Analytical Green's Function Approximation and Option Pricing

Closed-form solutions to Kolmogorov type equations are very important in many application areas, such as derivative pricing, quantum mechanics, statistical physics, etc. However, they are only available for some very special equations. This book considers general second order parabolic equations with coefficients that are dependent on both time and space. It extends the recently developed Dyson-Taylor commutator method for autonomous equations to non-autonomous equations. Closed-form approximations of the Green's functions that are accurate to any prescribed order are obtained. Consequently, the solutions to second order parabolic equations can be obtained by integrating the approximated Green's functions against initial data. For applications, this book considers Local Volatility models and Stochastic Volatility models that appear in option pricing theory, gives explicit formulas for European option prices, and carries out numerical tests for such formulas.