measure. We rely on the Robbins-Monro algorithm with projection, in order to approximate optimal change of measure parameter, for various levels of resolution in our multilevel algorithm. Furthermore, we propose incorporating discretization schemes with higher-order strong convergence, in order to simulate the underlying stochastic differential equations (SDEs) thereby achieving better accuracy. In order to do so, we study the Central Limit Theorem for the general multilevel algorithm. Further, we study the asymptotic behavior of our estimator, thereby proving the Strong Law of Large Numbers. Finally, we present numerical results to substantiate the efficacy of our developed algorithm.