replications and payoffs
TRANSCRIPT
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Payoff and Replications
Chapters 8, 10
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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
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Option Positions
Long (buy) callLong (buy) putShort (write) callShort (write) put
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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 ($)
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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 ($)
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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 ($)
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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 ($)
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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
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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
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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)
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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)
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Bull Spread Using Calls(Figure 10.2, page 225)
K1 K2
Profit
ST
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Bull Spread Using PutsFigure 10.3, page 226
K1 K2
Profit
ST
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Bear Spread Using PutsFigure 10.4, page 227
K1 K2
Profit
ST
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Bear Spread Using CallsFigure 10.5, page 229
K1 K2
Profit
ST
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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)
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Butterfly Spread Using CallsFigure 10.6, page 231
K1 K3
Profit
STK2
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Butterfly Spread Using PutsFigure 10.7, page 232
K1 K3
Profit
STK2
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Calendar Spread Using CallsFigure 10.8, page 232
Profit
STK
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Calendar Spread Using PutsFigure 10.9, page 233
Profit
STK
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A Straddle CombinationFigure 10.10, page 234
Profit
STK
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Strip & StrapFigure 10.11, page 235
Profit
K ST
Profit
K ST
Strip Strap
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A Strangle CombinationFigure 10.12, page 236
K1 K2
Profit
ST
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Standard contracts
Straddles Strangles Risk reversals Binary call or put Butterfly spread
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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