8.1

Properties ofStock Option PricesSome model-independent results

Chapter 8

8.2

Notation• c : European 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

8.3

Effect of Variables on Option Pricing (Table 8.1, page 168)

c p C PVariable

S0

KTrD

+ + –+

? ? + ++ + + ++ – + –

–– – +

– + – +

8.4

American vs European Options

An American option is worth at least as much as the corresponding European option

C cP p

8.5

Pricing bounds

8.6Calls: An Arbitrage

Opportunity?

• Suppose that

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

• Is there an arbitrage opportunity?• (Buy the call, sell underlying, deposit PV(K)!)

8.7Puts: An Arbitrage

Opportunity?• Suppose that

p = 1 S0 = 37 T = 0.5 r =5%

K = 40 D = 0

• Is there an arbitrage opportunity?• (Buy put, buy stock, borrow

PV(K)!)

8.8

Put-Call Parity; No Dividends (Equation 8.3, page 174)

• 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 + S0

8.9

Arbitrage Opportunities• Suppose that

c = 3 S0 = 31

T = 0.25 r = 10%

K =30 D = 0

• What are the arbitrage possibilities and how are they exploited when p = 2.25 ?

• p = 1 ?

8.10

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

8.11

• For an American call option:

S0 = 100; T = 0.25; K = 60; D = 0Should you exercise immediately?

• What should you do if 1 You feel confident about the performance of

the stock for the next 3 months? 2 You do not feel that the stock is worth holding

for the next 3 months?

An Extreme Situation

8.12

Reasons For Not Exercising a Call Early, Case 1)

(No Dividends )

• No income is sacrificed by holding the option

• We delay paying the strike price

• Holding the call provides insurance against stock price falling below strike price

• Everything else is the same

• => wait, don’t exercise

8.13

Case 2)

• The investor should rather sell the option

• He will receive more than the intrinsic value since other investors prefer to hold the stock. Otherwise price could not be S0.

8.14

Should Puts Be Exercised Early ?

Are there any advantages to exercising an American put when

S0 = 60; T = 0.25; r=10%

K = 100; D = 0

8.15

The Impact of Dividends on Lower Bounds to Option Prices

(Equations 8.5 and 8.6, page 179)

c S D Ke

p D Ke S

rT

rT

0

0

8.16

Extensions of Put-Call Parity

• American options; D = 0

• European options; D > 0

c + D + Ke -rT = p + S0

• American options; D > 0

rTKeSPCKS 00

rTKeSPCKDS 00

8.17

Trading Strategies Involving Options

Note: Net profit diagrams

Chapter 9

8.18

Three Alternative Strategies

• Take a position in the option and the underlying

• Take a position in 2 or more options of the same type (A spread)

• Combination: Take a position in a mixture of calls & puts (A combination)

8.19Positions in an Option & the Underlying (Figure 9.1, page 186)

Profit

STK

Profit

ST

K

Profit

ST

K

Profit

STK

(a) (b)

(c)

(d)

8.20

Bull Spread Using Calls(Figure 9.2, page 187)

K1 K2

Profit

ST

8.21

Bull Spread Using PutsFigure 9.3, page 189

K1 K2

Profit

ST

8.22

Bear Spread Using CallsFigure 9.4, page 189

K1 K2

Profit

ST

8.23

Bear Spread Using PutsFigure 9.5, page 190

K1 K2

Profit

ST

8.24

Butterfly Spread Using CallsFigure 9.6, page 191

K1 K3

Profit

STK2

8.25

Butterfly Spread Using PutsFigure 9.7, page 192

K1 K3

Profit

STK2

8.26

Calendar Spread Using CallsFigure 9.8, page 193

Profit

ST

K

8.27

Calendar Spread Using PutsFigure 9.9, page 193

Profit

ST

K

8.28

A Straddle CombinationFigure 9.10, page 194

Profit

STK

8.29

Strip & StrapFigure 9.11, page 195

Profit

K ST

Profit

K ST

Strip Strap

8.30

A Strangle CombinationFigure 9.12, page 196

K1 K2

Profit

ST

8.31

Warning!!

• Next week (but only Tuesday) our use of quantitative methods will accelerate. Chapter 10 onwards.

• We need models for uncertainty to price options.

• We will start using stochastic variables and processes.

• The fall break is a good opportunity to refresh/review your stat and prob knowledge.