Simpson Approximation Pricing And Its Efectiveness Of Zero-coupon Bonds

In order to use the mathematical tools to analyse and solve the financial interest rate models, many models contain continuous time interest rates. If the short-term interest rate yt satisfies the stochastic differential equation dyt=(α+βyt)dt+σytγdWt, by Ito formula and the no-arbitrage rule, we can derive that the pricing function P(t,T) for zero-coupon bonds satisfies a partial differential equation and it is of the exponential integral form P(t,T)=Et[exp(-∫tTysds)]. But it is very difficult to calculate this formula in real world problems. It at least has four difficulties. First, for γ≠0and γ≠1/2, we can not write an explicit formula for yt. Second, to calculate the definite integral∫tTysds we need all values of yt in [0,T], but we usual have discrete values of short-term interest rate yt. Third, if we use some approximation integral formula to approximate the definite integral, it is very difficult to calculate the partial sums in the formula. Fourth, for the continuous time interest rate yt, the calculation of expectation Et[exp(-∫tTysds)] is very difficult.In this paper, we want to overcome the above difficulties to establish an approxi-mate formula of the pricing function for zero-coupon bonds P(t,T)=Et[exp(∫tTysds)]. We use a sequence of discrete time interest rates to instead of the continuous time inter-est rate yt, then apply the Simpson integral approximate formula to the exponential in-tegral, we derive an approximation solution of pricing function for zero-coupon bonds. First, we get several lemmas by discretizing the integral, with which the approximation formula of pricing function can be obtained through comprehensive analysis. Second we discuss the impact of all parameters on the value of approximation pricing function utilizing numerical computation. In this way, we can observe that the approximation solution is of great reference value in practice. Finally we give the effectiveness of this solution by means of numerical simulation and computation.