30 Frequently Asked Questions In Quantitative Finance
practice though. If you do find such an arbitrage then
it usually disappears by the time you put the trade on.
See Kamara & Miller (1995) for details of the statistics
of no-arbitrage violations.
When there are dividends on the underlying stock dur-
ing the life of the options then we must adjust the
equation to allow for this. We now find that
C−P=S−Present value of all dividends−Ee−r(T−t).
This, of course, assumes that we know what the divi-
dends 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 expiration of
the option. Dividends should similarly be discounted.
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 an American put should
be exercised if the stock falls low enough.
References and Further Reading
Kamara, A & Millet, T 1995 Daily and Intradaily Tests of Euro-
pean Put-Call Parity.Journal of Financial and Quantitative
Analysis, December 519–539
