1、Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.1,Interest Rate Derivatives: Models of the Short RateChapter 23,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.2,Term Structure Models,Blacks model is concerned with describing the probability
2、distribution of a single variable at a single point in timeA term structure model describes the evolution of the whole yield curve,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.3,Use of Risk-Neutral Arguments,The process for the instantaneous short rate, r, in the trad
3、itional risk-neutral world defines the process for the whole zero curve in this worldIf P(t, T ) is the price at time t of a zero-coupon bond maturing at time T,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.4,Equilibrium Models,Options, Futures, and Other Derivatives,
4、5th edition 2002 by John C. Hull,23.5,Mean Reversion (Figure 23.1, page 539),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.6,Alternative Term Structuresin Vasicek & CIR (Figure 23.2, page 540),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23
5、.7,Equilibrium vs No-Arbitrage Models,In an equilibrium model todays term structure is an outputIn a no-arbitrage model todays term structure is an input,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.8,Developing No-Arbitrage Model for r,A model for r can be made to fi
6、t the initial term structure by including a function of time in the drift,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.9,Ho and Lee,dr = q(t )dt + sdzMany analytic results for bond prices and option pricesInterest rates normally distributedOne volatility parameter, sA
7、ll forward rates have the same standard deviation,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.10,Initial ForwardCurve,Short Rate,r,r,r,r,Time,Diagrammatic Representation of Ho and Lee,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.11,Hul
8、l and White Model,dr = q(t ) ar dt + sdzMany analytic results for bond prices and option pricesTwo volatility parameters, a and sInterest rates normally distributedStandard deviation of a forward rate is a declining function of its maturity,Options, Futures, and Other Derivatives, 5th edition 2002 b
9、y John C. Hull,23.12,Diagrammatic Representation of Hull and White,Short Rate,r,r,r,r,Time,Forward RateCurve,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.13,Options on Coupon Bearing Bonds,A European option on a coupon-bearing bond can be expressed as a portfolio of o
10、ptions 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 interest rate equals this critical value,Options,
11、 Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.14,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 the discount rate used varies from node to node
12、,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.15,Two-Step Tree Example (Figure 23.6, page 551),Payoff after 2 years is MAX100(r 0.11), 0pu=0.25; pm=0.5; pd=0.25; Time step=1yr,0.35*,1.11*,0.23,0.00,0.14 3,0.12 1,0.10 0,0.08 0,0.06 0,r,P,*: (0.253 + 0.501 + 0.250)e0.12
13、1 *: (0.251.11 + 0.500.23 +0.250)e0.101,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.16,Alternative Branching Processes in a Trinomial Tree(Figure 23.7, page 552),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.17,An Overview of the Tree B
14、uilding Procedure,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, 5th ed
15、ition 2002 by John C. Hull,23.18,Example,s = 0.01 a = 0.1 dt = 1 year The zero curve is as shown in Table 23.1 on page 556,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.19,The Initial Tree(Figure 23.8, page 554),Node,A,r,0.000%,1.732%,0.000%,-1.732%,3.464%,1.732%,0.000
16、%,-1.732%,-3.464%,p,u,0.1667,0.1217,0.1667,0.2217,0.8867,0.1217,0.1667,0.2217,0.0867,p,m,0.6666,0.6566,0.6666,0.6566,0.0266,0.6566,0.6666,0.6566,0.0266,p,d,0.1667,0.2217,0.1667,0.1217,0.0867,0.2217,0.1667,0.1217,0.8867,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.20,T
17、he Final Tree(Figure 23.9, Page 556),Node,A,B,C,D,E,F,G,H,I,r,3.824% 6.937%,5.205%,3.473%,9.716%,7.984%,6.252%,4.520%,2.788%,p,u,0.1667,0.1217,0.1667,0.2217,0.8867,0.1217,0.1667,0.2217,0.0867,p,m,0.6666,0.6566,0.6666,0.6566,0.0266,0.6566,0.6666,0.6566,0.0266,p,d,0.1667,0.2217,0.1667,0.1217,0.0867,0.
18、2217,0.1667,0.1217,0.8867,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.21,Extensions,The tree building procedure can be extended to cover more general models of the form:d(r ) = q(t ) a (r )dt + sdz,Options, Futures, and Other Derivatives, 5th edition 2002 by John C.
19、Hull,23.22,Other Models,These models allow the initial volatility environment to be matched exactlyBut the future volatility structure may be quite different from the current volatility structure,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.23,Calibration: a and s con
20、stant,The volatility parameters a and s are chosen so that the model fits the prices of actively traded instruments such as caps and European swap options as closely as possibleWe can choose a global best fit value of a and imply s from the prices of actively traded instruments. This creates a volat
21、ility surface for interest rate derivatives similar to that for equity option or currency options (see Chapter 15),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,23.24,Calibration: a and s functions of time,We 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,
