1、Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.1,Interest Rate Derivatives: More Advanced ModelsChapter 24,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.2,The Two-Factor Hull-White Model (Equation 24.1, page 571),Options, Futures, and Othe
2、r Derivatives, 5th edition 2002 by John C. Hull,24.3,Analytic Results,Bond prices and European options on zero-coupon bonds can be calculated analytically when f(r) = r,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.4,Options on Coupon-Bearing Bonds,We cannot use the sa
3、me procedure for options on coupon-bearing bonds as we do in the case of one-factor modelsIf we make the approximate assumption that the coupon-bearing bond price is lognormal, we can use Blacks modelThe appropriate volatility is calculated from the volatilities of and correlations between the under
4、lying zero-coupon bond prices,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.5,Volatility Structures,In the one-factor Ho-Lee or Hull-White model the forward rate S.D.s are either constant or decline exponentially. All forward rates are instantaneously perfectly correla
5、tedIn the two-factor model many different forward rate S.D. patterns and correlation structures can be obtained,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.6,Example Giving Humped Volatility Structure (Figure 24.1, page 572)a=1, b=0.1, s1=0.01, s2=0.0165, r=0.6,0.00,
6、0.20,0.40,0.60,0.80,1.00,1.20,1.40,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.7,Transformation of the General Model,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.8,Transformation of the General Model continued,Options, Futures, and Oth
7、er Derivatives, 5th edition 2002 by John C. Hull,24.9,Attractive Features of the Model,It is Markov so that a recombining 3-dimensional tree can be constructedThe volatility structure is stationaryVolatility and correlation patterns similar to those in the real world can be incorporated into the mod
8、el,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.10,HJM Model: Notation,P(t,T ): price at time t of a discount bond with principal of $1 maturing at TWt : vector of past and present values of interest rates and bond prices at time t that are relevant for determining bo
9、nd price volatilities at that timev(t,T,Wt ): volatility of P(t,T),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.11,Notation continued,(t,T1,T2): forward rate as seen at t for the period between T 1 and T 2F(t,T): instantaneous forward rate as seen at t for a contract
10、maturing at Tr(t): short-term risk-free interest rate at tdz(t): Wiener process driving term structure movements,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.12,Modeling Bond Prices,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.13,Modeli
11、ng Forward RatesEquation 24.7, page 575),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.14,Tree For a General Model,A non-recombining tree means that the process for r is non-Markov,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.15,The LIBO
12、R Market Model,The LIBOR market model is a model constructed in terms of the forward rates underlying caplet prices,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.16,Notation,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.17,Volatility Stru
13、cture,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.18,In Theory the Ls can be determined from Cap Prices,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.19,Example 24.1 (Page 579),If Black volatilities for the first threecaplets are 24%, 2
14、2%, and 20%, thenL0=24.00%L1=19.80%L2=15.23%,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.20,Example 24.2 (Page 579),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.21,The Process for Fk in a One-Factor LIBOR Market Model,Options, Futures,
15、 and Other Derivatives, 5th edition 2002 by John C. Hull,24.22,Rolling Forward Risk-Neutrality (Equation 24.16, page 579),It is often convenient to choose a world that is always FRN wrt a bond maturing at the next reset date. In this case, we can discount from ti+1 to ti at the di rate observed at t
16、ime ti. The process for Fk is,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.23,The LIBOR Market Model and HJM,In 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 ris
17、k-neutral world,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.24,Monte Carlo Implementation of BGM Cap Model (Equation 24.18, page 580),Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.25,Multifactor Versions of BGM,BGM can be extended so th
18、at there are several components to the volatilityA factor analysis can be used to determine how the volatility of Fk is split into components,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.26,Ratchet Caps, Sticky Caps, and Flexi Caps,A plain vanilla cap depends only on
19、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 factors,Options, Futures, and Other Derivatives, 5th edition 2002 by John C.
20、 Hull,24.27,Valuing European Options in the LIBOR Market Model,There is a good analytic approximation that can be used to value European swap options in the LIBOR market model. See pages 582 to 584.,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.28,Calibrating the LIBOR
21、 Market Model,In theory the LMM can be exactly calibrated to cap prices as described earlierIn practice we proceed as for the one-factor models in Chapter 23 and minimize a function of the formwhere Ui is the market price of the ith calibrating instrument, Vi is the model price of the ith calibratin
22、g instrument and P is a function that penalizes big changes or curvature in a and s,Options, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.29,Types of Mortgage-Backed Securities (MBSs),Pass-ThroughCollateralized Mortgage Obligation (CMO)Interest Only (IO)Principal Only (PO),Opt
23、ions, Futures, and Other Derivatives, 5th edition 2002 by John C. Hull,24.30,Option-Adjusted Spread(OAS),To calculate the OAS for an interest rate derivativewe value it assuming that the initial yield curve is the Treasury curve + a spreadWe use an iterative procedure to calculate the spread that makes the derivatives model price = market price. This is the OAS.,
