1. DERIVATIVES: OPTIONS
Reference: John C. Hull, Options, Futures and Other Derivatives, Prentice Hall

2. Types of Options A call is an option to buy A put is an option to sell
A European option can be exercised only at the end of its life
An American option can be exercised at any time 2

3. Option Positions
Long call
Long put
Short call
Short put 3

4. Long Call
Profit from buying one European call option: option price = $5, strike price = $100. 30 20 10 -5 70 80 90
100
110
120
130
Profit ($)
Terminal
stock price ($) 4

5. Short Call
Profit from writing one European call option: option price = $5, strike price = $100
-30
-20
-10 5 70 80 90
100
110
120
130
Profit ($)
Terminal
stock price ($) 5

6. Long Put
Profit from buying a European put option: option price = $7, strike price = $70 30 20 10 -7 70 60 50 40 80 90
100
Profit ($)
Terminal
stock price ($) 6

7. Short Put
Profit from writing a European put option: option price = $7, strike price = $70
-30
-20
-10 7 70 60 50 40 80 90
100
Profit ($)
Terminal
stock price ($) 7

8. K = Strike price, ST = Price of asset at maturity
Payoffs from Options (Terminal Value) European Options, Cost of option is NOT included
K = Strike price, ST = Price of asset at maturity
Payoff ST K
SHORT CALL
LONG CALL
SHORT PUT
LONG PUT 8

9. MAX (ST - K, 0) MIN (K- ST , 0) MAX (K- ST , 0) MIN (ST - K, 0)
Payoff ST K
SHORT CALL
LONG CALL
SHORT PUT
LONG PUT 9

10. Assets Underlying Exchange-Traded Options
Stocks
Foreign Currency
Stock Indices
Futures 10

11. Specification of Exchange-Traded Options
Expiration date
Strike price
European or American
Call or Put (option class) 11

12. Terminology Moneyness : At-the-money option In-the-money option
Out-of-the-money option 12

13. 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. 13

14. – – + + – – + + + + + + + + – – + + – – + + Variable S0 K T  r D c p
Effects, in the price of an option, when one variable increases, and all the others remain unaltered
European American
Variable S0 K T  r D c p C P – – + + – – + + + + ? + + + + – – + + – – + +
Option decreases in value if D increases
Option increases in value if S0 increases 14

15. Some Useful Relationships…
A call option can never be worth more than the stock
c ≤ S0
C ≤ S0

16. A put option can never be worth more than K
p ≤ K
P ≤ K

17. American vs European Options
An American option is worth at least as much as the corresponding European option
C  c
P  p 17

18. For a European put we know that at maturity the put option cannot be worth more than K
It follows then, that today, the price cannot be more than the present value of K, thus
p ≤ K e-rT

19. Lower Bound for The Value of a European Call Option When There Are No Dividends
c  S0 –Ke -rT
Proof on next page [using an arbitrage argument] 19

20. Consider two portfolios: [A] a European call option + cash = Ke –rT
and
[B] one share of the stock
In case of portfolio A, the cash can be invested at the risk-free rate, thus, at time = T, we will have K dollars
We have two possible outcomes at time =T
[1] If ST > K, I will exercise the option (spend K) and now I have a stock whose value is ST OR
[2] If ST < K there is no point in exercising the option (option=worthless); so I end up with K dollars
Thus, at time T, the value of portfolio A is MAX (K, ST)

21. c + Ke –rT > S0 (QED) Consider two portfolios:
[A] a European call option + cash = Ke –rT
and
[B] one share of the stock
The value of portfolio A is MAX (K, ST) (at time T)
AND the value of portfolio (B), at time = T, is obviously, ST
Clearly, (A) is more valuable than (B) at maturity ((in other words, B can be equal to A but never more)), therefore, in the absence of arbitrage opportunities the same situation has to be valid today (time = 0); namely, (A) is more valuable than (B), thus
c + Ke –rT > S0 (QED)

22. Lower Bound for The Value of a European Put Option When There Are No Dividends
p  Ke –rT - S0
Proof on next page [using an arbitrage argument] 22

23. Consider two portfolios: [C] a European put option + one share and
[D] cash = Ke –rT
In case of portfolio D, the cash can be invested at the risk-free rate, thus, at time = T, we will have K dollars
We have two possible outcomes at time =T for position [C]
[1] If ST < K, I will exercise the option (sell the stock at K) and now I have a portfolio whose value is K OR
[2] If ST > K there is no point in exercising the option (option=worthless); so I end up with a value = ST
Thus, at time T, the value of portfolio C is MAX (K, ST)

24. By a reasoning similar to the one employed before we conclude that
Portfolio C is worth more than portfolio D (whose value, at T, is K no matter what); and thus, the same should hold at time = 0 (today)
Thus,
p + S0 > Ke –rT
(QED)

25. Value of A and C Portfolios
Time = T
ST > K
ST < K
Portfolio A
Call option
ST − K
cash K
Total ST
Portfolio C
Put Option
K− ST
Share
All options are European (CANNOT be exercised before T), thus, since the value of the two portfolios( A and B) is the same at time = T, they must have the same value today (time = 0)
This leads to the so-called…. 25

26. The Put-Call Parity Theorem
Both portfolios are worth max(ST , K ) at maturity
They must therefore be worth the same today. This means that (today)
c + Ke -rT = p + S0
European Options; No dividends 26

27. S0 - K < C - P < S0 - Ke -rT
The Put-Call Parity Theorem (OR Result) Is Only Valid For EUROPEAN Options; For AMERICAN Options We Can Show That For The Case In Which There Are No Dividends
S0 - K < C - P < S0 - Ke -rT 27

28. European options; D > 0
If Dividends are included (D is the PV of the Dividend during the life of the option) then we have…
European options; D > 0
c + D + Ke -rT = p + S0
American options; D > 0
S0 - D - K < C - P < S0 - Ke -rT