1. Chapter 9
Properties of Stock Options

2. Factors affecting option price Assumptions and notation
Outline
Factors affecting option price
Assumptions and notation
Upper and lower bounds for option price
Put-call parity
Early exercise : calls on a non-dividends-paying stock
Early exercise : puts on a non-dividends-paying stock
Effect of dividends

3. 1.The current stock price, S0 2.The strike price, K
Factors affecting option prices
There are six factors affecting the price of a stock option:
1.The current stock price, S0
2.The strike price, K
3.The time to expiration, T
4.The volatility of the stock prive,  
5.The risk-free interest rate, r
6.The dividends expected during the life of the option

4. – – + + – – + + + + + + + + – – + + – – + +
Effect of Variables on Option Pricing
(Table 9.1, Page 206)
Summary of the effect on the price of a stock option of increasing one variable while keeping all others fixed.
Variable S0 K T  r D c p C P – – + + – – + + + + ? + + + + – – + + – – + +

5. 1.There are no transactions costs.
Assumptions and Notation(1/2)
We assume that there are some market participants, such as
large investment banks, for which the following statements are
true:
1.There are no transactions costs.
2.All trading profits (net of trading losses) are subject
to the same tax rate.
3.Borrowing and lending are possible at the risk-free
interest rate.

6. Assumptions and Notation(2/2)
S0 : Stock price today
K : Strike price
T : Life of option
: Volatility of stock price
r : Risk-free interest rate
D : Present value of dividends during option’s life
ST : Stock price at option maturity
p : European put option price
c : European call option price
P : American Put option price
C : American Call option price

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

8. American C≦S0 P≦K European c≦S0 (c≦C) p≦ke -rT (p≦P)
Upper Bounds for Options
-- No Dividends
Call option can never be worth more than the stock;
Put option can never be worth more than the strike price.
Hence, the stock price is an upper bound to the option price.
American
C≦S0
P≦K
European
c≦S0 (c≦C)
p≦ke -rT (p≦P)

9. Lower Bounds for European Call (1/2)
-- No Dividends
A lower bound for the price of a European call option on a non- dividend-paying stock is
S0 –Ke –rT
We first look at a numerical example and then consider a more formal argument.
Example
S0 =$20 , K=$18 , r = 10% per annum , T = 1 year

10. Lower Bounds for European Call (2/2)
-- No Dividends
Option is a right without obligation ; hence , it has positive value: c  0
For a more formal argument, we consider the following two portfolios:
Protfolio A : c + Ke –rT
Protfolio B : S0
Portfolio A is always worth no less than B!
c  max( S0 –Ke –rT, 0 ) (Equation9.1, p. 211)

11. Lower Bounds for European Put (1/2)
-- No Dividends
A lower bound for the price of a European put option on a non-dividend-paying stock is
Ke –rT – S0
We first look at a numerical example and then consider a more formal argument.
Example
S0 =$37 , K=$40 , r = 5% per annum , T = 0.5 year

12. Lower Bounds for European Put (2/2)
-- No Dividends
Option is a right without obligation ; hence , it has positive value: p  0
For a more formal argument, we consider the following two portfolios:
Protfolio C : p + S0
Protfolio D : Ke –rT
Portfolio C is always worth no less than D!
p  max( Ke -rT–S0 , 0 ) (Equation9.2, p. 212)

13. c + Ke -rT = p + S0 (Equation9.3, p. 211)
Put-Call Parity : No Dividends (1/2)
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 + S (Equation9.3, p. 211)

14. Put-Call Parity : No Dividends (2/2)
Portfolio A
ST≧K
ST＜K
c(K)
ST －Ｋ
Ke -rT Ｋ K
Total ST
Portfolio C
ST≧K
ST＜K
p(K)
K－ST S0 ST
Total K

15. Example - Arbitrage Opportunities
Suppose that
c = S0 = 31
T = r = 10%
K = D = 0
What are the arbitrage possibilities when
p = 2.25 ?
p = 1 ?

16. Should Calls be Exercised Early ? (p. 216)
Usually there is some chance that an American option will be exercised early
This should never be exercised early
∵ c≧S0－Ke -rT and C≧c
∴ C≧c≧S0－Ke -rT≧S0－K (exercise value)
(∵Ke -rT＜K)

17. Reasons for Not Exercising Call Early ( No Dividends )
No income from the stock is sacrificed
Payment of the strike price is delayed
Holding the call provides insurance against stock price falling below strike price

18. Should Puts be Exercised Early? (p. 217)
Are there any advantages to exercising an American put when
∵ p≧Ke -rT －S0 and P≧p
∴ P≧p ≧ Ke -rT －S0 ＞ K －S0
But P must be larger than KS0 because it is always possible to exercise before maturity

19. The Impact of Dividends on Lower Bounds to Option Prices (Page 218)
We will use D to denote the present value of the
dividends during the life of the option. In the
calaulation of D, a dividend is assumed to occur at
the time of its ex-dividend date.

20. Extensions of Put-Call Parity
American options; D = 0 (Equation9.4, p. 215)
S0 - K < C - P < S0 - Ke –rT
European options; D > 0 (Equation9.7, p. 219)
c + D + Ke -rT = p + S0
American options; D > 0 (Equation9.8, p. 219) S0 - D - K < C - P < S0 - Ke -rT