Payoff and Replications

Chapters 8, 10

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

Option Positions

Long (buy) callLong (buy) putShort (write) callShort (write) put

Long Call on eBay(Figure 8.1, Page 182)—Limited liability

Profit from buying one eBay European call option: option price = $5, strike price = $100, option life = 2 months

Correction: Focus on payoff, not “profit”30

20

10

0-5

70 80 90 100

110 120 130

Profit ($)

Terminalstock price ($)

Short Call on eBay (Figure 8.3, page 184) —Unlimited liability

Profit from writing one eBay European call option: option price = $5, strike price = $100

-30

-20

-10

05

70 80 90 100

110 120 130

Profit ($): Change to payoff

Terminalstock price ($)

Long Put on IBM (Figure 8.2, page 183) –Limited profit & liability

Profit from buying an IBM European put option: option price = $7, strike price = $70

30

20

10

0

-770605040 80 90 100

Profit ($): Change to payoff

Terminalstock price ($)

Short Put on IBM (Figure 8.4, page 184) –Limited liability

Profit from writing an IBM European put option: option price = $7, strike price = $70

-30

-20

-10

7

070

605040

80 90 100

Profit ($)Terminal

stock price ($)

Payoffs from OptionsWhat is the Option Position in Each Case? K = Strike price, ST = Price of asset at maturity

Payoff Payoff

ST STK

K

Payoff Payoff

ST STK

K

Which of the position has limited liability? Plot the payoff

Long stock Short stock Long call, put Short put Short call Short 1 call, long 1 put at the same strike Short 1 call, long 1 stock Short 1 call, short 1 put

Types of Derivative StrategiesChapter 11

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)

Positions in an Option & the Underlying (Figure 10.1, page 224)

Profit

STK

Profit

ST

K

Profit

ST

K

Profit

STK

(a) (b)

(c)

(d)

Bull Spread Using Calls(Figure 10.2, page 225)

K1 K2

Profit

ST

Bull Spread Using PutsFigure 10.3, page 226

K1 K2

Profit

ST

Bear Spread Using PutsFigure 10.4, page 227

K1 K2

Profit

ST

Bear Spread Using CallsFigure 10.5, page 229

K1 K2

Profit

ST

Box Spread

A combination of a bull call spread and a bear put spread

If all options are European a box spread is worth the present value of the difference between the strike prices

If they are American this is not necessarily so. (See Business Snapshot 10.1)

Butterfly Spread Using CallsFigure 10.6, page 231

K1 K3

Profit

STK2

Butterfly Spread Using PutsFigure 10.7, page 232

K1 K3

Profit

STK2

Calendar Spread Using CallsFigure 10.8, page 232

Profit

STK

Calendar Spread Using PutsFigure 10.9, page 233

Profit

STK

A Straddle CombinationFigure 10.10, page 234

Profit

STK

Strip & StrapFigure 10.11, page 235

Profit

K ST

Profit

K ST

Strip Strap

A Strangle CombinationFigure 10.12, page 236

K1 K2

Profit

ST

Standard contracts

Straddles Strangles Risk reversals Binary call or put Butterfly spread

A general replication formula

Prove this formula:

Try to replicate the terminal payoff that pays ln(ST) If you can replicate, you can price. Price variance swap in terms of European options,

assuming continuous underlying dynamics.

0

' '' ''t

t

S

T t t T t T TSf S f S f S S S f K K S dK f K S K dK