《金融期货与期权》（英文版） Chapter 8 Properties of Stock Option Prices

8.1 Properties of Stock Option Prices Chapter 8 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 8.1 Properties of Stock Option Prices Chapter 8

8.2 Notation C: European call C: American Call option option price prIce P: European put P: American Put option option price price So: Stock price today T: Stock price at option °K: Strike price maturity ·T: Life of option D: Present value of Volatility of stock dividends during options ife orIce Risk-free rate for maturity T with cont comp Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 8.2 Notation • c : European call option price • p : European put option price • S0 : Stock price today • K : Strike price • T : Life of option • : Volatility of stock price • C : American Call option price • P : American Put option price • ST :Stock price at option maturity • D : Present value of dividends during option’s life • r : Risk-free rate for maturity T with cont comp

Effect of Variables on Option Pricing (table8.1, page 168) Variable c P 十 KTor +一?++一 P—+?+一+ 十 D 十 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 8.3 Effect of Variables on Option Pricing (Table 8.1, page 168) Variable c p C P S0 K T  r D + + – + ? ? + + + + + + + – + – – – – + – + – +

8.4 American vs European Options An American option is worth at least as much as the corresponding european option P≥ Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 8.4 American vs European Options An American option is worth at least as much as the corresponding European option C  c P  p

Calls: An arbitrage 8.5 Opportunity? Suppose that =3 So=20 T=1 r=10% K=18 D=0 Is there an arbitrage opportunity? Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Calls: An Arbitrage 8.5 Opportunity? • Suppose that c = 3 S0 = 20 T = 1 r = 10% K = 18 D = 0 • Is there an arbitrage opportunity?

Lower Bound for European cal Option prices: No dividends Equation 8.1, page 173) C之 rT 0 Ke Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 8.6 Lower Bound for European Call Option Prices; No Dividends (Equation 8.1, page 173) c  S0 –Ke -rT

Puts: An Arbitrage 8.7 Opportunity Suppose that So 37 T=0.5 r=5% K=40 D=0 Is there an arbitrage opportunity? Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Puts: An Arbitrage 8.7 Opportunity? • Suppose that p = 1 S0 = 37 T = 0.5 r =5% K = 40 D = 0 • Is there an arbitrage opportunity?

Lower Bound for European P8.8 Prices: No dividends (Equation 8.2, page 174) P≥KeT 0 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 8.8 Lower Bound for European Put Prices; No Dividends (Equation 8.2, page 174) p  Ke-rT–S0

8.9 Put-Call Parity: no dividends (Equation 8.3, page 174) Consider the following 2 portfolios Portfolio A: European call on a stock Pv of the strike price in cash Portfolio C: European put on the stock the stock Both are worth MAX(ST, k) at the maturity of the options They must therefore be worth the same today This means that C+Ke-rl=p+ 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 8.9 Put-Call Parity; No Dividends (Equation 8.3, page 174) • Consider the following 2 portfolios: – Portfolio A: European call on a stock + PV of the strike price in cash – Portfolio C: European put on the stock + the stock • Both are worth MAX(ST , K ) at the maturity of the options • They must therefore be worth the same today – This means that c + Ke -rT = p + S0

8.10 Arbitrage Opportunities Suppose that C=3 S0=31 T=0.25 r=10% K=30 D=0 What are the arbitrage possibilities when P=2.25? D Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 8.10 Arbitrage Opportunities • Suppose that c = 3 S0 = 31 T = 0.25 r = 10% K =30 D = 0 • What are the arbitrage possibilities when p = 2.25 ? p = 1 ?