150 Mathematics for Finance
7.2 Put-Call Parity
In this section we shall make an important link between the prices of European
call and put options.
Consider a portfolio constructed by and writing and selling one put and
buying one call option, both with the same strike priceXand exercise dateT.
Adding the payoffs of the long position in calls and the short position in puts,
we obtain the payoff of a long forward contract with forward priceX and
delivery timeT.Indeed, ifS(T)≥X,then the call will payS(T)−Xand the
put will be worthless. IfS(T)<X, then the call will be worth nothing and the
writer of the put will need to payX−S(T). In either case, the value of the
portfolio will beS(T)−Xat expiry, the same as for the long forward position,
see Figure 7.2. As a result, the current value of such a portfolio of options
should be that of the forward contract, which isS(0)−Xe−rT, see Remark 6.3.
This motivates the theorem below. Even though the theorem follows from the
above intuitive argument, we shall give a different proof with a view to possible
generalisations.
Figure 7.2 Long forward payoff constructed from calls and putsTheorem 7.1 (Put-Call Parity)
For a stock that pays no dividends the following relation holds between the
prices of European call and put options, both with exercise priceXand exercise
timeT:
CE−PE=S(0)−Xe−rT. (7.1)
Proof
Suppose that
CE−PE>S(0)−Xe−rT. (7.2)
In this case an arbitrage strategy can be constructed as follows: At time 0
- buy one share forS(0);