Generalized Swap Market Model and the Valuation of Interest Rate Derivatives

Abstract

In this paper we will establish a generalized Swap Market Model (GSMM) by unifying the stochastic process of swap rates with constant tenors under a single swap measure. GSMM is a natural extension of Libor Market Model (LMM) for swap rates, and LMM can be considered as a special case of GSMM since Libors is a special swap rate with the constant tenor of one period. GSMM can be applied for pricing and hedging any interest rate derivatives, and is suited especially for CMS and swap rate products. There are a number advantages of GSMM: (1) GSMM models swap rates directly, and therefore achieves the best match between products and model. (2) GSMM can be calibrated to the term structure of swaption volatilities easily and quickly. (3) There is no translation of risk sensitivities with respect to swap rates within GSMM. In contrast, risk sensitives such as Vega for swap rates can not be derived directly, and must be translated in an inefficient, inaccurate and nontransparent manner in the most existing interest rate models. (4) All smile modelings for LMM can be taken over for GSMM since GSMM and LMM share an almost identical mathematical structure. (5) GSMM avoids the inconsistency of the market conventions in cap and swaptions markets. Accompanied by these favorite features, GSMM should be a promising interest rate model for pricing and hedging most traded swap rate structures in financial market.

Keywords: Swap Market Model, Libor Market Model, Numeraire change, Swaptions, CMS