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1、Chapter 30Interest Rate Derivatives:Model of the Short Rate,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,1,Term Structure Models,Blacks model is concerned with describing the probability distribution of a single variable at a single point in timeA term structure model
2、 describes the evolution of the whole yield curve,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,2,The Zero Curve,The process for the instantaneous short rate,r,in the traditional risk-neutral world defines the process for the whole zero curve in this worldIf P(t,T)is t
3、he price at time t of a zero-coupon bond maturing at time Twhere is the average r between times t and T,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,3,Equilibrium Models(Risk Neutral World),Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,4
4、,Mean Reversion(Figure 30.1,page 684),Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,5,Alternative Term Structures in Vasicek&CIR(Figure 30.2,page 686),Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,6,Properties of Vasicek and CIR,P(t,T)=A(
5、t,T)eB(t,T)rThe A and B functions are different for the two modelsThese can be used to provide alternative duration and convexity measures,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,7,Bond Price Processes in a Risk Neutral World,From Itos lemmaMarket price of intere
6、st rate risk appears to be about 1.2This can be used to convert a real world process to a risk-neutral process or vice versa,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,8,Equilibrium vs No-Arbitrage Models,In an equilibrium model todays term structure is an outputIn
7、a no-arbitrage model todays term structure is an input,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,9,Developing No-Arbitrage Model for r,A model for r can be made to fit the initial term structure by including a function of time in the drift,Options,Futures,and Other
8、 Derivatives,8th Edition,Copyright John C.Hull 2012,10,Ho-Lee Model,dr=q(t)dt+sdzMany analytic results for bond prices and option pricesInterest rates normally distributedOne volatility parameter,sAll forward rates have the same standard deviation,Options,Futures,and Other Derivatives,8th Edition,Co
9、pyright John C.Hull 2012,11,Diagrammatic Representation of Ho-Lee(Figure 30.3,page 691),Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,12,Hull-White Model,dr=q(t)ar dt+sdzMany analytic results for bond prices and option pricesTwo volatility parameters,a and sInterest ra
10、tes normally distributedStandard deviation of a forward rate is a declining function of its maturity,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,13,Diagrammatic Representation of Hull and White(Figure 30.4,page 692),Options,Futures,and Other Derivatives,8th Edition,C
11、opyright John C.Hull 2012,14,Black-Karasinski Model(equation 30.18),Future value of r is lognormalVery little analytic tractability,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,15,Options on Zero-Coupon Bonds(equation 30.20,page 694),In Vasicek and Hull-White model,pr
12、ice of call maturing at T on a zero-coupon bond lasting to s isLP(0,s)N(h)KP(0,T)N(hsP)Price of put isKP(0,T)N(h+sP)LP(0,s)N(h)where,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,16,Options on Coupon-Bearing Bonds,In a one-factor model a European option on a coupon-bea
13、ring 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 which the coupon-bearing bond price equals the strike price at maturityThe strike price for each zero-coupon bond is set equal to its value when the inte
14、rest rate equals this critical value,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,17,Interest Rate Trees vs Stock Price Trees,The variable at each node in an interest rate tree is the Dt-period rateInterest rate trees work similarly to stock price trees except that th
15、e discount rate used varies from node to node,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,18,Two-Step Tree Example(Figure 30.6,page 697),Payoff after 2 years is MAX100(r 0.11),0pu=0.25;pm=0.5;pd=0.25;Time step=1yr,Options,Futures,and Other Derivatives,8th Edition,Cop
16、yright John C.Hull 2012,19,10%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,Alternative Branching Processes in a Trinomial Tree(Figure 30.7,page 698),Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,20,Pr
17、ocedure for Building Tree,dr=q(t)ar dt+sdz 1.Assume q(t)=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 structure,Options,Futures,and Other Derivatives,8th Edition,Cop
18、yright John C.Hull 2012,21,Example(page 700 to 705),s=0.01 a=0.1 Dt=1 year,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,22,Building the First Tree for the Dt rate R,Set vertical spacing:Change branching when jmax nodes from middle where jmax is smallest integer greate
19、r 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,8th Edition,Copyright John C.Hull 2012,23,The First Tree(Figure 30.8,page 699),Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,2
20、4,Shifting Nodes,Work forward through treeRemember Qij the value of a derivative providing a$1 payoff at node j at time iDtShift nodes at time iDt by ai so that the(i+1)Dt bond is correctly priced,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,25,The Final Tree(Figure 3
21、0.9,Page 702),Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,26,Extensions,The tree building procedure can be extended to cover more general models of the form:d(r)=q(t)a(r)dt+sdzWe set x=f(r)and proceed similarly to before,Options,Futures,and Other Derivatives,8th Edit
22、ion,Copyright John C.Hull 2012,27,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 swap options as closely as possibleWe 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 s,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,28,
