In this paper we extend the standard LIBOR market model to
accommodate the pronounced phenomenon of implied volatility smiles/skews.
We adopt a multiplicative stochastic factor to the volatility functions of all
relevant forward rates. The stochastic factor follows a square-root diffusion
process, and it can be correlated to the forward rates. For any swap rate, we
derive an approximate process under its corresponding forward swap measure.
By utilizing the analytical tractability of the approximate process, we develop
a closed-form formula for swaptions in term of Fourier transforms. Extensive
numerical tests are carried out to support the swaptions formula. The extended
model captures the downward volatility skews by taking negative correlations
between forward rates and their volatilities, which is consistent with empirical