152 Mathematics for Finance
lying stock is $20.37 and the interest rate is 7.48%. Find an arbitrage
opportunity.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 difference 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 first sight seems consistent with intuition because
higher stock prices mean higher payoffs on calls, then the price of a put would
also grow. The latter, however, contradicts common sense because higher stock
prices mean lower payoffs 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 Black–Scholes formula is derived for
call and put options in Chapter 8.
Following the argument presented at the beginning of this section, we can
reformulate put-call parity as follows:
CE−PE=VX(0), (7.4)whereVX(0) is the value of a long forward contract, see (6.10). Note that ifX
is equal to the theoretical forward priceS(0)erTof the asset, then the value of
the forward contract is zero,VX(0) = 0, and soCE=PE.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 andT,then
VX(0) =S(0)−div 0 −Xe−rT, where div 0 is the present value of the dividend.
It follows that
CE−PE=S(0)−div 0 −Xe−rT. (7.5)
If dividends are paid continuously at a raterdiv,thenVX(0) =S(0)e−rdivT−
Xe−rT,so
CE−PE=S(0)e−rdivT−Xe−rT. (7.6)
Exercise 7.5
Outline an arbitrage proof of (7.5).Exercise 7.6
Outline an arbitrage proof of (7.6).