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