Picard approximation of stochastic differential equations and application to LIBOR models

TL;DR

This paper introduces a Picard iteration-based approximation scheme for LIBOR market models that enables faster, parallelizable Monte Carlo pricing of derivatives, reducing computational complexity and maintaining accuracy.

Contribution

It proposes a novel Picard iteration approach for LIBOR models that simplifies calculations and enables parallel computation, improving speed and scalability.

Findings

Comparable accuracy to Euler discretization

Significant speed-up in derivative pricing

Reduced complexity from exponential to quadratic growth

Abstract

The aim of this work is to provide fast and accurate approximation schemes for the Monte Carlo pricing of derivatives in LIBOR market models. Standard methods can be applied to solve the stochastic differential equations of the successive LIBOR rates but the methods are generally slow. Our contribution is twofold. Firstly, we propose an alternative approximation scheme based on Picard iterations. This approach is similar in accuracy to the Euler discretization, but with the feature that each rate is evolved independently of the other rates in the term structure. This enables simultaneous calculation of derivative prices of different maturities using parallel computing. Secondly, the product terms occurring in the drift of a LIBOR market model driven by a jump process grow exponentially as a function of the number of rates, quickly rendering the model intractable. We reduce this growth from exponential to quadratic using truncated expansions of the product terms. We include numerical illustrations of the accuracy and speed of our method pricing caplets, swaptions and forward rate agreements.