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Payoff and Replications Chapters 8, 10
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
