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

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)