Question

A trader buys a European call option and sells a European put option. The options have the same underlying asset, strike price, and maturity. Describe the trader’s position. Under what circumstances does the price of the call equal the price of the put?

Answer 1

The trader has a long European call option with strike price K and a short European put option with strike price K . Suppose the price of the underlying asset at the maturity of the option is S_T . If , the call option is exercised by the investor and the put option expires worthless. The payoff from the portfolio is then S_T-K. If S_T<K, the call option expires worthless and the put option is exercised against the investor. The cost to the investor is K-S_T.  

Alternatively we can say that the payoff to the investor in this case is S_T-K (a negative amount). In all cases, the payoff is S_T-K, the same as the payoff from the forward contract. The trader’s position is equivalent to a forward contract with delivery price K.

Suppose that F is the forward price. If K = F, the forward contract that is created has zero value. Because the forward contract is equivalent to a long call and a short put, this shows that the price of a call equals the price of a put when the strike price is F.