1. Chapter 9&10 Options: General Properties

2. Review of Option Types 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
Each option controls 100 shares of the underlying stocks
The date specified in the contract is know as the expiration date or the maturity date.
The price specified in the contract is known as the exercise price or the strike price. 2

3. Option Positions There are two sides to every option contract.
Long position: has bought the option
Short position: has sold or written the option.
Four types of option positions:
Long call
Long put
Short call
Short put
The writer of an option receives cash up front, but has potential liabilities later. 3

4. Long Call (Figure 9.1, Page 195)
Profit from buying one European call option: option price = $5, strike price = $100, option life = 2 months 30 20 10 -5 70 80 90
100
110
120
130
Profit ($)
Terminal
stock price ($) 4

5. Short Call (Figure 9.3, page 197)
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 (Figure 9.2, page 196)
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 (Figure 9.4, page 197)
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. Assets Underlying Exchange-Traded Options Page 198-199
Stocks
Foreign Currency
Stock Indices
Futures 8

9. Specification of Exchange-Traded Options
Expiration date
Strike price
European or American
Call or Put (option class)
The Precise Expiration date: Saturday Immediately following the third Friday of the expiration month.
An investor with a long position normally has until 4:30pm central time on that Friday to instruct a broker to exercise the option. The broker then has until 10:59pm the next day to complete the papework
January, February, and March cycle.
Jan, APR, July, and October
Feb, May, Aug, and Nov
Strike price: Spaced $2.5, $5.0 or $10.0 apart
$2.5 when price is between $5 and $25;
$5.0 when price is between $25 and $200
$10.0 when price is above $200 9

10. Terminology Moneyness : At-the-money option: S=K
In-the-money option: S>K (call) or S<K (put)
Out-of-the-money option: S<K (call) or S>K (put)
An option will be exercised only when it is in the money. 10

11. Terminology (continued)
Option class: all options of the same type(call or puts); e.g. IBM calls
Option series: options of a given class with the same T and K; e.g. IBM 70 October calls
Intrinsic value:
max(0, the value of the option if it were exercised immediately)
Time value: option value- intrinsic value
Option class: all options of the same type (Call or puts): IBM calls are one type; IBM puts are another
Option series: All options of a given class with the same expiration ate and strike price.(particular contract)
Intrinsic Value: the maximum of zero and the value of option would have if it were exercised immediately.
An in-the-money american option must be worth at least as much as its intrinsic value because can exercise immediately.
Often it is optimal for the holder of an in the money American option to wait rather than exercise immediately. Then time value of option
Total value=intrinsic value + time value 11

12. Dividends & Stock Splits (Page 202-204)
Suppose you own N options with a strike price of K :
No adjustments are made to the option terms for cash dividends
When there is an n-for-m stock split,
the strike price is reduced to mK/n
the no. of options is increased to nN/m
Stock dividends are handled similarly to stock splits
An n-for-m stock split should cause the stock price to go down to m/n of its previous value. 12

13. Dividends & Stock Splits (continued)
Consider a call option to buy 100 shares for $20/share
How should terms be adjusted:
for a 2-for-1 stock split?
K=1/2*20=$10; N=100*2=200
for a 5% stock dividend?
Equivalent to a 21-for-20 stock split
K=20*(20/21); N=100*(21/20)
½*K=$10/share; 200 shares
20% is a 6-for-1 split
5% is a 21 for 20 split
20/21*20=19.05 shares price
100*21/20=105 shares 13

14. Market Makers
Most exchanges use market makers to facilitate options trading
A market maker quotes both bid and ask prices when requested
The market maker does not know whether the individual requesting the quotes wants to buy or sell
The existence of the market maker ensures that buy and sell orders can always be executed at some price without any delays.
Market makers add liquidity to the market. 14

15. Margins (Page )
Buying on margin: borrow up to 50% of the price from the broker to buy shares.
If the share price declines so that the loan is substantially more than 50% of the stock’s current value, there is a margin call.
For options with maturities less than 9 months, option price must be paid in full.
Options already contain substantial leverage and buying on margin would raise the leverage.
Can borrow up to 25% of the option value for options with maturities greater than 9 months.

16. Margins (Page )
Margins are required when options are sold
Naked option: an option that is not combined with an offsetting position in the underlying stock.
When a naked option is written the margin is the greater of:
A total of 100% of the proceeds of the sale plus 20% of the underlying share price less the amount (if any) by which the option is out of the money
A total of 100% of the proceeds of the sale plus 10% of the underlying share price (call) or exercise price (put)
For other trading strategies there are special rules 16

17. Margins (Page )
Example: writes 4 naked option, OP=5, strike=40, stock price is 38. Then the option is 2 out of money. The 1st calculation gives 400X(5+0.2*38-2)=4240;
The 2nd calculation gives 400X(5+0.1X38)=3520; the initial margin is therefore 4240.
If it is a put: it would be 2 dollar in the money: 400X(5+0.2*38)=5040

18. Warrants
Warrants are options that are issued by a corporation or a financial institution
Predetermined number of options: The number of warrants outstanding is determined by the size of the original issue and changes only when they are exercised or when they expire 18

19. Warrants (continued)
The issuer settles up with the holder when a warrant is exercised
When call warrants are issued by a corporation on its own stock, exercise will usually lead to new treasury stock being issued 19

20. Employee Stock Options (see also Chapter 15)
Employee stock options are a form of remuneration issued by a company to its executives
They are usually at the money when issued
When options are exercised the company issues more stock and sells it to the option holder for the strike price
Expensed on the income statement 20

21. Convertible Bonds
Convertible bonds are regular bonds that can be exchanged for equity at certain times in the future according to a predetermined exchange ratio.
They are bonds with an embedded call option on the company’s stock. 21

22. Chapter 10 Properties of Stock Options

23. Notation c: European call option price p: European put option price
S0:
Stock price today K:
Strike price T:
Life of option s:
Volatility of stock price C:
American call option price P:
American put option price
ST:
Stock price at option maturity D:
PV of dividends paid during life of option r
Risk-free rate for maturity T with cont. comp. 23

24. Properties of Stock Option Prices
Effect of Variables on Option Pricing
Variable c p C P S0 + − K T ? s r D
Volatility: a measure of how uncertain we are about future stock price movements.
As volatility increases, the chance that the stock increase a lot or decrease a lot increases.
For owner of a stock, there two outcomes tend to offset each other.
The owner of a call benefits from price increases but has limited downside risk if the event of price decreases.
The owner of a put benefits from price decreases, but has limited downside risk in the event of price increases.

25. American vs European Options
An American option is worth at least as much as the corresponding European option C  c P  p 25

26. Upper Bounds c<=S0 C<=S0
If not: buy the stock and sell the call option
P<=K
p<=Ke-rT
If not: write the put option and invest the proceeds at r

27. Lower Bound for European Call Option Prices; No Dividends (Equation 10
Lower Bound for European Call Option Prices; No Dividends (Equation 10.4, page 220)
c  S0 –Ke -rT 27

28. Calls: An Arbitrage Opportunity?
Suppose that
Is there an arbitrage opportunity?
c = 3
S0 = 20
T = 1
r = 10%
K = 18
D = 0
Short the stock, buy the call: Cash inflow: 20-3=17.00
Invest for one year, then 17*exp(0.1)= Option expires at year 1
If stock price>18, then arbs exercise the option, get 18, close out the position. Profit is =0.79
If the stock is less than 18, then the stock is bought in the market and the short position is closed out: get even bigger profit. i.e., =1.79
Formally, Portfolio A: buy one Europ Call plus zero coupon bond that provides payoff of K at time T
Portfolio B: one share of stock
If St>K, then exercise and PA=St, if St<K, the call option expires worthless the profit is K. Hence PA=MAX(St,K); PB=St
c+Ke(-rt)>=S0
C>=max(S0-Ke(-rt),0) 28

29. Calls: An Arbitrage Opportunity?
Time 0: short stock and buy the call: 20-3=17
Invest 17 for 1 year at 10%
Time T:
$18.79 from cash investment
If ST>18, exercise the option for K=18
Profit= >0
If ST<=18. profit>= >0

30. Lower Bound for European Put Prices; No Dividends (Equation 10
Lower Bound for European Put Prices; No Dividends (Equation 10.5, page 221)
p  Ke -rT–S0 30

31. Put-Call Parity: No Dividends
Consider the following 2 portfolios:
Portfolio A: European call on a stock + zero-coupon bond that pays K at time T
Portfolio C: European put on the stock + the stock
The call and put have the same strike price and time to maturity. 31

32. Values of Portfolios ST > K ST < K Portfolio A Call option
Zero-coupon bond K
Total ST
Portfolio C
Put Option
K− ST
Share 32

33. The Put-Call Parity Result (Equation 10.6, page 222)
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 33

34. Arbitrage Opportunities
Suppose that
What are the arbitrage possibilities when
p = 2.25 ?
p = 1 ?
c= 3
S0= 31
T = 0.25
r = 10%
K =30
D = 0
C+ke(-rT)=32.26; p+s0=33.25; PC is overpriced
Buying the call and shorting both the put and the stock, generating: =30.25 upfront
Invest at the risk free rate: e (0.1*0.25)=31.02 in three months
If St>30, the call will be exercised; if less than 30, the put will be exercised, either case, arbitrageur end up with buying one share for 30, this share can be used to close out the positions. The net profit is =1.02
if p=1, then c+ke-rt=32.26; p+so=32; PA is overpriced, shorting the call, buying both the put and the stock. Initial investment: =29. financing the payment at risk free rate: 29e(0.1*0.25)= As in the early case, with call or put will be exercised, which leads to the stock sold for 30. the net profit is =0.27 34

35. Bound for European Options: No Dividends
Call
Put
Put-Call Parity

36. 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
This should never be exercised early
Consider an American call on a non-dividend-paying stock with 3 months to expiration . S0 = 40. K=70.
What should you do if
You want to hold the stock for the next 3 months?
No income is sacrificed
You delay paying the strike price
Holding the call provides insurance against stock price falling below strike price

37. Early Exercise
What if you do not feel that the stock is worth holding for the next 3 months?
Exercise the option and sell the stock immediately?
In this case, the investor is better off selling the option than exercising it.
Payoff from exercising: S0 – K
Payoff from selling the option: C
C>=S0 –Ke-rT C>S0 –K when T>0
It’s better to sell the option rather than exercising it!

38. Reasons For Not Exercising a Call Early (No Dividends)
No income is sacrificed
You delay paying the strike price
Holding the call provides insurance against stock price falling below strike price 38

39. Bounds for European or American Call Options (No Dividends)
Max(S0–Ke-rT , 0) <= c, C <=S0

40. Bounds for a put option European put: Max(Ke-rT - S0 )<=p<=Ke-rT
American put: Max(K - S0 )<=p<=K

41. Bounds for European and American put

42. The Impact of Dividends on Lower Bounds to Option Prices
A: one European call plus cash equal to D+ Ke-rT
B: one share
C: one European put plus one share
D: cash equal to D+ Ke-rT

43. Extensions of Put-Call Parity
European options: D>0
C+D+Ke-rT = P + S0