An integral equation representation approach for valuing Russian options with a finite time horizon

• We describe a general analytic solution for the inhomogeneous Black–Scholes partial differential equation with mixed boundary conditions using Mellin transform techniques.
• We derive integral equation satisfied by Russian option values by using the analytic formula.
• We present some numerical solutions and plots of the integral equation of Russian options using recursive integration methods.
• We also demonstrate the computational accuracy and efficiency of our method compared to other competing approaches.

In this paper, we first describe a general solution for the inhomogeneous Black–Scholes partial differential equation with mixed boundary conditions using Mellin transform techniques. Since Russian options with a finite time horizon are usually formulated into the inhomogeneous free-boundary Black–Scholes partial differential equation with a mixed boundary condition, we apply our method to Russian options and derive an integral equation satisfied by Russian options with a finite time horizon. Furthermore, we present some numerical solutions and plots of the integral equation using recursive integration methods and demonstrate the computational accuracy and efficiency of our method compared to other competing approaches.