Put-Call Parity

Put-Call Parity     

                                                                                          

The prices of European puts and calls on the same stock with identical exercise prices and expiration dates have a special relationship. The put price, call price, stock price, exercise price, time to expiration, and risk-free rate are all related by a formula called put-call parity. Let us see how this formula is derived.

Imagine a portfolio, called portfolio A, consisting of one share of stock and one European put. This portfolio will require an investment of S₀ + P_e(S₀, T, X). Now consider a second portfolio, called portfolio B, consisting of a European call with the same exercise price and risk-free pure discount bonds with a face value off. That  portfolio will require an investment of C_e(S₀, T, X) + X(1 +r)^-T. Now let us look at what happens at expiration. Table 2.1 presents the outcomes.

Table 2.1

				Payoffs from Portfolio
Portfolio		Current Value	ST ≤ X	ST > X
A	Long put	Pe(S0, T, X)	ST – X	0
	Stock	S0	ST	ST
			X	ST
B	Long call	Ce(S0, T, X)	0	ST – X
	Long bond	X(1 +r)-T	X	X
				ST

The stock is worth S_T regardless of whether S_T is more or less than X. Likewise the risk-free bonds are worth X regardless of the outcome. If S_T exceeds X, the call expires in the money and is worth S_T – X and the put expires worthless. If S_T is less than or equal to X, the put expires in-the-money worth S_T – X and the call expires worthless.

The total values of portfolios A and B are equal regardless of the outcome of the stock       price. However, given the Law of One Price, the current value of the two portfolios must be equal. Thus we require that:

   S₀ + P_e(S₀, T, X) = C_e(S₀, T, X) + X(1 +r)^-T

This statement is referred to as put-call-parity and it is probably one of the most important results in understanding options. It says that a share of stock plus a put is equivalent to a call plus risk-free bonds. It shows the relationship between the call and put prices, the stock price, the time to expiration, and the exercise price.

Suppose the combination of the put and the stock is worth less than the combination of the call and the bonds. Then you can create an arbitrage portfolio by buying the put and stock and short selling the call and the bonds. Selling short a call just means writing the call, and selling short the bonds simply means to borrow the present value of X and promise to pay back X at the option's expiration. Since everyone would start doing this transaction, the prices would be forced back in line with the put-call parity equation.

By observing the signs in front of each term, we can easily determine which combinations replicate others. If the sign is positive, we should buy the option, stock or bond. If the sign is negative, we should sell. For example, suppose we isolate the call price:

      C_e(S₀, T, X) =  S₀ + P_e(S₀, T, X) - X(1 +r)^-T

Then owning a call is equivalent to owning a put, owning the stock, and selling the bonds (borrowing). If the stock pay dividends, once again, we simply insert

which is the stock price minus the present value of the dividends.

While put-call parity is an extremely important and useful result, it does not hold so neatly if the options are American. The put-call parity rule for American options must be stated as inequalities:

where S’₀ is, again, the stock price minus the present value of the dividends.

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