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

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 wellknown, 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.

Keywords: Path integrals, Forced quantum harmonic oscillator, Stochastic processes, Fokker-Planck equation, Maximum-likelihood estimation, Arrow-Debreu pricing, Zero-coupon bonds, Derivative pricing, Black-Karasinski model