Exact Solution to CEV Model with Uncorrelated Stochastic Volatility

Stochastic volatility models are widely used in interest rate modeling to match the option smiles -- the two most popular are the Heston model and the SABR one. These have been incorporated into arbitrage-free term structure frameworks, Heston-LMM and SABR-LMM respectively.In this paper we consider the CEV model with a general stochastic volatility. Assuming that rate-volatility correlation is zero we are able to obtain an exact integral representation of the option price provided that we have a closed form for a Moment Generating Function of the cumulative stochastic variance or of its inverse. Using this result we derive explicit solutions in terms of two-dimensional integral for both CEV-CIR model (or power generalization of the Heston) and the SABR one. Moreover the results in this paper may be easily extended to any affine process (possibly multi-factor and including jumps) leading to numerous practical applications