MATH 238 WINTER 2009
PROBLEM SET 1  SOLUTIONS
Problem 1
: Let
S
be the current stock price,
K
the strike price of the
option,
T
the expiration time of the option,
t
the current time,
S
T
the stock
price at time
T
,
r
the riskfree interest rate,
c
the price of a European call
option and
p
the price of a European put option.
Explain why the two
portfolios
(a) one European call option plus cash equal to
Ke

r
(
T

t
)
(b) one European put option plus one share
have the same payoff at time
T
: max(
S
T
, K
). Deduce that the value of the
portfolios today must be the same, so that
(1)
c
+
Ke

r
(
T

t
)
=
p
+
S
Explain this
putcall parity
relation with figures. (Ch. 8, Hull.)
Solution
: At any time
t
≤
T
, we are considering the two portfolios
(a):
one European call option plus an amount of
Ke

r
(
T

t
)
cash put
in the bank
(b):
one European put option and one share of the underlying asset
S
t
.
Then the value of portfolio (a) at time
T
equals
c
T
+
K
= max(
S
T

K,
0) +
K
= max(
S
T
, K
)
while the value of portfolio (b) at time
T
equals
p
T
+
S
T
= max(
K

S
T
,
0) +
S
T
= max(
S
T
, K
)
Therefore both portfolios have indeed the same payoff max(
S
T
, K
) at the
expiration time
T
of the options.
We now claim that this implies that the values of these portfolios today,
i.e. at any time
t
≤
T
, must be the same, so that we have the
putcall parity
relation
c
t
+
Ke

r
(
T

t
)
=
p
t
+
S
t
for all
t
≤
T
.
One way to see this is from the assumption that there is no arbitrage
which we now present.
Assume that at some time
t < T
portfolio (a) is cheaper than portfolio (b)
(i.e.
c
t
+
Ke

r
(
T

t
)
< p
t
+
S
t
). Then we buy (a) (i.e. we buy a call option
and we put an amount of
Ke

r
(
T

t
)
in the bank ) and sell (b) (i.e. write
a put option and short sell a share of the stock).
This creates a positive
cash flow at time t, and no other flow afterwords (i.e. an arbitrage). To see
this assume that at time T it happens that
S
T
< K
. The call option is not
1
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MATH 238 WINTER 2009
PROBLEM SET 1  SOLUTIONS
S
T
K
K
Payoff
Figure 1.
Payoff: One Call minus one Put equals the Stock

K
exercised. The cash in the bank is now worth K and we use it to buy the
one share of the stock (for K) from the person to whom we wrote the put
option. Finally we take this one share and return to the broker who made
the initial short sale possible. Similarly if
S
T
≥
K
we are able at time T to
fulfill all our obligations with zero cost. If portfolio (b) is cheaper than (a)
we buy (b), sell (a) and we show that this is again an arbitrage.
Mathematically, we simply have to take conditional expectations of the
discounted payoffs in the equation
e

r
(
T

t
)
max(
K

S
T
,
0)+
Ke

r
(
T

t
)
=
e

r
(
T

t
)
max(
K

S
T
,
0)+
e

r
(
T

t
)
S
T
with respect to the risk neutral probability law, given
S
t
=
S
. This gives
C
(
t, S
) +
Ke

r
(
T

t
)
=
P
(
t, S
) +
S
for
t < T
. We have used here the fact that the discounted price of the risky
asset is a martingale under the risk neutral probability:
E
⋆
[
e

rT
S
T

S
t
=
S
] =
e

rt
S
.
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 Winter '08
 Papanicolaou
 Math, Strike price, risk neutral probability

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