Arbitrage Proofs for PutCall Parity and Minimum Value
1
I. PutCall Parity
Putcall parity states that
C
=
S

Ee

rT
+
P
(1)
To prove this statement, assume that it doesn’t hold and show that it is possible
to make riskless proﬁts. We will use numbers for concreteness. Assume
S
= $110,
E
= $100,
t
= 1,
r
= 0. Also assume
C
= $12 and
P
= $5. Thus the call is
undervalued and/or the put is overvalued, given
S
= $110 and
E
= $100.
Construct an arbitrage by buying the “cheap” call at $12 and selling the “expen
sive” put at $5. Recall that long a call and short a put both proﬁt when
S
rises.
Therefore, we complete the arbitrage by selling
S
short at $110. Our portfolio today:
Cash ﬂow
Buy 1 call
 $ 12
Short (write) 1 put
+ $ 5
Sell short 1 share
+ $110
Invest proceeds
$103
Net
0
Examine what happens on expiration (1 year from now) if
S > E
and if
S < E
.
If
S
≥
E
:
Exercise call
$100
Deliver against short

Put expires worthless
0
Receive proceeds from investment
$103
$3
1
Notes for Finance 100 (sections 301 and 302) prepared by Jessica A. Wachter.
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 Fall '09
 Finance, Arbitrage, minimum value

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