A Path-Integral Approximation for Non-Linear Diffusions

Abstract

Using the path-integral formalism we develop an accurate and easy-to-compute semi-analytical approximation to transition probabilities and Arrow-Debreu densities for arbitrary diffusions. We illustrate the accuracy of the method by presenting results for the Black-Karasinski model for which the proposed approximation provides remarkably accurate results, even in regimes of high volatility and for multi-year time horizons. The accuracy and the computational efficiency of the proposed approximation makes it a viable alternative to fully numerical schemes for a variety applications, ranging from maximum-likelihood estimation in econometrics to derivatives pricing.

Keywords: Path integrals; Stochastic processes; Maximum-likelihood estimation; Arrow-Debreu pricing; Zero-coupon bonds; Derivative pricing; Black-Karasinski model