Integral Transforms of Distribution Densities

Analytical tractability is a desirable property of stochastic stock price models. Informally speaking, a stochastic model is analytically tractable if various important characteristics of the model can be represented explicitly or asymptotically in terms of standard functions of mathematical analysis. Classical stochastic volatility models (Hull-White, Stein-Stein, Heston) are analytically tractable. In this chapter, explicit formulas are obtained for Laplace transforms of mixing densities and Mellin transforms of stock price densities in classical stochastic volatility models. For example, an alternative proof of an explicit formula for the Laplace transform of the distribution density of an integrated geometric Brownian motion due to L. Alili and J.C. Gruet is given. Chapter 4 also contains an explicit formula for the stock price density in the correlated Hull-White model with driftless volatility obtained by Y. Maghsoodi. In addition, Chap. 4 provides explicit formulas for the Mellin transform of the stock price density in the correlated Heston and Stein-Stein models.