Analytical solutions of interest rate path-dependent options based on the shifted libor market model

Abstract

The LIBOR market model of Brace, Gatarek, and Musiela is widely used to price interest rate derivatives. Plain vanilla-type options are valuated analytically, but the other exotic options are usually valuated through simulation on this model. The LIBOR market model assumes that the volatility functions just depend on time; that is, it uses the volatility term structure. In this paper, we derive a spot LIBOR process based on the shifted LIBOR market model with volatility skews under the assumptions of deterministic and separable volatility functions. We then deduce approximately explicit solutions of interest rate path-dependent options, such as barrier option, lookback option, and Asian option by assuming a deterministic integrand in the Radon-Nikodym derivative. To test the accuracy of these solutions, we compare them with solutions from the Monte Carlo method.