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) call
 Long (buy) put
 Short (write) call
 Short (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 = $100, option life = 2 months
Correction: Focus on payoff, not “profit”
30 Profit ($)
20
10
70
0
-5
80
90
100
Terminal
stock price ($)
110 120 130
Short Call on eBay
(Figure 8.3, page 184) —Unlimited liability
Profit from writing one eBay European call option: option
price = $5, strike price = $100
Profit ($): Change to payoff
5
0
-10
-20
-30
110 120 130
70
80
90 100
Terminal
stock 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 = $70
30 Profit ($): Change to payoff
20
10
0
-7
Terminal
stock price ($)
40
50
60
70
80
90 100
Short Put on IBM
(Figure 8.4, page 184) –Limited liability
Profit from writing an IBM European put option: option
price = $7, strike price = $70
Profit ($)
7
0
-10
-20
-30
40
50
Terminal
stock price ($)
60
70
80
90 100
Payoffs from Options
What is the Option Position in Each Case?
K = Strike price, ST = Price of asset at maturity
Payoff
Payoff
K
K
ST
Payoff
ST
Payoff
K
K
ST
ST
Which of the position has limited liability? Plot the
payoff
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






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 Strategies
Chapter 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
Profit
K
K
ST
ST
(a)
(b)
Profit
Profit
K
ST
(c)
K
(d)
ST
Bull Spread Using Calls
(Figure 10.2, page 225)
Profit
ST
K1
K2
Bull Spread Using Puts
Figure 10.3, page 226
Profit
K1
K2
ST
Bear Spread Using Puts
Figure 10.4, page 227
Profit
K1
K2
ST
Bear Spread Using Calls
Figure 10.5, page 229
Profi
t
K1
K2
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 Calls
Figure 10.6, page 231
Profit
K1
K2
K3
ST
Butterfly Spread Using Puts
Figure 10.7, page 232
Profit
K1
K2
K3
ST
Calendar Spread Using Calls
Figure 10.8, page 232
Profit
ST
K
Calendar Spread Using Puts
Figure 10.9, page 233
Profit
ST
K
A Straddle Combination
Figure 10.10, page 234
Profit
K
ST
Strip & Strap
Figure 10.11, page 235
Profit
Profit
K
Strip
ST
K
Strap
ST
A Strangle Combination
Figure 10.12, page 236
Profit
K1
K2
ST
Standard contracts
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Straddles
Strangles
Risk reversals
Binary call or put
Butterfly spread
A general replication formula

Prove this formula:

f  ST   f  St   f '  St  ST  St    f ''  K  K  ST  dK   f ''  K  ST  K  dK
St
0




St
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.
