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 = − −
