1、interest rate derivatives: model of the short ratechapter 30options, futures, and other derivatives, 7th international edition, copyright john c. hull 20081term structure models blacks model is concerned with describing the probability distribution of a single variable at a single point in time a te

2、rm structure model describes the evolution of the whole yield curveoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 20082the zero curve the process for the instantaneous short rate, r, in the traditional risk-neutral world defines the process for the whole z

3、ero curve in this world if p(t, t ) is the price at time t of a zero-coupon bond maturing at time twhere is the average r between times t and toptions, futures, and other derivatives, 7th international edition, copyright john c. hull 20083 p t te er t t( , )()requilibrium modelsoptions, futures, and

4、 other derivatives, 7th international edition, copyright john c. hull 20084rendleman & bartter: vasicek: cox, ingersoll, & ross (cir): drr dtr dzdra br dtdzdra br dtr dz()()mean reversion (figure 30.1, page 675)options, futures, and other derivatives, 7th international edition, copyright joh

5、n c. hull 20085interestratehigh interest rate has negative trendlow interest rate has positive trendreversionlevelalternative term structures in vasicek & cir (figure 30.2, page 676) options, futures, and other derivatives, 7th international edition, copyright john c. hull 20086zero ratematurity

6、zero ratematurityzero ratematurityequilibrium vs no-arbitrage models in an equilibrium model todays term structure is an output in a no-arbitrage model todays term structure is an inputoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 20087developing no-arbit

7、rage model for ra model for r can be made to fit the initial term structure by including a function of time in the driftoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 20088 ho-lee modeldr = q(t)dt + dz many analytic results for bond prices and option price

8、s interest rates normally distributed one volatility parameter, all forward rates have the same standard deviationoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 20089diagrammatic representation of ho-lee (figure 30.3, page 679)options, futures, and other d

9、erivatives, 7th international edition, copyright john c. hull 200810short raterrrrtimehull-white model dr = q(t ) ar dt + dz many analytic results for bond prices and option prices two volatility parameters, a and interest rates normally distributed standard deviation of a forward rate is a declinin

10、g function of its maturity options, futures, and other derivatives, 7th international edition, copyright john c. hull 200811diagrammatic representation of hull and white (figure 30.4, page 680) options, futures, and other derivatives, 7th international edition, copyright john c. hull 200812short rat

11、errrrtimeforward ratecurveblack-karasinski model (equation 30.18) future value of r is lognormal very little analytic tractability options, futures, and other derivatives, 7th international edition, copyright john c. hull 200813dztdtrtatrd)()ln()()()ln(qoptions on zero-coupon bonds (equation 30.20,

12、page 682)in vasicek and hull-white model, price of call maturing at t on a bond lasting to s islp(0,s)n(h)-kp(0,t)n(h-p)price of put iskp(0,t)n(-h+p)-lp(0,s)n(h)whereoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200814ttsklaeeaktpslphpattsappp)( lee-hofor

13、 price. strike theis and principal theis 2112), 0(), 0(ln12)(options on coupon bearing bonds in a one-factor model a european option on a coupon-bearing bond can be expressed as a portfolio of options on zero-coupon bonds. we first calculate the critical interest rate at the option maturity for whic

14、h the coupon-bearing bond price equals the strike price at maturity the strike price for each zero-coupon bond is set equal to its value when the interest rate equals this critical valueoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200815interest rate tre

15、es vs stock price trees the variable at each node in an interest rate tree is the dt-period rate interest rate trees work similarly to stock price trees except that the discount rate used varies from node to nodeoptions, futures, and other derivatives, 7th international edition, copyright john c. hu

16、ll 200816two-step tree example (figure 30.6, page 684)payoff after 2 years is max100(r 0.11), 0pu=0.25; pm=0.5; pd=0.25; time step=1yroptions, futures, and other derivatives, 7th international edition, copyright john c. hull 20081710%0.35*12% 1.11*10% 0.238% 0.0014% 312% 110% 08% 06% 0 *: (0.253 + 0

17、.501 + 0.250)e0.121 *: (0.251.11 + 0.500.23 +0.250)e0.101alternative branching processes in a trinomial tree(figure 30.7, page 685)options, futures, and other derivatives, 7th international edition, copyright john c. hull 200818(a)(b)(c)procedure for building treedr = q(t ) ar dt + dz 1.assume q(t )

18、 = 0 and r (0) = 02.draw a trinomial tree for r to match the mean and standard deviation of the process for r3.determine q(t ) one step at a time so that the tree matches the initial term structureoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200819exampl

19、e (page 686 to 691) = 0.01 a = 0.1 dt = 1 year the zero curve is as shown in table 30.1 on page 697options, futures, and other derivatives, 7th international edition, copyright john c. hull 200820building the first tree for the d dt rate r set vertical spacing: change branching when jmax nodes from

20、middle where jmax is smallest integer greater 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 200821trdd3tdthe first tree(figure 30.8, page 687)options

21、, futures, and other derivatives, 7th international edition, copyright john c. hull 200822abcdefghinodeabcdefghir0.000% 1.732% 0.000% -1.732% 3.464% 1.732% 0.000% -1.732% -3.464%pu0.1667 0.1217 0.16670.22170.8867 0.1217 0.16670.22170.0867pm0.6666 0.6566 0.66660.65660.0266 0.6566 0.66660.65660.0266pd

22、0.1667 0.2217 0.16670.12170.0867 0.2217 0.16670.12170.8867shifting 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 pricedoptions, futures, and other derivatives, 7th

23、 international edition, copyright john c. hull 200823the final tree(figure 30.9, page 689)options, futures, and other derivatives, 7th international edition, copyright john c. hull 200824abcdefghinodeabcdefghir3.824% 6.937%5.205%3.473%9.716% 7.984% 6.252%4.520%2.788%pu0.16670.12170.16670.22170.88670.12170.16670.22170.0867pm0.66660.65660.66660.65660.02660.65660.66660.65660.0266pd0.16670.22170.16670.12170.08670.22170.16670.12170.8867extensionsthe tree building procedure c