# American Institute of Mathematical Sciences

April  2006, 2(2): 199-227. doi: 10.3934/jimo.2006.2.199

## LIBOR market model with stochastic volatility

 1 Department of Mathematics, University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China, China

Received  December 2005 Revised  February 2006 Published  April 2006

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 findings.
Citation: Lixin Wu, Fan Zhang. LIBOR market model with stochastic volatility. Journal of Industrial & Management Optimization, 2006, 2 (2) : 199-227. doi: 10.3934/jimo.2006.2.199
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