Put-Call Parity and Synthetic Options

Put–call parity specifies a relationship between the prices of call and put options with the identical strike price K and expiry T. To derive the put–call parity relationship, we must assume that the call and put options involved are European options. Perhaps the most important feature of put–call parity is that it must be satisfied at all times, in a model independent manner. A violation of this leads to arbitrage opportunities.

 

Call (K,T) +KerT  = Put (K,T) + S0eqT

 

Portfolio A: Synthetic Option: Purchase one call on an underlying asset S, struck at K and expiring at T. Sell a put on the same underlying, with the same strike price and maturity.

 

Portfolio B: Purchase a forward contract that gives the obligation to buy S at a price K at maturity date T.

 

In all states, Portfolio A has the same payoff at maturity as Portfolio B.

For European options, early exercise is not possible. If the values of these two portfolios are the same at the expiry of the options, then the present values of these portfolios must also be the same, otherwise, an investor can arbitrage and make a risk-free profit by purchasing the less expensive portfolio, selling the more expensive one and holding the long-short position to maturity.

 

Accordingly, we have the price equality:     Call (K, T) – Put (K, T) = Forward (K, T) =  S0 e−qT − Ke−rT

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