《固定收益证券chap08estimatingthebinomialtree》由会员分享，可在线阅读，更多相关《固定收益证券chap08estimatingthebinomialtree（28页珍藏版）》请在金锄头文库上搜索。
1、Chapter08,ESTIMATING THE BINOMIAL TREE,Interest Rate Tree Binomial Model,we assumed a simple binomial approach where in each period the one-period spot rate would either increase to equal to a proportion u times its initial value or decrease to equal to a proportion d times the initial rate, with pr
2、obability of the increase in one period being q = .5. At the end of n periods, this binomial process yields a distribution of n+1 possible spot rates (e.g., for n = 3, there are four possible rates: Suuu = u3S0, Suud = u2d S0, Sudd = ud2 S0, and Sddd = d3S0 ).,Interest Rate Tree Binomial Model,Gener
3、al binomial model Given current level of short-term rate r, next-period short rate, can take on only two possible values: an upper value Su and a lower value Sd, with equal probability 0.5 In period 2, the short-term interest rate can take on four possible values: Suu, Sud, Sdu, Sdd More generally,
4、in period n, the short-term interest rate can take on 2n values = very time-consuming and computationally inefficient Recombining trees Means that an upward-downward sequence leads to the same result as a downward-upward sequence For example, Sud = Sdu Only (n+1) different values at period n,SUBDIVI
5、DING THE BINOMIAL TREE,h = length of the period in years; n = number of periods of length h defining the maturity of the bond, where n = (maturity in years)/h,ESTIMATING THE BINOMIAL TREE,u and d Estimation Approach Calibration Model,distribution of logarithmic returns,This distribution, though, is
6、not normally distributed since spot rates cannot be negative (i.e., we normally do not have negative interest rates). However, the distribution of spot rates can be converted into a distribution of logarithmic returns, gn , where:,u and d Estimation Approach,Probability Distribution Resulting from a
7、 Binomial Process,let and e and Ve be the estimated mean and variance of the logarithmic return of spot rates for a period equal in length to n periods,Solving for u and d,for large n (n = 30),Annualized Mean and Variance,Annualized parameters are obtained by simply multiplying the estimated paramet
8、ers of a given length by the number of periods of that length that make up a year. For example, if quarterly data is used to estimate the mean and variance (qe and Vqe ), then we simply multiply those estimates by four to obtain the annualized parameters (Ae = 4qe and (VAe = 4Vqe ).,example,If the a
9、nnualized mean and variance of the logarithmic return of one-year spot rates were .044 and .0108, and we wanted to evaluate a three-year bond with six-month periods (h = of a year), then we would use a six-period tree to value the bond (n = (3 years)/() = 6 periods) and u and d would be 1.1 and .95:
10、,Interest Rate Tree PDE,where are independent variables taking on values (+1,-1) with proba (1/2,1/2) Continuous-time limit (Merton (1973),Drawbacks of u and d estimation approach,A binomial interest rates tree generated using the u and d estimation approach is constrained to have an end-of-the-peri
11、od distribution with a mean and variance that matches the analysts estimated mean and variance. The tree is not constrained, however, to yield a bond price that matches its equilibrium value.,Calibration Model,Calibration Model,The calibration model generates a binomial tree by first finding spot ra
12、tes which satisfy a variability condition between the upper and lower rates. Given the variability relation, the model then solves for the lower spot rate which satisfies a price condition where the bond value obtained from the tree is consistent with the equilibrium bond price given the current yie
13、ld curve.,Variability Condition,Price Condition,One of the problems with using just a variability condition (or equivalently just u and d estimates) is that it does not incorporate all of the information.,example,To see this, suppose the current yield curve has one-, two-, and three-year spot rates
14、of y1 = 10%, y2 = 10.12238%, and y3 = 10.24488%, respectively. Furthermore, suppose that we estimate the annualized logarithmic mean and variance to be .048167 and .0054, respectively.,Su = 11% and Sd = 9.5%,Two-Period Binomial Tree,example,Interest Rate Tree PDE,Calibration of the model is performed so as to make model consistent with the current term structure,
