Put-call parity is a relationship between the prices of a European-style call option and a European-style put option, as long as they have the same strike and expiration:

Call price — Put price = Stock price

— Strike price (present valued from expiration).

Example

Stock price is $98, a European call option struck at $100 with an expiration of nine months has a value of $9.07. The nine-month, continuously compounded, interest rate is 4.5%. What is the value of the put option with the same strike and expiration?

By rearranging the above expression we find

The put must therefore be worth $7.75.

Long answer

This relationship,

between European calls (value C) and puts (value P)with the same strike (K) and expiration (T)valued at time t is a result of a simple arbitrage argument. If you buy a call option, at the same time write a put, and sell stock short, what will your payoff be at expiration? If the stock is above the strike at expiration you will have S — K from the call, 0 from the put and — S from the stock. A total of —K. If the stock is below the strike at expiration you will have 0 from the call, —S again from the stock, and —(K — S) from the short put. Again a total of — K. So, whatever the stock price is at expiration this portfolio will always be worth — K, a guaranteed

amount. Since this amount is guaranteed we can discount it back to the present. We must have

This is put-call parity.

Another way of interpreting put-call parity is in terms of implied volatility. Calls and puts with the same strike and expiration must have the same implied volatility.

The beauty of put-call parity is that it is a model-independent relationship. To value a call on its own we need a model for the stock price, in particular its volatility. The same is true for valuing a put. But to value a portfolio consisting of a long call and a short put (or vice versa), no model is needed. Such model-independent relationships are few and far between in finance. The relationship between forward and spot prices is one, and the relationships between bonds and swaps is another.

In practice options don't have a single price, they have two prices, a bid and an offer (or ask). This means that when looking for violations of put-call parity you must use bid (offer) if you are going short (long) the options. This makes the calculations a little bit messier. If you think in terms of implied volatility then it's much easier to spot violations of put-call parity. You must look for non-overlapping implied volatility ranges. For example, suppose that the bid/offer on a call is 22%/25% in implied volatility terms and that on a put (same strike and expiration) is 21%/23%. There is an overlap between these two ranges (22-23%) and so no arbitrage opportunity. However, if the put prices were 19%/21% then there would be a violation of put-call parity and hence an easy arbitrage opportunity. Don't expect to find many (or, indeed, any) of such simple free-money opportunities in practice though. If you do find such an arbitrage then it usually disappears by the time you put the trade on. See Kamara and Miller (1995) for details of the statistics of no-arbitrage violations.

When there are dividends on the underlying stock during the life of the options, we must adjust the equation to allow for this. We now find that

C — P = S — Present value of all dividends — Ee—r(T—1).

This, of course, assumes that we know what the dividends will be.

If interest rates are not constant then just discount the strike back to the present using the value of a zero-coupon bond with maturity the same as the expiration of the option. Dividends should similarly be discounted.

When the options are American, put-call parity does not hold, because the short position could be exercised against you, leaving you with some exposure to the stock price. Therefore you don't know what you will be worth at expiration. In the absence of dividends it is theoretically never optimal to exercise an American call before expiration, whereas an American put should be exercised if the stock falls low enough.

References and Further Reading

Kamara, A & Miller, T 1995 Daily and Intradaily Tests of European Put-Call Parity. Journal of Financial and Quantitative Analysis, December 519-539.

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