Put-call parity

From WilmottWiki

Put-call parity is a model-independent arbitrage relationship between European-style call prices and put prices with the same strike price and expiration :

,

where is underlying stock price, is the current value of a zero-coupon bond with maturity , when it pays $1 and PV(Dividends) is the present value of the dividends received by the stock owner over the holding period.

The only assumptions needed to derive this relation are that:

• there is a zero-coupon bond of the same maturity as the option's expiration and
• the dividends to be received are known and certain.

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.

An empirical study of the put-call parity relationship was performed by Kamara and Miller (1995). They found that, in practice, there are many small violations. But in investigating all violations in intraday transaction data, they found that almost half of the arbitrages result in a loss when execution delays are accounted for. They also found that the mean ex post profit in trying to exploit the violations was negative.

Early exercise

When the options are American, put-call parity does not hold. This is 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 as American put should be exercised if the stock falls low enough.

References

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