# Introduction to call_and_put_options

Jun 28th, 2015
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Consider the following portfolios: Portfolio 1: A European call option, and cash at time t equal to Ke

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Put-Call Paritychris bemisMay 22, 2006Recall that a replicating portfolio of a contingent claim determines the claims price.This was justified by the no arbitrage principle.Using this idea, we obtain a relationship between a European call and a European putoption.1Consider the following portfolios:Portfolio 1: A European call option, and cashat time t equal to KerTPortfolio 2: A European put option, and oneshare of the underlying, SHere, K is the strike price for both the calland the put, and T is the time to expirationfor both options.2At maturity, Portfolio 1 is worthK + max(ST K, 0) = max(ST , K),and Portfolio 2 is worthST + max(K ST , 0) = max(K, ST ).Since the options are European, the prices today of these portfolios must be the same.If the prices were different, there would be anarbitrage opportunity (just like the gold example).3We have now thatc0 + KerT = p0 + S0(1)This relationship is put-call parity, and holdsfor European options.Notice that this relationship is not model dependentwe only relied on the no arbitrage principle.4We got a hint that puts and calls are relatedwhen we saw the implied volatility skew forcrude oil:0.7Put & Call Volatility Smiles: March 29, 2006 for Exercise May 29,2006(Call is dashed/Put is solid)0.650.6Implied Volatility0.550.50.450.40.350.30.254045505560Strike Prices657075805Recall that the only unobservable quantity inthe Bla

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