PDE formulation of some SABR/LIBOR market models and its numerical solution with a sparse grid combination technique

TL;DR

This paper introduces a PDE-based approach for pricing SABR/LIBOR market models, utilizing a sparse grid technique to efficiently handle high-dimensional problems and reduce computational time compared to Monte Carlo methods.

Contribution

It presents the first PDE formulations for SABR/LIBOR models and applies a sparse grid combination technique to overcome high-dimensional computational challenges.

Findings

Sparse grid method achieves comparable accuracy to Monte Carlo simulations.

The PDE approach significantly reduces computational time for high-dimensional models.

The method extends PDE pricing to models with more than three underlying interest rates.

Abstract

SABR models have been used to incorporate stochastic volatility to LIBOR market models (LMM) in order to describe interest rate dynamics and price interest rate derivatives. From the numerical point of view, the pricing of derivatives with SABR/LIBOR market models (SABR/LMMs) is mainly carried out with Monte Carlo simulation. However, this approach could involve excessively long computational times. For first time in the literature, in the present paper we propose an alternative pricing based on partial differential equations (PDEs). Thus, we pose original PDE formulations associated to the SABR/LMMs proposed by Hagan \cite{haganSABRLIBOR}, Mercurio \& Morini \cite{mercurioMorini} and Rebonato \cite{rebonatoWhite}. Moreover, as the PDEs associated to these SABR/LMMs are high dimensional in space, traditional full grid methods (like standard finite differences or finite elements) are not able to price derivatives over more than three or four underlying interest rates. In order to overcome this curse of dimensionality, a sparse grid combination technique is proposed. A comparison between Monte Carlo simulation results and the ones obtained with the sparse grid technique illustrates the performance of the method.