Chapter 9

Mechanics of Options

Markets

1

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

2

Option Positions

Long call

Long put

Short call

Short put

3

Long Call (Figure 9.1, Page 195)

Profit from buying one European call option: option

price = $5, strike price = $100, option life = 2 months

4

30

20

10

0 -5

70 80 90 100

110 120 130

Profit ($)

Terminal

stock price ($)

Short Call (Figure 9.3, page 197)

Profit from writing one European call option: option

price = $5, strike price = $100

5

-30

-20

-10

0 5

70 80 90 100

110 120 130

Profit ($)

Terminal

stock price ($)

Long Put (Figure 9.2, page 196)

Profit from buying a European put option: option

price = $7, strike price = $70

6

30

20

10

0

-7 70 60 50 40 80 90 100

Profit ($)

Terminal

stock price ($)

Short Put (Figure 9.4, page 197)

Profit from writing a European put option: option price

= $7, strike price = $70

7

-30

-20

-10

7

0 70

60 50 40

80 90 100

Profit ($) Terminal

stock price ($)

Payoffs from Options What is the Option Position in Each Case?

K = Strike price, ST = Price of asset at maturity

Long call: max(0,ST-K) Short call: -max(0,ST-K)

Long put: max(0,K-ST) Short call: - max(0,K-ST) 8

Payoff Payoff

ST ST K

K

Payoff Payoff

ST ST K

K

Assets Underlying

Exchange-Traded Options Page 198-199

Stocks

Foreign Currency

Stock Indices

Futures

9

Terminology

Moneyness :

At-the-money option

In-the-money option

Out-of-the-money option

10

Terminology (continued)

Option class: all options of the same type (calls

or puts) traded on a certain asset

Option series: all the options of a given class with

the same expiration date and strike price

Intrinsic value: maximum of zero and the value of

exercise if it were exercised immediately

Time value: premium a rational investor would

pay over its current exercise value (intrinsic

value), based on the probability it will increase in

value before expiry.

11

Stock Splits (Page 202-204)

Example: before the stock split the value of a firm is 100$ and is divided into 100 shares worth 1$ each

After a 2 for 1 stock split the value of the firm is divided into 200 shares worth 0.5$ each

Suppose you own N options with a strike price of K :

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

12

Dividends & Stock Splits: example

Consider a call option to buy 100 shares

for $20/share

How should terms be adjusted:

for a 2-for-1 stock split? Call option to buy

200 shares for $10/share

for a 5% stock dividend? Equivalent to a 21

for 20 stock split (21/20=1.05): the holder of

the option has the right to buy 100*21/20=105

share for 20/21*20$=19.05$ each

13

Market Makers

Typically trading of options in an exchange market

takes place through a market maker.

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 offer price is higher than the bid and

exchanges can set up upper bounds to the bid-

offer spread

Market makers make the market more liquid

14

Margins (Page 205-206)

Margins are required when options are sold

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

15

Employee Stock Options and

Convertible Bonds 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

Convertible bonds are regular bonds that can be

exchanged for equity at certain times in the future

according to a predetermined exchange ratio

16

Chapter 10

Properties of Stock Options

17

Notation

18

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.

Effect of Variables on Option

Pricing (Table 10.1, page 215)

Variable c p C P

S0 + − + −

K − + − +

T ? ? + +

s + + + +

r + − + −

D − + − +

19

American vs European Options

20

An American option is worth at least as much

as the corresponding European option

C c

P p

Calls: An Arbitrage Opportunity? Suppose that

Is there an arbitrage opportunity?

Short the stock and buy the call to get a cash flow of 20$ - 3$ = 17$. Invest for one year and get 17e0.1 = 18.79$

If ST > 18$, exercise the call and get a profit of 0.79$

If ST < 18$, for example, 17$, buy the stock back and make a profit of 18.79$ - 17$ = 1.79$

21

c = 3; S0 = 20; K = 18; T = 1; r = 10%; D = 0

71.371.31820 1.0

0 ceKeS rT

Lower Bound for European Call

Option Prices; No Dividends (Equation 10.4, page 220)

c S0 –Ke -rT

22

Puts: An Arbitrage Opportunity? Suppose that

Is there an arbitrage opportunity?

Borrow 38$ to buy the put and the stock, and repay after 6 months 38e0.05*0.5=38.96$

If ST < 40 the arbitrageur exercises the put and sells for 40$. The profit is 40$ - 38.96$ = 1.04$

If ST > 40, for example 42$, the arbitrageur discards the option, sells the stock and makes a profit of 42$ - 38.96$ = 3.04$

23

p= 1; S0 = 37; K = 40; T = 0.5; r =5%; D = 0

$01.2$01.2$37$40 5.005.0

0 peSKe rT

Lower Bound for European Put

Prices; No Dividends (Equation 10.5, page 221)

p Ke -rT–S0

24

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

25

Values of Portfolios

26

ST > K ST < K

Portfolio A Call option ST − K 0

Zero-coupon bond K K

Total ST K

Portfolio C Put Option 0 K− ST

Share ST ST

Total ST K

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

c – p = S0 - Ke -rT

27

Suppose that

What are the arbitrage possibilities when p = 2.25? In this case

the put-call parity condition does not hold:

Portf. A: c + Ke–rT =3+ 30e-0.1*0.25=32.26$ and

Portf. C: p+S0=2.25+31=33.25$

Portfolio C is overpriced: short C and buy A.

Buy the call and short both the put and the stock to generate the

cash flow -3 + 2.25 + 31= 30.25$

Invest this cash flow at 10% to get 30.25e0.1*0.25=$31.02 in 3 months

If ST>30, exercise the call and buy stock for 30.00$. If ST<30 the put

is exercised and the counterparty sells at 30.00$.

So independently of ST the net profit is: =31.02-30=1.02$

28

Arbitrage Opportunities

c= 3; S0= 31; K =30; r =10%; T = 0.25; D = 0

Suppose that

What are the arbitrage possibilities when p = 1? In this case the

put-call parity condition does not hold:

Portf.A: c + Ke –rT =3+ 30e-0.1*0.25=32.26$ and

Portf C: p+S0=1+31=32$

Portfolio A is overpriced: short A and buy C.

Short the call to finance the purchase of the put and the stock which

lead to a cash outflow 31+1-3=29$.

Borrow 29$ at the 10%p.a. to repay 29e0.1*0.25=$29.73 in 3 months

If ST<30, exercise the put and sell at 30.00$. If ST>30, the call will

be exercised and the arbitrageur will sell at 30.00$.

So independently of ST the net profit is: = 30.00 - 29.73 =0.27$

29

Arbitrage Opportunities

c= 3; S0= 31; K =30; r =10%; T = 0.25; D = 0

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

30

Reasons For Not Exercising a Call

Early (No Dividends) Consider an American call option:

S0 = 100; T = 0.25; K = 60; D = 0

Suppose you think the stock is a good investment. Should you exercise immediately? No, for the following reasons:

– No income is sacrificed (the stock bears no dividends)

– You would forgo interest payment (time value of money)

– You would be worse off in the case that the spot price falls below 60 at maturity (holding the call provides an insurance against stock price falling below strike)

31

Reasons For Not Exercising a

Call Early (No Dividends) Suppose you do not feel that the stock is a good investment (it is overpriced). Should you exercise the option and sell the stock?

Doing so would generate a profit of S0 – K =40$ (intrinsic value of the option)

No, it is better to sell the call option because it has both intrinsic value and time value: C > S0 – K

Remember the lower bound for call prices:

c ≥ S0 – Ke-rT

. Because C ≥ c, C ≥ S0 – Ke-rT.

Because Ke-rT <K, it follows that C > S0 – K

32

Bounds for European or American Call

Options (No Dividends)

33

Should Puts Be Exercised

Early ?

• Are there any advantages to exercising an American put early?

• In contrast to call options it can be optimal to exercise put options early if deep in the money.

• Exercising the option earlier means getting money sooner than later (time value of money, i.e. interest rate)

• Because asset prices cannot be negative, if stock prices are sufficiently low the time value of holding the option is small

34

Bounds for European and American

Put Options (No Dividends)

35