An Approximate Swaption Formula in Heath–Jarrow–Morton Models

Abstract

This article provides an analytical approximation formula for a swaption price when the instantaneous forward rate follows a Heath–Jarrow–Morton (HJM) model. The author’s approximation strategy, based on the chaos expansion approximation, is to replicate the probability density function of the complex quasi-Gaussian process from a simpler one, which has a semi-closed form solution. It is not restricted to the linear approximation, as is the technique proposed by the existing literature, but can be extended to higher order approximations. Moreover, computation of the approximation is fast; hence, it is suitable for calibration purposes. The author illustrates results through numerical implementation and calibration done using market data.

TOPICS: Options, interest-rate and currency swaps, derivatives

Key Findings

• • The authors approximation strategy, based on the chaos expansion approximation, is to replicate the probability density function of the complex quasi-Gaussian process from a simpler one, which has a semi-closed form solution.

• • It is not restricted to the liner approximation, as is the technique proposed by the existing literature, but can be extended to higher-order approximations.

• • Computation of the approximation is fast; hence, it is suitable for calibration purposes.