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 of 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

Acknowledgments

It is a pleasure to acknowledge Toby Falk for stimulating discussions and a careful reading of the manuscript, and the anonymous referees for many valuable suggestions. The author is indebted to Prof. Valerio Tognetti for teaching him path integrals and many other things. The views and opinions expressed in this article are those of the author and do not represent the views of his employers.

Disclosure statement

No potential conflict of interest was reported by the author.