7.1 Properties of Stock Option Prices Chapter 7 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 7.1 Properties of Stock Option Prices Chapter 7
72 Notation European call C: American Call option option price price p: European put P: American Put option option price prIce So: Stock price today ST: Stock price at time T X: Strike price D: Present value of T: Life of option dividends during options :Motl波动率)ofe stock price r: Risk-free rate for maturity T with cont comp 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 7.2 Notation • c : European call option price • p : European put option price • S0 : Stock price today • X : Strike price • T : Life of option • : Volatility(波动率) of stock price • C : American Call option price • P : American Put option price • ST :Stock price at time T • D : Present value of dividends during option’s life • r : Risk-free rate for maturity T with cont comp
73 Effect of Variables on Option Pricing (Table 7. 1, page 169) Variable c P 十 T 十 D 十 十 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 7.3 Effect of Variables on Option Pricing (Table 7.1, page 169) Variable c p C P S0 X T r D + + – + ? ? + + + + + + + – + – – – – + – + – +
7.4 American vs European Options An American option is worth at least as much as the corresponding European option P≥ Options, Futures, and Other Derivatives, 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 7.4 American vs European Options An American option is worth at least as much as the corresponding European option C c P p
7.5 Upper bound for Options prices It should be relatively easy to see that C<So and c≤S Otherwise, you could make a risk-less profit by buying the stock and selling the option Likewise p≤ Xe-ri and P≤X Otherwise, you could make a risk-less profit by selling the option and investing the proceeds at r 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 7.5 Upper Bound for Options Prices • It should be relatively easy to see that c S0 and C S0 Otherwise, you could make a risk-less profit by buying the stock and selling the option • Likewise, p Xe-rT and P X Otherwise, you could make a risk-less profit by selling the option and investing the proceeds at r
7.6 Lower bound for European Call Prices (NO Dividends) Consider the following positions t=0 SX Portfolio a Buy ci S-X Lend Xe-rT at r -Xe-TT Ⅹ Net flows -C-Xe Ⅹ T Portfolio b Buy one share A is worth more than b, so it must cost more to set it up initially. So c+ Xe-IT> So c> max[So Xe-r, 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 7.6 Lower Bound for European Call Prices (NO Dividends) • Consider the following positions t = 0 ST X Portfolio A Buy Call -c 0 ST - X Lend Xe-rT at r -Xe-rT X X Net Flows -c-Xe-rT X ST Portfolio B Buy one share -S0 ST ST • A is worth more than B, so it must cost more to set it up initially. So c + Xe-rT > S0 c > max[S0 -Xe -rT , 0]
77 Calls: An Arbitrage Possibility Suppose that c=3 So=52 7=1 广5% =50 D=0 Is there an arbitrage possibility? 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 7.7 Calls: An Arbitrage Possibility? • Suppose that c = 3 S0= 52 T = 1 r= 5% X= 50 D= 0 • Is there an arbitrage possibility?
78 Calls: An Arbitrage Possibility? (continued sC>S 0 ⅩeT? S-Xe-rT= =52-50e005(100 52-47.56 4.44 Yes, an arbitrage is possible as 3< 4.44 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 7.8 Calls: An Arbitrage Possibility? (continued) • Is c > S0 - Xe -rT ? S0 - Xe -rT = = 52 - 50e-0.05(1.00) = 52 - 47.56 = 4.44 • Yes, an arbitrage is possible as 3 < 4.44
7.9 Calls: An Arbitrage Possibility? (continued Yes, an arbitrage is possible as 3 50 Buy the call 3 S50 Se/ stock 52 S Lend Xe-rT 50e0.051 50 50 Net flows-3+52-50e0051=14450-S7 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 7.9 Calls: An Arbitrage Possibility? (continued) • Yes, an arbitrage is possible as 3 X Buy the Call -c 0 ST -X Sell Stock S0 -ST -ST Lend Xe -rT -Xe-rT X X Net Flows -c+S-Xe -rT X-ST 0 • Numerically, t = 0 ST50 Buy the Call -3 0 ST -50 Sell Stock 52 -ST -ST Lend Xe -rT -50e-0.05*1 50 50 Net Flows -3+52-50e-0.05*1=1.44 50-ST 0 Now Possibly More Later
710 Lower bound for European Put prices NO Dividends) Consider the following positions t=0 SX Portfolio c Buy Put Ⅹ-S Buy Stock Net flows Portfolio d 0SSX Lend xe-rt at r Xe C is worth more than d, so it must cost more to set it up initially. So, p+So>Xe-rT p>maxkXe-r-So, 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 7.10 Lower Bound for European Put Prices (NO Dividends) • Consider the following positions t = 0 ST X Portfolio C Buy Put -p X - ST 0 Buy Stock -S0 ST ST Net Flows -p-S0 X ST Portfolio D Lend Xe-rT at r -Xe-rT X X • C is worth more than D, so it must cost more to set it up initially. So, p+S0 > Xe-rT p > max[Xe-rT - S0 , 0]