Explicit Density Approximations for Local Volatility Models Using Heat Kernel Expansions

Stephen Taylor, Scott Glasgow, James Taylor, Jan Vecer

Research output: Contribution to journalArticlepeer-review

Abstract

Heat kernel perturbation theory is a tool for constructing explicit approximation formulas for the solutions of linear parabolic equations. We review the crux of this perturbative formalism and then apply it to differential equations which govern the transition densities of several local volatility processes. In particular, we compute all the heat kernel coefficients for the CEV and quadratic local volatility models; in the later case, we are able to use these to construct an exact explicit formula for the processes’ transition density. We then derive low order approximation formulas for the cubic local volatility model, an affine-affine short rate model, and a generalized mean reverting CEV model. We finally demonstrate that the approximation formulas are accurate in certain model parameter regimes via comparison to Monte Carlo simulations.

Original language English (US) 847-867 21 Methodology and Computing in Applied Probability 18 3 https://doi.org/10.1007/s11009-015-9463-6 Published - Sep 1 2016 Yes

All Science Journal Classification (ASJC) codes

• Statistics and Probability
• Mathematics(all)

Keywords

• CEV model
• Heat kernel expansion
• Local volatility
• Short rate models