华东师范大学：《金融工程》英文版 Chapter 9 Introduction to Binomial Trees

9.1 Introduction to Binomial trees Chapter g Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University 9.1 Introduction to Binomial Trees Chapter 9

9.2 A Simple binomial model of stock Price movements In a binomial model the stock price at the BEGINNING of a period can lead to only 2 stock prices at the ENd of that period Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University 9.2 A Simple Binomial Model of Stock Price Movements • In a binomial model, the stock price at the BEGINNING of a period can lead to only 2 stock prices at the END of that period

93 Option pricing Based on the assumption of No arbitrage opportunities Procedures i Establish a portfolio of stock and option → Value the portfolio no arbitrage opportunities no uncertainty at maturity no risk with the portfolio risk-free interest earned i Value the option Risk-free interest =value of portfolio today Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University 9.3 Option Pricing Based on the Assumption of No Arbitrage Opportunities • Procedures: ➔ Establish a portfolio of stock and option ➔ Value the Portfolio ➢ no arbitrage opportunities ➢ no uncertainty at maturity ➢ no risk with the portfolio ➢ risk-free interest earned ➔ Value the option ➢ Risk-free interest = value of portfolio today

94 A Simple binomial model: Example 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, 4th edition@ 2000 by John C. Hull Tang Yincai, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University 9.4 A Simple Binomial Model: Example • 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

9.5 A Call option a3-month call option on the stock has a strike price of $21 Figure 9.1(P202) Stock Price $22 Option price= $1 Stock price= $20 Option Price=? Stock Price =$18 Option price $0 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University 9.5 Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price=? A Call Option • A 3-month call option on the stock has a strike price of $21. • Figure 9.1 (P.202)

9.6 Setting Up a riskless portfolio Consider the portfolio lonG A shares SHORT 1 call option Figure 9.1 becomes 22△ S=20 18∧ Portfolio is risk/ess when 224-1=18A or△=0.25 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University 9.6 • Consider the Portfolio: LONG D shares SHORT 1 call option • Figure 9.1 becomes • Portfolio is riskless when 22D – 1 = 18D or D = 0.25 22D – 1 18D Setting Up a Riskless Portfolio S0 = 20

Valuing the portfolio ( with risk-Free Rate 1290) 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=18*0.25 The value of the portfolio today is 4.50e012025=4.3670 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University 9.7 Valuing the Portfolio ( with Risk-Free Rate 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 = 18 * 0.25 • The value of the portfolio today is 4.50e-0.12*0.25=4.3670

98 Valuing the option The portfolio that is: LONG 0.25 shares SHORT 1 call 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 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University 9.8 Valuing the Option • The portfolio that is: LONG 0.25 shares SHORT 1 call 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

99 Generalization Consider a derivative that lasts for time t and that is dependent on a stock Figure 9.2(P. 203) < fd Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University 9.9 Generalization • Consider a derivative that lasts for time T and that is dependent on a stock • Figure 9.2 (P.203) S0u ƒu S0d ƒd S0 ƒ

9.10 Generalization(continued) Consider the portfolio that is: LONG A shares ShORT 1 derivative Figure 9.2 becomes SouA-f so-f Sod△-fa The portfolio is riskless when Sou△-fn=Sod△-fa or when △ f susd Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University 9.10 Generalization (continued) • Consider the portfolio that is: LONG D shares SHORT 1 derivative • Figure 9.2 becomes • The portfolio is riskless when S0uD – ƒu = S0d D – ƒd or when S u S d f f u d 0 − 0 − D = S0uD – ƒu S0 dD – ƒd DS0 - f