《金融期货与期权》（英文版） Chapter 23 Interest Rate derivatives Models of the short rate

3.1 Interest rate derivatives Models of the short rate Chapter 23 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 23.1 Interest Rate Derivatives: Models of the Short Rate Chapter 23

23.2 Term Structure models Blacks model is concerned with describing the probability distribution of a single variable at a single point in time aterm structure model describes the evolution of the whole yield curve Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 23.2 Term Structure Models • Black’s model is concerned with describing the probability distribution of a single variable at a single point in time • A term structure model describes the evolution of the whole yield curve

23.3 Use of risk-Neutral Arguments The process for the instantaneous short rate. r. in the traditional risk-neutral world defines the process for the whole zero curve in this world If P(t, T)is the price at time t of a zero coupon bond maturing at time t P()=E| Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

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 traditional risk-neutral world defines the process for the whole zero curve in this world • If P(t, T ) is the price at time t of a zero￾coupon bond maturing at time T P t T E e  r T t ( , )  ( ) = − −

Equilibrium models Rendleman bartter. dr= ur dt +or dz Vasicek: b-r)dt+odz Cox, Ingersoll, ROSS CIR) dr=a( b-r)dt +ovr dz Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 23.4 Equilibrium Models Rendleman & Bartter: Vasicek: Cox, Ingersoll, & Ross (CIR): dr r dt r dz dr a b r dt dz dr a b r dt r dz = + = − + = − +     ( ) ( )

23.5 Mean reversion (Figure 23. 1, page 539) Interest rate HIGH interest rate has negative trend Reversion Level LOW interest rate has positive trend Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 23.5 Mean Reversion (Figure 23.1, page 539) Interest rate HIGH interest rate has negative trend LOW interest rate has positive trend Reversion Level

Alternative Term structures 23.6 in vasicek cir (Figure 23. 2, page 540 Zero rate Zero rate Maturity Maturity Zero rate Maturity Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Alternative Term Structures 23.6 in Vasicek & CIR (Figure 23.2, page 540) Zero Rate Maturity Zero Rate Maturity Zero Rate Maturity

23.7 Equilibrium vs No-Arbitrage Models In an equilibrium model today's term structure is an output In a no-arbitrage model today' s term structure is an input Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 23.7 Equilibrium vs No-Arbitrage Models • In an equilibrium model today’s term structure is an output • In a no-arbitrage model today’s term structure is an input

23.8 Developing No-arbitrage Model for r a model for r can be made to fit 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

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 fit the initial term structure by including a function of time in the drift

23.9 Ho and lee dr =e(tat odz Many analytic results for bond prices and option prices Interest rates normally distributed One volatility parameter, o all forward rates have the same standard deviation Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 23.9 Ho and Lee dr = q(t )dt + dz • Many analytic results for bond prices and option prices • Interest rates normally distributed • One volatility parameter,  • All forward rates have the same standard deviation

Diagrammatic Representation of 23.10 Ho and lee Short Rate Initial forward Cu urve Time Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 23.10 Initial Forward Curve Short Rate r r r r Time Diagrammatic Representation of Ho and Lee