CMS, CMS Spreads and Similar Options in the Multi-Factor HJM Framework

Abstract

Constant maturity swaps (CMS) and CMS spread options are analyzed in the multi-factor HJM framework. For Gaussian models, which include some Libor Market Models and the G2 model, explicit approximated pricing formulae are provided. Two approximating approaches are proposed: an exact solution to an approximated equation and an approximated solution to the exact equation. The first approach borrows from previous literature on other models; the second approach is new. For this latter, the price approximation errors are smaller than in the previous literature and negligible in practice. These approaches are being used here to price standard CMS and CMS spreads and can be used for some European exotic products.

Keywords: CMS, CMS spread, Heath-Jarrow-Morton, multi-factor model, Gaussian models, G2, Libor Market Model, analytical formula, efficient approximation