Libor Market Model with Stochastic Volatility

Four papers introducing LIBOR market model (LMM) were published in 1997. They seemed to unify market practice with arbitrage-free framework - it came out that for one year only. The next year, after Russian crisis, cap and swaption markets started to show evident volatility smile and skew. Several attempts were made to capture that phenomenon into the arbitrage-free framework. Our note is strongly inspired by papers and conference talks by Mark Joshi and Riccardo Rebonato. We share their opinions that: - Since smiles and skews are caused by different market features, it is more natural to model smile and skew separately, rather then to use unified framework of implied smile. - Displaced Diffusion approach is easier in treatment then Constant Elasticity of Variance (CEV) approach for interest rate derivatives and gives the same modelling possibilities. - Displaced Diffusion and Stochastic Volatility are perfectly suited to work together. Since our attention is fixed more on swaptions then on caps/floors, we would like to opt for another version of the LIBOR market model with stochastic volatility and displaced diffusion (SVDDLMM) then Joshi and Rebonato: - We use various random displacement factors for various LIBOR rates. - For Stochastic Volatility we propose a new simple non mean reverting multi-lognormal model. We also try to convince the Reader that mean reversion in stochastic volatility models excludes correct modelling of long term options - swaptions are canonical example. Easy closed form formulae are given for caps/floors and European swaptions what makes calibration procedure more effective and transparent - at least we are not "prisoners of Monte Carlo". We are able to calibrate model to various smile/skew shapes for caps/floors and swaptions with various length and of various maturities.