《金融期货与期权》（英文版） Chapter 24 Interest Rate derivatives More advanced models

24.1 Interest rate derivatives More advanced models Chapter 24 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 24.1 Interest Rate Derivatives: More Advanced Models Chapter 24

24.2 The two-Factor hul white Model (equation 24.1, page 571) e()+-ax」+o;dz du=-budt + dz where x=f(r)and the correlation between dz, and dz2 is p The short rate reverts to a level dependent on u, and u Itself Is mean reverting Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

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) dx  t u axdt dz du budt dz x f r dz dz u u = + − + = − + =     ( ) ( ) 1 1 2 2 1 2 where and the correlation between and is The short rate reverts to a level dependent on , and itself is mean reverting

24.3 Analytic Results Bond prices and European options on zero-coupon bonds can be calculated analytically when fr)=r Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other 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 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 blacks model The appropriate volatility is calculated from the volatilities of and correlations between the underlying zero-coupon bond prices Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 24.4 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

24.5 Volatility structures In the one-factor ho-lee or hull-White model the forward rate s ds are either constant or decline exponentially. Al forward rates are instantaneously perfectly correlated In 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

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 correlated • In the two-factor model many different forward rate S.D. patterns and correlation structures can be obtained

Example giving Humped Valatility 24.6 Structure(Figure 24. 1, page 572) a=1,b=0.1,o1=0.01,∝2=0.0165,p=0.6 100 0.60 040 0.20 0.00 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

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, 1=0.01, 2=0.0165, =0.6 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40

24.7 ransformation of the General model dx=o(r)+u-ax]dt +0, d= du =-budt +o,dz2 Where x= f(r) and the correlation between dz, and dz, is p We define y=x+u/(b-a)so that y=[()-ay] du=-budt +o2dz2 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 24.7 Transformation of the General Model     dx t u ax dt dz du budt dz x f r dz dz y x u b a dy t ay dt dz du budt dz = + − + = − + = = + − = − + = − +        ( ) ( ) ( ) ( ) 1 1 2 2 1 2 3 3 2 2 where and the correlation between and is We define so that

24.8 ransformation of the General model continued 2po12 0 b-a)2 b The correlation between dz and po, +02 (b-a) Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 24.8 Transformation of the General Model continued         3 2 1 2 2 2 2 1 2 2 3 2 3 2 = + − + − + − ( ) ( ) b a b a dz dz b a The correlation between and is 1

24.9 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 ncorporated into the model Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other 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 constructed • The volatility structure is stationary • Volatility and correlation patterns similar to those in the real world can be incorporated into the model

24.10 hJM Model: notation P(t, T): price at time t of a discount bond with principal of $1 maturing at T Q2 ,: vector of past and present values of interest rates and bond prices at time t that are relevant for determining bond price volatilities at that time V(t, T, Q2,): volatility of P(t, T) Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

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 T Wt : vector of past and present values of interest rates and bond prices at time t that are relevant for determining bond price volatilities at that time v(t,T,Wt ): volatility of P(t,T)