1. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.1 Interest Rate Derivatives: More Advanced Models Chapter 22

2. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.2 The Two-Factor Hull-White Model (Equation 22.1, page 602)

3. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.3 Analytic Results Bond prices and European options on zero-coupon bonds can be calculated analytically when f(r) = r

4. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.4 Options on Coupon-Bearing Bonds We cannot use the same procedure for options on coupon-bearing bonds as we do in the case of one-factor models If we make the approximate assumption that the coupon-bearing bond price is lognormal, we can use Black’s model The appropriate volatility is calculated from the volatilities of and correlations between the underlying zero-coupon bond prices

5. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.5 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 correlated In the two-factor model many different forward rate S.D. patterns and correlation structures can be obtained

6. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.6 Example Giving Humped Volatility Structure (Figure 22.1, page 603) a=1, b=0.1,  1 =0.01,  2 =0.0165,  =0.6 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40

7. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.7 Transformation of the General Model

8. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.8 Transformation of the General Model continued

9. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.9 Attractive Features of the Model It is Markov so that a recombining 3- dimensional tree can be constructed The volatility structure is stationary Volatility and correlation patterns similar to those in the real world can be incorporated into the model

10. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.10 HJM Model: Notation P(t,T ) : price at time t of a discount bond with principal of $1 maturing at T  t  : vector of past & present values of interest rates & bond prices at time t that are relevant for determining bond price volatilities at that time v(t,T,  t  ): volatility of P(t,T)

11. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.11 Notation continued ƒ(t,T 1,T 2 ) : forward rate as seen at t for the period between T 1 and T 2 F(t,T) : instantaneous forward rate as seen at t for a contract maturing at T r(t) : short-term risk-free interest rate at t dz(t) : Wiener process driving term structure movements

12. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.12 Modeling Bond Prices

13. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.13 Modeling Forward Rates Equation 22.7, page 606)

14. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.14 Two Factor Model (Equation 22.12, page 608)  1 (t, T )  2 (t, T ) T – t

15. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.15 Tree For a General Model A non-recombining tree means that the process for r is non-Markov

16. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.16 Monte Carlo Implementation (Equations 22.13 and 22.14, page 609) The HJM model is usually implemented using Monte Carlo simulation

17. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.17 The LIBOR Market Model The LIBOR market model is a model that is constructed so that it is exactly consistent with either cap prices (but not both)

18. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.18 Notation

19. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.19 Volatility Structure

20. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.20 Volatility Structure continued (Equation 22.19, page 611)

21. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.21 Example 22.1 (Page 611) If Black volatilities for the first three caplets are 24%, 22%, and 20%, then  0 =24.00%  1 =19.80%  2 =15.23%

22. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.22 Example 22.2 (Page 611)

23. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.23 The Process for F k in a One- Factor LIBOR Market Model

24. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.24 Rolling Forward Risk- Neutrality (Equation 22.20, page 612) 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 t i+1 to t i at the  i rate observed at time t i. The process for F k is

25. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.25 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

26. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.26 Monte Carlo Implementation of BGM Cap Model (Equation 22.22, page 612)

27. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.27 Multifactor Versions of BGM BGM can be extended so that there are several components to the volatility A factor analysis can be used to determine how the volatility of F k is split into components

28. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.28 Pros and Cons of LIBOR Market Model It is automatically calibrated to cap prices It can be implemented using Monte Carlo simulation Unfortunately the model is non-Markov and trees produced for it do not in general recombine

29. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.29 Types of Mortgage-Backed Securities (MBSs) Pass-Through Collateralized Mortgage Obligation (CMO) Interest Only (IO) Principal Only (PO)

30. Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 22.30 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 spread We use an iterative procedure to calculate the spread that makes the derivative’s model price = market price. This is the OAS.