期货期权及其衍生品配套课件全34章Ch30

1、Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,1,Interest Rate Derivatives: Model of the Short Rate,Chapter 30,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,2,Term Structure Models,Blacks model is conc

2、erned with describing the probability distribution of a single variable at a single point in time A term structure model describes the evolution of the whole yield curve,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,3,The Zero Curve,The process for t

3、he instantaneous short rate, r, in the traditional risk-neutral world defines the process for the whole zero curve in this world If P(t, T ) is the price at time t of a zero-coupon bond maturing at time T where is the average r between times t and T,Options, Futures, and Other Derivatives, 7th Inter

4、national Edition, Copyright John C. Hull 2008,4,Equilibrium Models,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,5,Mean Reversion (Figure 30.1, page 675),Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,

5、6,Alternative Term Structures in Vasicek pm=0.5; pd=0.25; Time step=1yr,10% 0.35*,12% 1.11*,10% 0.23,8% 0.00,14% 3,12% 1,10% 0,8% 0,6% 0,*: (0.253 + 0.501 + 0.250)e0.121 *: (0.251.11 + 0.500.23 +0.250)e0.101,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2

6、008,18,Alternative Branching Processes in a Trinomial Tree(Figure 30.7, page 685),Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,19,Procedure for Building Tree,dr = q(t ) ar dt + sdz 1.Assume q(t ) = 0 and r (0) = 0 2.Draw a trinomial tree for r to ma

7、tch the mean and standard deviation of the process for r 3.Determine q(t ) one step at a time so that the tree matches the initial term structure,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,20,Example (page 686 to 691),s = 0.01 a = 0.1 Dt = 1 year

8、The zero curve is as shown in Table 30.1 on page 697,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,21,Building the First Tree for the Dt rate R,Set vertical spacing: Change branching when jmax nodes from middle where jmax is smallest integer greater

9、than 0.184/(aDt) Choose probabilities on branches so that mean change in R is -aRDt and S.D. of change is,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,22,The First Tree(Figure 30.8, page 687),Options, Futures, and Other Derivatives, 7th Internationa

10、l Edition, Copyright John C. Hull 2008,23,Shifting Nodes,Work forward through tree Remember Qij the value of a derivative providing a $1 payoff at node j at time iDt Shift nodes at time iDt by ai so that the (i+1)Dt bond is correctly priced,Options, Futures, and Other Derivatives, 7th International

11、Edition, Copyright John C. Hull 2008,24,The Final Tree(Figure 30.9, Page 689),Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,25,Extensions,The tree building procedure can be extended to cover more general models of the form: d(r ) = q(t ) a (r )dt + s

12、dz We set x=f(r) and proceed similarly to before,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,26,Calibration to Determine a and s,The volatility parameters a and s (perhaps functions of time) are chosen so that the model fits the prices of actively traded instruments such as caps and European s as closely as possible We minimize a function of the fo