Put-Call Parity

Put-call parity is a principle that defines the relationship between the price of European put options and European call options of the same class, that is, with the same underlying asset, strike price and expiration date. Put-call parity states that simultaneously holding a short European put and long European call of the same class will deliver the same return as holding one forward contract on the same underlying asset, with the same expiration, and a forward price equal to the option's strike price. If the prices of the put and call options diverge so that this relationship does not hold, an arbitrage opportunity exists, meaning that sophisticated traders can theoretically earn a risk-free profit. Such opportunities are uncommon and short-lived in liquid markets.

The equation expressing put-call parity is:

C + PV(x) = P + S

where:

C = price of the European call option

PV(x) = the present value of the strike price (x), discounted from the value on the expiration date at the risk-free rate

P = price of the European put

S = spot price or the current market value of the underlying asset

Put-call parity applies only to European options, which can only be exercised on the expiration date, and not American options, which can be exercised before. 

Say that you purchase a European call option for TCKR stock. The expiration date is one year from now, the strike price is $15, and purchasing the call costs you $5. This contract gives you the right—but not the obligation—to purchase TCKR stock on the expiration date for $15, whatever the market price might be. If one year from now, TCKR is trading at $10, you will not exercise the option. If, on the other hand, TCKR is trading at $20 per share, you will exercise the option, buy TCKR at $15 and break even, since you paid $5 for the option initially. Any amount TCKR goes above $20 is pure profit, assuming zero transaction fees. 

Say you also sell (or "write" or "short") a European put option for TCKR stock. The expiration date, strike price, and cost of the option are the same. You receive $5 from writing the option, and it is not up to you to exercise or not exercise the option, since you don't own it. The buyer has purchased the right, but not the obligation, to sell you TCKR stock at the strike price; you are obligated to take that deal, whatever TCKR's market share price. So if TCKR trades at $10 a year from now, the buyer will sell you the stock at $15, and you will both break even: you already made $5 from selling the put, making up your shortfall, while the buyer already spent $5 to buy it, eating up his or her gain. If TCKR trades at $15 or above, you have made $5 and only $5, since the other party will not exercise the option. If TCKR trades below $10, you will lose money—up to $10, if TCKR goes to zero.

The profit or loss on these positions for different TCKR stock prices is graphed below. Notice that if you add the profit or loss on the long call to that of the short put, you make or lose exactly what you would have if you had simply signed a forward contract for TCKR stock at $15, expiring in one year. If shares are going for less than $15, you lose money. If they are going for more, you gain. Again, this scenario ignores all transaction fees.

Another way to imagine put-call parity is to compare the performance of a protective put and a fiduciary call of the same class. A protective put is a long stock position combined with a long put, which acts to limit the downside of holding the stock. A fiduciary call is a long call combined with cash equal to the present value (adjusted for the discount rate) of the strike price; this ensures that the investor has enough cash to exercise the option on the expiration date. Before, we said that TCKR puts and calls with a strike price of $15 expiring in one year both traded at $5, but let's assume for a second that they trade for free:

In the two graphs above, the y-axis represents the value of the portfolio, not the profit or loss, because we're assuming that traders are giving options away. They are not, however, and the prices of European put and call options are ultimately governed by put-call parity. In a theoretical, perfectly efficient market, the prices for European put and call options would be governed by the equation:

C + PV(x) = P + S

Let's say that the risk-free rate is 4% and that TCKR stock is currently trading at $10. Let's continue to ignore transaction fees and assume that TCKR does not pay a dividend. For TCKR options expiring in one year with a strike price of $15 we have:

C + (15 ÷ 1.04) = P + 10

4.42 = P - C

In this hypothetical market, TCKR puts should be trading at a $4.42 premium to their corresponding calls. This makes intuitive sense: with TCKR trading at just 67% of the strike price, the bullish call seems to have the longer odds. Let's say this is not the case, though for whatever reason, the puts are trading at $12, the calls at $7.

7 + 14.42 < 12 + 10

21.42 fiduciary call < 22 protected put

When one side of the put-call parity equation is greater than the other, this represents an arbitrage opportunity. You can "sell" the more expensive side of the equation and buy the cheaper side to make, for all intents and purposes, a risk-free profit. In practice, this means selling a put, shorting the stock, buying a call and buying the risk-free asset (TIPS, for example).

In reality, opportunities for arbitrage are short-lived and difficult to find. In addition, the margins they offer may be so thin that an enormous amount of capital is required to take advantage of them.