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33 We have written the values as fractions to make it clear that the value at
time 0 of these two options are exactly the same. Once you realize this it is
easy to prove.
Theorem 6.6. The values VP and VC of the put and call options with the same
strike K and expiration N are related by
VP VC = K
(1 + r)N S0 In particular if K = (1 + r)N S0 then VP = VC .
Proof. The key observation is that
SN + (K SN )+ (SN K )+ = K Consider the two cases SN K and SN K . Dividing by (1 + r)N , taking E ⇤
expected value and using the fact that Sn /(1 + r)n is a martingale
S0 + E ⇤ (K SN )+
(1 + r)N E⇤ (K SN )+
(1 + r)N
(1 + r)N Since the second term on the left is VP and the third is VC the desired result
follows. 191 6.4. CAPITAL ASSET PRICING MODEL It is important to note that the last result can be used to compute the value
of any put from the corresponding call. Returning to the previous example and
looking at the time 1 node where the price is 54, the formula above says
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
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