《金融期货与期权》（英文版） Chapter 10 Introduction to Binomial trees

Introduction to Binomial trees Chapter 10 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 10.1 Introduction to Binomial Trees Chapter 10

10.2 A Simple binomial model A stock price is currently $20 In three months it will be either $22 or $18 Stock Price $22 Stock price $20 Stock Price =$18 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 10.2 A Simple Binomial Model • A stock price is currently $20 • In three months it will be either $22 or $18 Stock Price = $22 Stock Price = $18 Stock price = $20

10.3 A Call option(Figure 10.1, page 200 A 3-month call option on the stock has a strike price of 21 Stock Price= $22 Option Price $1 Stock price $20 Option price=? Stock Price=$18 Option price sO Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 10.3 Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price=? A Call Option (Figure 10.1, page 200) A 3-month call option on the stock has a strike price of 21

10.4 Setting Up a riskless portfolio Consider the portfolio: long a shares short 1 call option 22△-1 18∧ Portfolio is riskless when 22A-1=18 on △=0.25 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 10.4 • Consider the Portfolio: long D shares short 1 call option • Portfolio is riskless when 22D – 1 = 18D or D = 0.25 22D – 1 18D Setting Up a Riskless Portfolio

10.5 Valuing the portfolio (Risk-Free Rate is 12%) The riskless portfolio is long 0. 25 shares short 1 call option The value of the portfolio in 3 months is 220.25-1=4.50 The value of the portfolio today is 4.5e-0.12025=4.3670 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 10.5 Valuing the Portfolio (Risk-Free Rate is 12%) • The riskless portfolio is: long 0.25 shares short 1 call option • The value of the portfolio in 3 months is 22´0.25 – 1 = 4.50 • The value of the portfolio today is 4.5e – 0.12´0.25 = 4.3670

10.6 Valuing the option The portfolio that is long 0.25 shares short 1 option is worth 4,367 The value of the shares is 5.000(=0.2520) The value of the option is therefore 0.633(=5000-4367) Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 10.6 Valuing the Option • The portfolio that is long 0.25 shares short 1 option is worth 4.367 • The value of the shares is 5.000 (= 0.25´20 ) • The value of the option is therefore 0.633 (= 5.000 – 4.367 )

10.7 Generalization( Figure 10.2, page 202) a derivative lasts for time and is dependent on a stock fd Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 10.7 Generalization (Figure 10.2, page 202) • A derivative lasts for time T and is dependent on a stock S0 ƒu S0d ƒd S0 ƒ

108 Generalization (continued) Consider the portfolio that is long A shares and short 1 derivative Soda-f The portfolio is riskless when SouA-fu= Sod 4-fa or fuf △ Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 10.8 Generalization (continued) • Consider the portfolio that is long D shares and short 1 derivative • The portfolio is riskless when S0uD – ƒu = S0d D – ƒd or D = − − ƒu d f S0 u S0 d S0 uD – ƒu S0dD – ƒd S0– f

10.9 Generalization (continued Value of the portfolio at time Tis △-f Value of the portfolio today is (S0△-fn)e7 Another expression for the portfolio value today is SoA-f ence f=Soa-Soua-fue-rt Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 10.9 Generalization (continued) • Value of the portfolio at time T is S0u D – ƒu • Value of the portfolio today is (S0u D – ƒu )e –rT • Another expression for the portfolio value today is S0D – f • Hence ƒ = S0D – (S0u D – ƒu )e –rT

10.10 Generalization (continued) Substituting for a we obtain f=lpf+(1-pfale-rt Where rT Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 10.10 Generalization (continued) • Substituting for D we obtain ƒ = [ p ƒu + (1 – p )ƒd ]e –rT where p e d u d rT = − −