Modifying the LMM to Price Constant Maturity Swaps

Abstract

An interest rate swap involves an exchange of cash payments calculated by applying two different interest rates to the same notional principal. In a constant maturity swap (CMS), one rate is the rate on the fixed-rate leg of the swap of a specified maturity and the other is a money market rate, such as LIBOR. For example, a CMS swap might call for exchanging the current 10-year swap rate against 90-day LIBOR. The problem in valuing the contract is that a CMS swap is normally priced under the forward measure (where forward short rates are assumed to be lognormal), while a 10-year swap would be priced under the forward swap measure (in which the swap rates are lognormal). The two measures are incompatible because neither one allows both rates to be lognormal. Wu and Chen develop a lognormal “approximation” to the swap rate within the LIBOR market model, allowing the pricing of CMS swaps of several types to be done more easily and accurately than with the convexity-adjustment alternative.