Mich 2010
Mathematical Finance: Short Solutions
3
15. Apply the Law of One Price to some suitable portfolios. The inequalities for the put
option are (
Ke

rT

S
)
+
≤
P
≤
Ke

rT
. For the final inequality compare portfolios
containing put options and some cash.
16. Adapt the argument for the American call option i.e. compare exercising the call at
T
1
with selling short.
17. Buy the calls with strike prices
K
1
and
K
3
and sell two calls with strike
K
2
. Whatever
the stock price
S
(
T
) the return is nonnegative so the portfolio requires a positive
amount to setup.
Buying the
K
1
and
K
3
puts and selling two
K
2
puts also guarantees a positive return
so
P
2
≤
(
P
1
+
P
3
)
/
2.
18. Start with putcall parity for both parts: (a) is not true (think about large
K
and
small
S
), (b) is true.
19. Modify the argument used to establish Theorem 1.3 (b).
20. The initial odds have
∑
(1 +
o
i
)

1
= 1 so there is no arbitrage. With natural labels
for the three new bets we find
o
12
= 1
/
5,
o
13
= 1
/
2 and
o
23
= 1.
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 Spring '10
 DrI.M.MacPhee
 Math, Inequalities, short solutions, geometric BM write

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