Extracting Of Yield Curves And Applying In The Parameter Estimation Of Gaussian HJM Model

If we can find enough spot rate in the market, we can get couples of real numbers which take term as independent variable and take spot rate as dependent variable. Then if a model is given,we can estimate the parameters by statistical methods. But in fact, the number of zero coupon securities in the market is so small that it is not easy to find enough spot rates. In order to get the spot rates with different terms, we turn to the method of coupon's peeling off. Thus, it is necessary for us to determine the pattern y_t^T of interest rate term structure model at time t, where y is spot rate and T is the maturity.According to the method of securities' pricing, we can depart some certain security with fixed interest rate into some cash flow used to pay for interest or cost and take the spot rate function supposed above as the discount function to get the theoretic price of this security. Of course, there is difference between the theoretic price B^mdl and the market price B^mkt , and they are not equal currently.The formula are as follows:Bmkt = i,where ￡mnexp{-afjn - W(l - e-W') ctine-^9})2.?=1 n=lThen let us pay attention to how to set the pattern of interest rate term structure model. Some scholars argue that the figures of spot rates with different maturities are different, so they partition the whole interest rate term structure into several segments and different segment has different function. This method is called spline method. According to the different patterns of the functions, the figure of interest rate term structure can be summed up as polynomial spline, exponential spline, polynomial spline, NS, Svensson(the improved NS) and so on. For the interest rate term structure gain by polynomial spline and exponential spline, the far interest will show the growing trend as the grow of term. This doesn't adapt to the factsituations, that is, the far interest should be calm relatively. The reference [6]shows that NS model can describe the interest rate term structure more well and truly. When I extract yield curves with this method, I made use of the function in Matlab named the nonlinear least square estimation to get a series of values of parameters. NS model's form isy{T) = a + b—------\-ce *,1where a, b, c, 0 are parameters estimated and r is residual term.The corresponding discount function isB(t) = exp{-ar - 60(1 - e"*) - cre^*}. Put it into the following equation,Nn=lthen we haveN?? p{in ( - en=XThis is a nonlinear regressive equation. UuUz,■■ ?,￡,-j\rdenote terms from today(date t)to the each one in the aggregate made up of all the payment dates of bond i, Yin denotes the payment of bond i during the tin segment. Estimating must depend on the method of the generalized least square, that is nonliear regressive technique.We estimate the parameters a, b, c, 0 using the datas of Shanghai Stock Exchange during 2005.9.19-2005.10.20., 10 days, 18 nationaldebt securities. See the interrelated information and the intraday closing price in the appendix 1. Here m = 18, w< = 1/18, then we have18N- c-W') a.n=lProgramme and use the least square method in Matlab to get parameters a,b,c,9. See the programme code in appendix 2. In this way the N-S model adaptive to our country market is determined. The parameters corresponding to 10 days are listed in the following form.Here SSE is the residual sum of squares and SST is the total sum of squares,./?2 = 1—SSE/SST belongs to [0,1], and approximates 1. From this we can see the model explains the data well.B?abceSSESSTyi9.19.0.0417-0.04350.00471.774716.5204433.3590y29.23.0.0433-0.04170.00272.084427.6048418.8358y39.26.0.0403-0.05640.00701.084132.2066441.6196y49.28.0.0436-0.04310.00322.317413.0361422.4825y59.30.0.0440-0.04410.00392.385613.9940421.8036y?10.10.0.0440-0.04520.00482.282814.5070428.2940y710.12.0.0435-0.04970.01492.1582174.2403407.7814y810.14.0.0412-0.04860.01611.9360167.9026393.5726y?10.18.0.0435-0.04880.01382.4076169.8620418.9729ylO10.20.0.0443-0.04950.01332.5603173.9271437.2226The corresponding graph of interest term structure of Shanghai Stock Exchange is the following graph. Here yl,y2,... ,yl0 cor-respond to the yield curves of 10 days during 2005.9.19-2005.10.20. Prom the graph we can see the yield increases as the term increases and tends to be calm in the end. The graph approximate horizon near t = 0. This is consistent with the fact, i.e. if term tends to 0, spot rate doesn't change apparently. In this way, we get the yield curves in the market indirectly.0.0580.0420.04After the yield curves are determined, we can make use of spot rates' values with different maturities to estimate the parameters of other interest rate term structure models, such as HJM model. Note that HJM model is a dynamic forward interest rate model. Inmodern finance analysis, forward interest rates are widely used.They can promise the expected situation of future trend of interest in the market, so they are all along the reference tools used by central bank to constitute the money policy.And the more important is that in a mature market, almost all the pricing of interest derivatives depend on the forward rate. Although there are no interest derivatives yet in our country, as the development of finance's globality and the progress of finance market reform, it can be certainly said that these derivatives' introducing is on the way.Thus, this paper does some study about a kind of forward rate HJM model-two factor Gaussian HJM model. Two factor Gaussian HJM model is(u,T)dt* f/ here supposing the market price of risk are constants A*.A;= 1,2,2 rTVL(t,T) = ^ak(t,T)( ak(t,y)dy - Xk).Spot rat R(t,T) isR(t,T) =-^ ￡ f(t,v)dv,Using Taylor expansion on t, we getdR R(t + At, T) - R(t, T) = ^{^}At + o(At),wheredtBy using of the corresponding theory of derivatives, we can work out fl/?Tydv. In the end, we getAt,T) = R(t,T)-{r(t)-R(t,T))At/{T-t)U - t)where o%(t, T) = — Jt ak(t,v)dv.In this way we turn the parameters estimation problem into estimating stochastic volatility and the market price of risk using the datas of spot rate. After a series of transform mentioned in the 5.1 section, we take the point's coordinate on the yield curve as the spot rate's data. Then use the technique of principal component analysis to get the structure of volatility. In the end we use the least square method again to determine the market price of risk. In this way the interest rate term structure represented by two factor Gaussian HJM model is obtained.