A Path Integral Approach to Option Pricing with Stochastic Volatility: Some Exact Results

Abstract

The Black-Scholes formula for pricing options on stocks and other securities\nhas been generalized by Merton and Garman to the case when stock volatility is\nstochastic. The derivation of the price of a security derivative with\nstochastic volatility is reviewed starting from the first principles of\nfinance. The equation of Merton and Garman is then recast using the path\nintegration technique of theoretical physics. The price of the stock option is\nshown to be the analogue of the Schrodinger wavefuction of quantum mechacnics\nand the exact Hamiltonian and Lagrangian of the system is obtained. The results\nof Hull and White are generalized results for pricing stock options for the\ngeneral correlated case are derived.\n