《金融期货与期权》（英文版） Chapter 22 Interest Rate derivatives The standard market models

22.1 Interest Rate Derivatives The standard market models Chapter 22 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 22.1 Interest Rate Derivatives: The Standard Market Models Chapter 22

22.2 Why Interest rate Derivatives are Much more difficult to Value Than Stock Options We are dealing with the whole term structure of interest rates; not a single variable The probabilistic behavior of an individual interest rate is more complicated than that of a stock price Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 22.2 Why Interest Rate Derivatives are Much More Difficult to Value Than Stock Options • We are dealing with the whole term structure of interest rates; not a single variable • The probabilistic behavior of an individual interest rate is more complicated than that of a stock price

Why Interest Rate Derivatives are 22.3 Much more difficult to value than Stock Options Volatilities of different points on the term structure are different nterest rates are used for discounting as well as for defining the payoff Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 22.3 Why Interest Rate Derivatives are Much More Difficult to Value Than Stock Options • Volatilities of different points on the term structure are different • Interest rates are used for discounting as well as for defining the payoff

22,4 Main approaches to Pricing Interest Rate Options Use a variant of blacks model Use a no-arbitrage ( yield curve based)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 22.4 Main Approaches to Pricing Interest Rate Options • Use a variant of Black’s model • Use a no-arbitrage (yield curve based) model

22.5 Blacks model its extensions Blacks model is similar to the Black-Scholes model used for valuing stock options It assumes that the value of an interest rate, a bond price, or some other variable at a particular time T in the future has a lognormal distribution Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 22.5 Black’s Model & Its Extensions • Black’s model is similar to the Black-Scholes model used for valuing stock options • It assumes that the value of an interest rate, a bond price, or some other variable at a particular time T in the future has a lognormal distribution

22.6 Black's model its extensions (continued The mean of the probability distribution is the forward value of the variable The standard deviation of the probability distribution of the log of the variable is G√T Where o is the volatility The expected payoff is discounted at the T-maturity rate observed today Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 22.6 Black’s Model & Its Extensions (continued) • The mean of the probability distribution is the forward value of the variable • The standard deviation of the probability distribution of the log of the variable is where s is the volatility • The expected payoff is discounted at the T-maturity rate observed today s T

22.7 Blacks model (egn 22. 1 and 22.2, p 509) c=P(O,T)IFON(d-KN(d2) p=P(0,T[KN(-d2)-FN(-d1) m(F/K)+7/2 K: strike price T: option maturity forward value of .o: volatility variable Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 22.7 Black’s Model (Eqn 22.1 and 22.2, p 509) d d T T F K T d p P T KN d F N d c P T F N d KN d = −s s + s = = − − − = − 2 1 2 0 1 2 0 1 0 1 2 ; ln( / ) / 2 (0, )[ ( ) ( )] (0, )[ ( ) ( )] • K : strike price • F0 : forward value of variable • T : option maturity • s : volatility

The black's model: Payoff Later 22.8 Than Variable Being Observed c=P(O, T[FON(d-KN(d2) p=P(0,T)KN(-a2)-F0N(-d1) n(F/K)+27/2 T K: strike price °T: time when Fo forward value of variable is observed variable T*: time of payoff °σ: volatility Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull The Black’s Model: Payoff Later 22.8 Than Variable Being Observed • K : strike price • F0 : forward value of variable • s : volatility • T : time when variable is observed • T * : time of payoff d d T T F K T d p P T KN d F N d c P T F N d KN d = −s s + s = = − − − = − 2 1 2 0 1 2 0 1 * 0 1 2 * ; ln( / ) / 2 (0, )[ ( ) ( )] (0, )[ ( ) ( )]

22.9 Validity of blacks Model Black's model appears to make two approximations 1. The expected value of the underlying variable is assumed to be its forward price 2. Interest rates are assumed to be constant for discounting We will see that these assumptions offset each other Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 22.9 Validity of Black’s Model Black’s model appears to make two approximations: 1. The expected value of the underlying variable is assumed to be its forward price 2. Interest rates are assumed to be constant for discounting We will see that these assumptions offset each other

22.10 European Bond options When valuing European bond options it is usual to assume that the future bond price is lognormal We can then use black,'s model (equations 22.1and22 Both the bond price and the strike price should be cash prices not quoted 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 22.10 European Bond Options • When valuing European bond options it is usual to assume that the future bond price is lognormal • We can then use Black’s model (equations 22.1 and 22.2) • Both the bond price and the strike price should be cash prices not quoted prices