Remark 7.1 We can make a simple but powerful observation based on Theorem 7.1: The prices of European calls and puts depend in the same way on any variables absent in the put-call parity relation (7.1). In other words, the dierence of these prices does not depend on such variables. As an example, consider the expected return on stock If the price of a call should grow along with the expected return, which on rst sight seems consistent with intuition because higher stock prices mean higher payos on calls, then the price of a put would also grow. The latter, however, contradicts common sense because higher stock prices mean lower payos on puts. Because of this, one could argue that put and call prices should be independent of the expected return on stock. We shall see that this is indeed the case once the BlackScholes formula is derived for call and put options in Chapter 8. Following the argument presented at the beginning of this section, we can

4) where VX (0) is the value of a long forward contract, see (6.10). Note that if X is equal to the theoretical forward price S(0)erT of the asset, then the value of the forward contract is zero, VX (0) = 0, and so C E = P E. Formula (7.4) allows us to generalise put-call parity by drawing on the relationships established in Remark 6.3. Namely, if the stock pays a dividend between times 0 and T , then VX (0) = S(0) div0 XerT , where div0 is the present value of the dividend. It follows that C E P E = S(0) div0 XerT . (7.5) If dividends are paid continuously at a rate rdiv, then VX (0) = S(0)erdivT XerT , so C E P E = S(0)erdivT XerT . (7.6) Exercise 7.5 Outline an arbitrage proof of (7.5). Exercise 7.6 Outline an arbitrage proof of (7.6). 7. Options: General Properties

Exercise 7.7 For the data in Exercise 6.5, nd the strike price for European calls and puts to be exercised in six months such that C E = P E. For American options put-call parity gives only an estimate, rather than a strict equality involving put and call prices. Theorem 7.2 (Put-Call Parity Estimates) The prices of American put and call options with the same strike price X and expiry time T on a stock that pays no dividends satisfy S(0) XerT C A P A S(0) X.

Proof Suppose that the rst inequality fails to hold, that is, C A P A S(0) + XerT > 0. Then we can write and sell a call, and buy a put and a share, nancing the transactions on the money market. If the holder of the American call chooses to exercise it at time t T , then we shall receive X for the share and settle the money market position, ending up with the put and a positive amount X + (C A P A S(0))ert = (Xert + C A P A S(0))ert (XerT + C A P A S(0))ert > 0. If the call option is not exercised at all, we can sell the share for X by exercising the put at time T and close the money market position, also ending up with a positive amount X + (C A P A S(0))erT > 0. Now suppose that C A P A S(0) + X < 0.