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Define p
1
, p
2
and p
3
as the prices of puts with strike prices K
1
, K
2
and K
3
.
With the usual notation
1
1
1
rT
c
K e
p
S
2
2
2
rT
c
K e
p
S
3
3
3
rT
c
K e
p
S
Hence
1
3
2
1
3
2
1
3
2
2
(
2
)
2
rT
c
c
c
K
K
K
e
p
p
p
Because
2
1
3
2
K
K
K
K
, it follows that
1
3
2
2
0
K
K
K
and
1
3
2
1
3
2
2
2
c
c
c
p
p
p
The cost of a butterfly spread created using European calls is therefore
exactly the same as the cost of a butterfly spread created using European
puts.
Problem 11.19
A butterfly spread is created by buying the $55 put, buying the $65 put
and selling two of the $60 puts. This costs
3
8
2
5
1
$
initially. The
following table shows the profit/loss from the strategy.
Stock Price
Payoff
Profit
65
T
S
0
1
60
65
T
S
65
T
S
64
T
S
55
60
T
S
55
T
S
56
T
S
55
T
S
0
1
The butterfly spread leads to a loss when the final stock price is greater
than $64 or less than $56.

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