(1+2)-dimensional Black-Scholes equations with mixed boundary conditions

Abstract

In this paper, we investigate (1+2)-dimensional Black-Scholes partial differential equations(PDE) with mixed boundary conditions. The main idea of our method is to transform the given PDE into the relatively simple ordinary differential equations(ODE) using double Mellin transforms. By using inverse double Mellin transforms, we derive the analytic representation of the solutions for the (1+2)-dimensional Black-Scholes equation with a mixed boundary condition. Moreover, we apply our method to European maximumquanto lookback options and derive the pricing formula of this options. 2000 Mathematics Subject Classification. 35K20. Key words and phrases. (1+2)-dimensional Black-Scholes equations, parabolic partial differential equations, double-Mellin transform, mixed boundary condition, option pricing.