Incompatibility of lognormal forward-Libor and Swap market models

Abstract

The lognormal forward-Libor and Swap market models were formulated to price caps and swaptions. However, the prices computed by these two models, under equivalent measures, are reported to be unequal. This study investigates this incompatibility by computing the prices of caps and swaptions under both the forward Libor measure and the forward Swap measure, in both the Libor and Swap market models. This was done by building a computer program that implements the Monte Carlo versions of the models, using data from caps and swaptions traded in the South African market. It was found that the actual price of caps, using the same implied volatility, were simulated accurately in both the Libor and Swap market models under both the forward Libor measure and the forward Swap measure. On the other hand, although the actual swaption prices were also simulated accurately in both the Libor and Swap market models under both the forward Libor measure and the forward Swap measure, a different implied volatility was used for each model. Therefore, the swaption price computed by the Libor market model was inconsistent with the price generated by the Swap market model; the two models are indeed incompatible. In order to price interest rate derivatives consistently, either the Libor market model or the Swap market model must be chosen. Since the Libor market model priced consistently under the two forward measures, and the time taken to simulate a price in the Libor market model was much less than in the Swap market model, the practice in the market to use the Libor market model in favour of the Swap market model is justified.