﻿ 数学与系统科学研究院

University of North Carolina at Charlotte

Abstract: In finance we need to deal with various random variables. In practice, a random variable usually takes values on a finite domain. For example, the value of an interest rate is positive and may not exceed 0.3. It is hard for the volatility of a stock price to be higher than 0.5. Therefore it is very useful, sometimes it is necessary, to design a stochastic model on a finite domain. Here we first derive intuitively the conditions which quarantee random variables to be defined on a finite domain.

When such conditions are satisfied, the parabolic partial differential equation for the financial derivatives based on such stochastic models is degenerated on the boundary. A final value problem for such a degenerate parabolic partial differential equation has a unique solution, which has been proven in our papers.

According to such results, we have provided a two-factor model for stock prices, in which both the stock price and the volatility are random variables and the volatility is guaranteed on a finite domain. For interest rate derivatives, we suggest a three-factor model, in which all the random variables are guaranteed on a finite domain. When such a model is used, pricing is reduced to finding solutions of a final value problem for such a degenerate parabolic partial differential equation. Since the formulation of the final value problem is well-posed, we can determine the value at the boundaries by the PDE. Our problems are defined on a finite small domain, we can find quite good results by spending less CPU time. For the two-factor model for stockes, we suggest a second-order finite difference method. Examples show that if the singularity-separating method and extrapolation techniques are used, then for American options, very good solutions can be obtained even on very coarse meshes. The three-factor interest rate model has been applied to calculating American swaption problem. Results show that this model can give very reliable results for such complicated problems. Both cases show that pricing can be done very efficiently if such models are used.