hroughout this chapter we have analyzed the no-arbitrage relationships among stock, bond and option prices that must hold in the single period binomial model (see topic 2.4, Call Option Valuation: A Riskless Hedge Approach, topic 2.5, Put Option Valuation:  A Riskless Hedge Approach), and topic 2.6 Option Valuation: A Synthetic Option Approach).

Besides these relationships, an additional arbitrage relationship must hold between the stock price and put and call option prices.  This relationship is called put-call parity.

In its simplest form, when the interest rate is zero and both the put P and the call C have the same strike price X, and S is the stock price, this relationship is:

P - C + S  =  X.

That is, a portfolio that consists of long one put with strike X, short one call with strike X, and long one stock is riskless.  Furthermore its value equals the strike price X.

Why must this be the case?  First, observe that if both the put and the call have the same strike price, then only one of them can finish in-the-money.  If the put is in-the-money, then S < X, so the call is out-of-the-money.  Similarly, if the call is in-the-money, then S>X, so the put is out-of-the-money.  As a result, P - C  equals X - S if S <  X, and equals S-X if S>X.  In either case, we get

P - C = X - S

The only other possibility is that S = X, in which case both P and C are zero, and the relationship then holds trivially.

How would you take advantage of this relationship to make an arbitrage profit if it did not hold?  The answer depends upon whether the appropriate bids and asks are available.

If this relationship does not hold, it implies that either the portfolio trades at more than X or less than X.

Suppose P - C + S > X and consider the following trading strategy.  Sell a stock, sell a put, and buy a call.  If you undertake these trades, you receive the amount P - C + S in cash.  If the stock price is greater than X, the put is worthless, and the call can be exercised to obtain a stock at price X.  This stock covers your short position, so at the end of the period, you have to pay X.

If the stock price is less than X, the call is worthless, but the put will be exercised  against  you, so you will again have to buy a stock at price X.  This stock will cover your short position.

Thus, independent of whether the stock  price  is less than X or more than X, you will end up paying X at  the  end  of the period.  But you received P - C + S > X earlier on, so you  have  locked in a sure profit.  Hence these prices cannot persist.

Similarly, if P - C + S < X, you can make a sure profit by buying a stock, buying  a  put, and selling a call.  Together, these imply that P - C + S = X, which is  the parity relationship.

If 1 plus the risk-free interest rate is r  >  1, then the parity relationship for European options says the current prices must equal the present value of the strike price discounted at the risk-free interest rate (i.e.,  P - C + S = X/r).

A further implication of put-call parity is that if you have solved for the value of a European call option, you can always determine the value of the European put option (having the same strike price) by applying the put-call parity relationship.

In Chapter 3, Binomial Model:  Two-Period Analysis, we extend the one-period binomial option pricing model to two periods.