Solution of Stochastic Volatility Models Using Variance Transition Probabilities and Path Integrals

Abstract

In this paper, we solve the problem of solution of stochastic volatility models in which the volatility diffusion can be solved by a one dimensional Fokker-planck equation. We use one dimensional transition probabilities for the evolution of PDE of variance. We also find dynamics of evolution of expected value of any path dependent function of stochastic volatility variable along the PDE grid. Using this technique, we find the conditional expected values of moments of log of terminal asset price along every node of one dimensional forward Kolmogorov PDE. We use the conditional distribution of moments of above path integrals along the variance grid and use Edgeworth expansions to calculate the density of log of asset price. Main result of the paper gives dynamics of evolution of conditional expected value of a path dependent function of volatility (or any other SDE) at any node on the PDE grid using just one dimensional PDE if we can describe its one step conditional evolution between different nodes of the PDE.

Keywords: stochastic volatility, Fokker-Planck, Forward Kolmogorov, path integral, transition probabilities

JEL Classification: G12, G13