Cap and Swaption Approximations in Libor Market Models with Jumps

Posted: 8 Dec 2001

See all articles by Paul Glasserman

Paul Glasserman

Columbia Business School

Nicolas Merener

Columbia University - Department of Applied Physics and Applied Mathematics

Date Written: November 2001

Abstract

This paper develops formulas for pricing caps and swaptions in LIBOR market models with jumps. The arbitrage-free dynamics of this class of models were characterized in Glasserman and Kou (1999) in a framework allowing for very general jump processes. For computational purposes, it is convenient to model jump times as Poisson processes; however, the Poisson property is not preserved under the changes of measure commonly used to derive prices in the LIBOR market model framework. In particular, jumps cannot be Poisson under both a forward measure and the spot measure, and this complicates pricing. To develop pricing formulas, we approximate the dynamics of a forward rate or swap rate using a scalar jump-diffusion process with time-varying parameters. We develop an exact formula for the price of an option on this jump-diffusion through explicit inversion of a Fourier transform. We then use this formula to price caps and swaptions by choosing the parameters of the scalar diffusion to approximate the arbitrage-free dynamics of the underlying forward or swap rate. We apply this method to two classes of models: one in which the jumps in all forward rates are Poisson under the spot measure, and one in which the jumps in each forward rate are Poisson under its associated forward measure. Numerical examples demonstrate the accuracy of the approximations.

Suggested Citation

Glasserman, Paul and Merener, Nicolas, Cap and Swaption Approximations in Libor Market Models with Jumps (November 2001). Available at SSRN: https://ssrn.com/abstract=292853 or http://dx.doi.org/10.2139/ssrn.292853

Paul Glasserman (Contact Author)

Columbia Business School ( email )

3022 Broadway
403 Uris Hall
New York, NY 10027
United States
212-854-4102 (Phone)
212-316-9180 (Fax)

Nicolas Merener

Columbia University - Department of Applied Physics and Applied Mathematics ( email )

New York, NY 10027
United States

Here is the Coronavirus
related research on SSRN

Paper statistics

Abstract Views
1,536
PlumX Metrics