1. Properties of Stock Options
Chapter 10

2. Notation c : European call option price C : American 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.

3. Effect of Variables on Option PricingK T  r D c p C P – – + + – – + + + + ? + + + + – – + + – – + +

4. American vs European Options
An American option is worth at least as much as the corresponding European option
C  c
P  p

5. Calls: An Arbitrage Opportunity?
Suppose that
c = S0 = 20
T = r = 10%
K = D = 0
Is there an arbitrage opportunity? Yes!

6. Lower Bound for European Call Option Prices with No Dividends
c  S0 –Ke –rT
3 < 3.713
Arbitrage! buy low, sell high
Buy call, short stock, deposit $16.29 (PV of K)

7. Arbitrage the gap! (between the 2 sides of the inequality)
T=0: buy call (-3), short stock (=20), deposit $ NCF=0.71 (the gap)
T=1 & S>=18: exercise call (-18), use stock obtained to cover short, close out bank account (+18). NCF=0
T=1 & S<18: allow call to lapse unexercised, buy stock to cover short (-S), close out bank account (+18) NCF=18-S>0

8. @ Expiry: Call vs. Portfolio (long stock & borrow debt service = K)

9. c  (S0 - D )–Ke –rT c  (S0 e –dT)–Ke –rT
Lower Bound for European Call Option Prices with Dividends D=PV of Dividends, d=cc dividend yield
c  (S0 - D )–Ke –rT
c  (S0 e –dT)–Ke –rT

10. Puts: An Arbitrage Opportunity?
Suppose that
p = S0 = 37 T = r =5%
K = D = 0
Is there an arbitrage opportunity? Yes!

11. Lower Bound for European Put Prices with No Dividends
p  Ke -rT–S0
1 < 2.01
Arbitrage! Buy low, sell high
Buy put, buy stock, borrow $39.01

12. Arbitrage the gap! (between the 2 sides of the inequality)
T=0: buy put (-1), borrow 39.01, buy stock (-37). NCF= 1.01 (this is the gap)
T=0.5 & S<=40: exercise put (+40) delivering stock owned, payoff loan (-40). NCF=0.
T=0.5 & S>40: Allow put to lapse unexercised, sell stock (+S), payoff loan (-40). NCF=(S-40)>0.

13. @ Expiry: Put vs. Portfolio (short stock & deposit account value = K)

14. p  Ke –rT – (S0 - D ) p  Ke –rT – (S0 e –dT)
Lower Bound for European Put Option Prices with Dividends D=PV of Dividends, d=cc dividend yield
p  Ke –rT – (S0 - D )
p  Ke –rT – (S0 e –dT)

15. Put-Call Parity with No Dividends
Consider the following 2 portfolios:
Portfolio A: European call on a stock + PV of the strike price in cash
Portfolio B: 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

16. Arbitrage Opportunities (when put-call parity is violated)
Suppose that
c = S0 = 31
T = r = 10%
K = D = 0
p = 1.26: arbitrage possibilities when
p = 2.25 ? p too high
p = 1 ? p too low

17. Arbitrage when p not = 1.26 Buy low, sell high!
p = 2.25 (too high) T=0: sell put (2.25), buy call (-3), short stock (31), deposit NCF=0.99 (the gap)
p = 1 (too low) T= 0: buy put (-1), sell call (3), buy stock (-31), borrow NCF=0.26 (the gap)

18. Early Exercise
Usually there is some chance that an American option will be exercised early
Exception is an American call on a non-dividend paying stock; it should never be exercised early
But if stock price is sufficiently low, American put on non-dividend paying stock should be exercised early

19. An Extreme Situation For an American call option:
S0 = 100; T = 0.25; K = 60; D = 0
Should you exercise immediately?
What should you do if
You want to hold the stock for the next 3 months?
You do not feel that the stock is worth holding for the next 3 months?

20. Reasons For Not Exercising a Call Early (No Dividends)
No income is sacrificed
We delay paying the strike price
Holding the call provides insurance against stock price falling below strike price

21. Should Puts Be Exercised Early ? Yes when stock price is very low.
Note: stock price can’t be lower than 0.
Are there any advantages to exercising
an American put when
S0 = 10; T = 0.25; r=10%
K = 100; D = 0 or > 0

22. The Impact of Dividends on Lower Bounds to Option Prices

23. Extensions of Put-Call Parity
American options; D = 0
S0 - K < C - P < S0 - Ke -rT
European options; D > 0
c + D + Ke -rT = p + S0
American options; D > 0
S0 - D - K < C - P < S0 - Ke -rT