Integral Representation of Probability Density of Stochastic Volatility Models and Timer Options

This paper contributes a generic probabilistic method to derive explicit exact probability densities for stochastic volatility models. Our method is based on a novel application of the exponential measure change in Palmowski & Rolski (2002). With this generic approach, we first derive explicit probability densities in terms of model parameters for several stochastic volatility models with non-zero correlations, namely the Heston (1993), 3/2, and a special case of the α-Hypergeometric stochastic volatility models recently proposed by Da Fonseca & Martini (2016). Then, we combine our method with a stochastic time change technique to develop explicit formulae for prices of timer options in the Heston model, the 3/2 model and a special case of the α-Hypergeometric model