- Options: General Properties 153
Exercise 7.7
For the data in Exercise 6.5, find the strike price for European calls and
puts to be exercised in six months such thatCE=PE.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 priceXand
expiry timeTon a stock that pays no dividends satisfy
S(0)−Xe−rT≥CA−PA≥S(0)−X.Proof
Suppose that the first inequality fails to hold, that is,
CA−PA−S(0) +Xe−rT> 0.Then we can write and sell a call, and buy a put and a share, financing the
transactions on the money market. If the holder of the American call chooses
to exercise it at timet≤T, then we shall receiveXfor the share and settle
the money market position, ending up with the put and a positive amount
X+(CA−PA−S(0))ert=(Xe−rt+CA−PA−S(0))ert
≥(Xe−rT+CA−PA−S(0))ert> 0.If the call option is not exercised at all, we can sell the share forXby exercising
the put at timeTand close the money market position, also ending up with a
positive amount
X+(CA−PA−S(0))erT> 0.
Now suppose that
CA−PA−S(0) +X< 0.In this case we can write and sell a put, buy a call and sell short one share,
investing the balance on the money market. If the American put is exercised
at timet≤T, then we can withdrawXfrom the money market to buy a share
and close the short sale. We shall be left with the call option and a positive
amount
(−CA+PA+S(0))ert−X>Xert−X≥ 0.