Ch07_Properties of Stock Option Prices(金融工程学,华东师大).

7.11
Puts: An Arbitrage Possibility?
• Suppose that p= 3 S0= 48 T= 0.25 r= 5% X= 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
• 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
7.6
(NO Dividends)
• 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]
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
7.13
Call Early Exercise on Non-Dividend Paying Stock
• From earlier we know that c > S0 - Xe-rT • Given that C c • C > S0 - Xe-rT > S0 - X • Thus, C > S0 - X, but if exercised early C = S0 - X • Hence, no reason to exercise early
The risk-less profit 0.37 can be obtained by borrowing Xe-rT, buying Put and Stock (See slide 7.10)
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
Variable c
7.3
S0 X T r D
+ – ? + + –
– + ? + – +
p
C
+ – + + + –
– + + + – +
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
7.14
Early Exercise
• Usually, there is some chance that an American option will be exercised early • An exception is an American call on a non-dividend paying stock
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
7.15
An Extreme Situation
• For an American call option with S0 = 100, T = 0.25, X = 60, r = 5%, D = 0
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
7.9
Calls: An Arbitrage Possibility?
Lower Bound for European Call Prices
• Consider the following positions
t=0 Portfolio A Buy Call Lend Xe-rT at r Net Flows Portfolio B Buy one share -c -Xe-rT -c-Xe-rT -S0 ST <X 0 X X ST ST >X ST - X X ST ST
7.12
Puts: An Arbitrage Possibility?
(continued)
• Is p > Xe-rT - S0 ? Xe-rT - S0 = = 50e-0.05(0.25) - 48 = 49.37 - 48 = 1.37 • No arbitrage possibility 3 > 1.37 • An arbitrage is possible if p=1
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
ST>50 ST-50 -ST 50 0
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
Lower Bound for European Put Prices
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?
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
– This should NEVER be exercised early 1. NO income is sacrificed 2. Strike price is paid later 3. Holding the call provides insurance against stock price falling below strike price
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
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,
7.4
American vs European Options
An American option is worth at least as much as the corresponding European option
C c P p
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
(NO Dividends) • Consider the following positions
t=0 Portfolio C Buy Put Buy Stock Net Flows Portfolio D Lend Xe-rT at r -p -S0 -p-S0 -Xe-rT ST <X X - ST ST X X ST >X 0 ST ST X
p Xe-rT 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
(continued) • Yes, an arbitrage is possible as 3 < 4.44
Buy the Call Sell Stock Lend Xe-rT Net Flows
t=0 -c S0 -Xe-rT -c+S-Xe-rT
• Numerically,
ST<X 0 -ST X X-ST
7.10
• 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]
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
– Should you exercise if …
•
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
Effect of Variables on Option Pricing (Table 7.1, page 169)
ST&gssibly More Later
t=0 ST<50 Buy the Call -3 0 Now Sell Stock 52 -ST Lend Xe-rT -50e-0.05*1 50 Net Flows -3+52-50e-0.05*1=1.44 50-ST