Zero-Coupon Bond Pricing Under The Stochastic Logistic Interest Rate Model

Zero coupon Bond pricing problem is one of the most basic problems in the financialmarkets, there are two ways to calculate derivative securities prices:（i）Monte carlo sim-ulation produces the path of the underlying securities under the risk neutral measures, byusing these paths to estimate discounted payment risk neutral expectations;（ii）Use the nu-merical method to solve partial diferential equations. My article mainly aims at the secondapproach, clarifying how to associate the risk neutral pricing problems with partial difer-ential equation. Such problem corresponding to the interest rate model is from the basicstochastic interest rate model, Vasicek model, Hull-White model, CIR model, and then toHJM model, and has become the core problem in the market.In this article, we explain howto apply Feyman-Kac formula to get solution of zero coupon pricing equation when interestrate model is Logistic equation,and We give probability of the equations from the point viewof probability;From the perspective of market price risk, zero coupon bond pricing equationsare given in the specific form.Under the general rate modeldR（t）=β（t, R（t））dt+γ（t, R（t））dW（t）,Hypothesis （2.3）ú（2.4） is constant, the price of zero coupon bond satisfies partial diferen-tial equationf_t（t,r）+β（t, r） fr（t,r）+1/2γ²ï¼ˆt,r） f_rr（t,r）=rf（t, r）.Under the Hull-white rate modeldR（u）=（a（u） b（u）R（u））du+σ（u）dW（u）,Hypothesis （2.3）å'Œï¼ˆ2.4）is constant, the price of zero coupon bond satisfies partial diferential equationf_t（t,r）+（a（t） b（t）r） fr（t,r）+1/2σ²ï¼ˆt） f_rr（t,r）=r f （t, r）.We might as well assume that the solution has this formf （t,r）=e^{-rC（t,T） A（t,T）,}where C（t,t）,A（t,t）is a random function,we can getft（t,r）=（rC（t, T）’ A（t, T）’） f （t, r）,fr（t,r）=C（t, T） f （t, r）,frr（t,r）=C2（t,T） f （t,r）,where C（t,T）’=（?）/（?）tC（t,T）, A（t,T）’=tA（t,T）.Hence the price of zero coupon bond satisfiesthe partial diferential equation[（-C （t,T）+b（t）C（t, T）1）r A’（t, T） a（t）C（t, T）+1/2σ²ï¼ˆt）C²ï¼ˆt,T）] f （t,r）=0.Due to the equation holds for r, we can get-C ’（t,T）+b（t）C（t, T）-1=0,-A’（t,T） a（t）C²ï¼ˆt,T）+1/2σ²ï¼ˆt）C²ï¼ˆt,T）=0,solve this equation,C（t,T）=integral from n=t to Te^{-=integral from n=t to s b（v）dv}dsA（t,T）==integral from n=t to T a（s）C（s,T）1/2σ²ï¼ˆs）C²ï¼ˆs,T）d_s.We can get the solution of the partial diferential equation.Under the CIR interest rate modeldR（t）=（α βR（t））dt+σ （R（t））dW（t）,Hypothesis （2.3）ú（2.4） is constant, the price of zero coupon bond satisfies partial diferen- tial equation We might as well just have such assumptions on the type of the solution where C（t, t）, A（t, t）is not a random function,we can get where C（t, T）’=（?）/（?）tC（t, T）, A（t, T）’=（?）/（?）ttA（t, T）.Hence the price of zero coupon bond satisfies the partial differential equation Due to the equation holds for （?）r, we can get solve this equation, where γ=1/2（?）, sinhu=eu-e-u/2, coshu=eu+e-u/2.We can get the solution of the partial differential equation.This paper briefly introduces the zero coupon bond pricing problems under stochastic logistic rate model.The following four theorems are the main results of this paper. Theorem1Under Logistic interest rate modelAssume that （2.3） and （2.4） hold. Then the price of zero coupon bond satisfies partial difer-ential equationTheorem2For any given initial valuer（0）=r0>0,the Logistic equation has aunique global continuous solution r（t）:Therefore, the probability type of the solution isTheorem3Under logistic interest rate model,by （2.5）（3.9） and （3.11） can get themarket price risk equation for zero coupon priceTheorem4By （2.7） and （3.18） one can get the solution of the market price riskequation...