A Path Integral Approach for Time-Dependent Hamiltonians with Applications to Derivatives Pricing

TL;DR

This paper extends a semi-classical path integral method to handle time-dependent Hamiltonians, demonstrating its effectiveness in pricing complex financial derivatives like the Black-Karasinski interest rate model.

Contribution

It generalizes existing path integral techniques to time-dependent cases, enabling more accurate and efficient derivatives pricing methods.

Findings

Accurately models the Black-Karasinski interest rate dynamics.

Offers a computationally efficient alternative to numerical schemes.

Extends the scope of path integral methods in finance.

Abstract

We generalize a semi-classical path integral approach originally introduced by Giachetti and Tognetti [Phys. Rev. Lett. 55, 912 (1985)] and Feynman and Kleinert [Phys. Rev. A 34, 5080 (1986)] to time-dependent Hamiltonians, thus extending the scope of the method to the pricing of financial derivatives. We illustrate the accuracy of the approach by presenting results for the well-known, but analytically intractable, Black-Karasinski model for the dynamics of interest rates. The accuracy and computational efficiency of this path integral approach makes it a viable alternative to fully-numerical schemes for a variety of applications in derivatives pricing.