On the pricing implications of the joint lognormal assumption for the swaption and cap markets

Riccardo Rebonato

Abstract

ABSTRACT

The inconsistency of the joint lognormal assumption (in their own forward measure) for forward and swap rates has often been mentioned in the literature. This inconsistency has (correctly) prompted some researchers to deny the Black model the no-arbitrage status. To the knowledge of the author, however, no study has been carried out of the magnitude of the price discrepancies introduced by the joint lognormal assumption; nor has the issue been addressed of the arbitrageability of these price deviations. The present work addresses these issues by means of Monte Carlo simulations and a suitable switch of numéraire, and does so by focusing on the swap-rate (as opposed to forward-rate) formalism. This choice of state variables has been motivated by the fact that formulas for drifts and volatilities for swap rates are less well known than the corresponding expressions for forward rates and require more careful handling of the input market quantities. Despite being somewhat more cumbersome, a swap-rate-based formalism can, if correctly implemented, provide a natural and powerful tool for problems such as Bermudan swaptions and options on constant maturity swaps. The no-arbitrage drifts of swap rates needed for the task are obtained using a new, simpler formalism and are expressed in terms of directly observable market quantities.