1、interest rate derivatives: hjm and lmmchapter 31options, futures, and other derivatives, 7th international edition, copyright john c. hull 20081hjm model: notationoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 20082 p(t,t ): wt :v(t,t,wt ): price at time t

2、 of a discount bond with principal of $1 maturing at tvector of past and present values of interest rates and bond prices at time t that are relevant for determining bond price volatilities at that timevolatility of p(t,t)notation continuedoptions, futures, and other derivatives, 7th international e

3、dition, copyright john c. hull 20083(t,t1,t2): f(t,t): r(t): dz(t): forward rate as seen at t for the period between t1 and t2instantaneous forward rate as seen at t for a contract maturing at tshort-term risk-free interest rate at twiener process driving term structure movementsmodeling bond prices

4、 (equation 31.1, page 704)options, futures, and other derivatives, 7th international edition, copyright john c. hull 20084 ) for process a get we approach letting for process the determine to lemma sito use can we because all for providing function any choose can we t,tftt),tf(t,tttttpttp),tf(t,tttt

5、vvtdzttpttvdtttptrttdptt(.),(ln),(ln0),()(),(),(),()(),(1221122121the process for f(t,t)equation 31.4 and 31.5, page 705)options, futures, and other derivatives, 7th international edition, copyright john c. hull 20085factor one than more is there whenhold results similar have must we(),( write weif

6、that means result thisds(t,)t,s(t,)t, m(t,)dzt,s(t,dtt,t,mttdftdzttvdtttvttvttdftttttttttttt),)(),(),(),(),(tree evolution of term structure is non-recombining options, futures, and other derivatives, 7th international edition, copyright john c. hull 20086tree for the short rate r is non-markov (see

7、 figure 31.1, page 714)the libor market model the libor market model is a model constructed in terms of the forward rates underlying caplet pricesoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 20087notationoptions, futures, and other derivatives, 7th inter

8、national edition, copyright john c. hull 20088tkf tttm tttf ttv tp t ttttkkkkkkkkkkk: th reset date forward rate between times and : index for next reset date at time volatility of at time volatility of ( , at time ( ):( )( ):( )( ):):11volatility structureoptions, futures, and other derivatives, 7t

9、h international edition, copyright john c. hull 20089we assume a stationary volatility structurewhere the volatility of depends only on the number of accrual periods between the next reset date and i.e., it is a function onlyof f ttkm tkk( )( )in theory the l ls can be determined from cap pricesopti

10、ons, futures, and other derivatives, 7th international edition, copyright john c. hull 200810yinductivel determined be to s the allows thishave must weprices cap to fit perfect a provides model the if caplet. the for volatility the is if when of volatility the as definei),()()(11221lllkiiikkkkkkkttt

11、itmktfexample 31.1 (page 708) if black volatilities for the first threecaplets are 24%, 22%, and 20%, thenl0=24.00%l1=19.80%l2=15.23%options, futures, and other derivatives, 7th international edition, copyright john c. hull 200811example 31.2 (page 708)options, futures, and other derivatives, 7th in

12、ternational edition, copyright john c. hull 200812n12345n(%)15.5018.2517.9117.74 17.27ln-1(%)15.5020.6417.2117.22 15.25n678910n(%)16.7916.3016.0115.7615.54ln-1(%)14.1512.9813.8113.6013.40the process for fk in a one-factor libor market modeloptions, futures, and other derivatives, 7th international e

13、dition, copyright john c. hull 200813dff dzp t tkkm tki l( )( ,),the drift depends on the world chosenin a world that is forward risk -neutralwith respect to the drift is zero1rolling forward risk-neutrality (equation 31.12, page 709)it is often convenient to choose a world that is always frn wrt a

14、bond maturing at the next reset date. in this case, we can discount from ti+1 to ti at the i rate observed at time ti. the process for fk isoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200814dffffdtdzkkiii m tk m tiij m tik m tlll( )( )( )( )1the libor m

15、arket model and hjmin the limit as the time between resets tends to zero, the libor market model with rolling forward risk neutrality becomes the hjm model in the traditional risk-neutral worldoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200815monte carl

16、o implementation of lmm model (equation 31.14, page 709)options, futures, and other derivatives, 7th international edition, copyright john c. hull 200816we assume no change to the drift between reset dates so thatf tf tf tlkjkjiiji jkjjjkjj kikkjj()( )exp( )1212 llllmultifactor versions of lmm lmm c

17、an be extended so that there are several components to the volatility a factor analysis can be used to determine how the volatility of fk is split into componentsoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200817ratchet caps, sticky caps, and flexi caps

18、 a plain vanilla cap depends only on one forward rate. its price is not dependent on the number of factors. ratchet caps, sticky caps, and flexi caps depend on the joint distribution of two or more forward rates. their prices tend to increase with the number of factorsoptions, futures, and other der

19、ivatives, 7th international edition, copyright john c. hull 200818valuing european options in the libor market modelthere is an analytic approximation that can be used to value european swap options in the libor market model. see equations 31.18 and 31.19 on page 713 options, futures, and other deri

20、vatives, 7th international edition, copyright john c. hull 200819calibrating the libor market model in theory the lmm can be exactly calibrated to cap prices as described earlier in practice we proceed as for short rate models to minimize a function of the formwhere ui is the market price of the ith calibrating instrument, vi is the model price of the ith calibrating instrument and p is a function that penalizes big changes or curvature in a and options, futures, and other derivatives, 7th