Chapter 14
Option Market – Part II
Call Options Payoffs and Profits
at Expiration
‰
Payoff to call holder (buyer)
max {ST – X, 0}
‰
Payoff to call writer (seller)
- max {ST – X, 0}
‰
Profit to call holder
max {ST – X, 0} - Premium
‰
Profit to Call Writer
Premium - max {ST – X, 0}
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1
Payoff Profiles for Calls
Profit
OTM
ITM
Call Holder
0
Stock Price
at time T
X
Call Writer
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Profit/Loss Profiles for Calls
Profit
Call Holder
OTM
ITM
C
0
Stock Price
at time T
X
-C
Call Writer
Stock Price
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Put Options Payoffs and Profits
at Expiration
‰
Payoff to put holder (buyer)
max {X - ST, 0}
‰
Payoff to put writer (seller)
- max {X - ST, 0}
‰
Profit to put holder
max {X - ST, 0} - Premium
‰
Profit to put Writer
Premium - max {X - ST, 0}
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Payoff Profiles for Puts
Profits
ITM
OTM
Put Holder
0
X
Stock Price
at time T
Put Writer
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Profit/Loss Profiles for Puts
Profits
ITM
OTM
Put Writer
P
0
Stock Price
at time T
X
-P
Put Holder
Stock Price
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Put-Call Parity Relationship
ST < X
ST > X
0
ST - X
Payoff for
Holding a Call
Payoff for
Writing A Put
- (X - ST)
Total Payoff
ST - X
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0
ST - X
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Payoff of a Long Call and A Short Put
Payoff
Long Call
Combined Payoff
X
Stock Price
Short Put
-X
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Payoff of a Leverage Equity
Payoff
Long Stock
Combined Payoff
X
-X
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Stock Price
Short Bond
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Put-Call Parity
‰
We can replicate the payoff from a long call
and a short put by:
¾
¾
‰
‰
Long 1 share of stock today and hold it to T;
Borrow a margin loan in the amount of X / (1 + rf)T
Since the payoff on a long call and a short put
are equivalent to leveraged equity, the prices
must be equal today:
C - P = S0 - X / (1 + rf)T
If the prices are not equal, arbitrage will be
possible
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An Example – Put Call Parity Arbitrage
Q:
A:
Stock Price = 110
Call Price = 17
Risk Free = 5%
Put Price = 5
Maturity
= 1 yr Strike Price = 105
Is there any arbitrage opportunity?
C - P > S0 - X / (1 + rf)T
17- 5 > 110 - (105/1.05)
12 > 10
Arbitrage opportunity:
¾
¾
The leveraged equity is less expensive
Buy (long) the low cost portfolio and sell (short) the high cost
alternative
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An Example – Put-Call Parity Arbitrage
Position
Immediate
Cashflow
Buy Stock
-110
Cashflow in Six Months
ST <105
ST > 105
ST
ST
Borrow
X/(1+r)T = 100 +100
-105
-105
Sell Call
+17
0
Buy Put
-5
Total
2
-(ST - 105)
105 - ST
0
0
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0
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Option Strategy I – Protective Put
‰
Long stock + Long (ATM) put
¾
Pay put premium for downside protection
Payoff from
A Protective put
Payoff from
Long a stock
X
P/L from
Holding stock
ST
P/L from
A Protective put
X
- S0
X
-P
X
Payoff from
Long a put
X
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ST
ST
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Option Strategy II – Covered Call
‰
Long stock + short (sell / write) call
¾
Sacrifice upside potential for call premium
Covered Call
Write a call
Payoff at T
X
ST
payoff
X
profit
Payoff at T
X
Long a stock
C-S0
ST
X
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ST
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Option Strategy III - Straddle
‰
Long call and put with same strike price
¾
Benefit from big jumps in stock prices
Payoff at T
Long Dec 160 call
X=160
ST
X
Straddle
X-P-C
Long Dec 160 Put
X
X
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X
ST
ST
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Wrap-up
‰
‰
‰
Payout and P/L for holders and sellers of
put or call options
Put-call parity
Three option strategies
¾
¾
¾
Protective put
Covered call
Straddle
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