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:: Put-Call Parity Relationship::

Consider buying a put and selling a call option on the same underlying, at the same strike price E and same expiry time T. This will guarantee a payoff of E-S(T) at expiry:

So if we further also consider buying the underlying we can guarantee a payoff of E. This is a riskless investment, and so according to there being no arbitrage opportunities this investment must at all times be the same as having invested E exp(-r T) (i.e. the discounted value of E), assuming a constant riskless interest rate r. So at any time t between the writing of the options and expiry, the following relation must hold:

or equivalently:

Where C(S,t) and P(S,t) are the respective values of the call and the put at time t.

This is the Put-Call Parity relation. So having priced a call option one can use this to price a put option on the same underlying with the same strike and expiry.

Written by Raffaello Vardavas.

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