Realized Volatility and Mixing Distributions

Stock price densities in an uncorrelated stochastic volatility model can be represented as certain mixtures of Black-Scholes densities. The role of a mixing factor in such a representation is played by the distribution of a realized volatility (a time-average of the volatility process). For a correlated model, mixing distributions may be higher-dimensional. For example, in the correlated Heston model and the correlated Hull-White model with driftless volatility, mixing distributions are two-dimensional, while in the general correlated Hull-White model and in the correlated Stein-Stein model, they are three-dimensional. The higher-dimensional mixing distributions in the models mentioned above are joint distributions of different combinations of the volatility, the variance, the integrated volatility, and the integrated variance. This chapter provides various representations of the stock price density in stochastic volatility models as special integral transforms of mixing distributions.