Black-Scholes Equation with the Variable Risk-free Interest Rate

Abstract

As for B-S model with constant coefficients, it can be obtained analytical solution directly by converting into parabolic partial differential equation with constant coefficients, however, the research on the explicit solution of the parabolic partial differential equation under the condition of time-varying is not yet mature. The risk-free interest rate as the discount rate is not always fixed, which is changeable. Consequently, the real option pricing method palys an important role both in theoretical and practical. There are two different methods which dealing with the B-S equation under the condition of the risk-free interest rate is time-varying, one of them is the method of infinite series expansion which is put forward by Kazemi. Another is the method based on decomposition of the path which is put forward by Yun Feng. The two methods both are the summary and improvement of the predecessors' experience. In this paper, we studied the Black-Scholes equation under certain assumptions, we obtain the solution to the problem. And we have proved the compare principle by using advantage function method. Under the compare principle, which is the innovation of this paper, we can find the solution of the Black-Scholes equation with the variable risk-free interest rate by combining with the idea of stage by stage.