The equation for Put Call Parity is
C + PV(X) = P + S
where
C = the current market value of the call;
PV(x) = the present value of the strike price x, discounted from the
expiration date at a suitable risk free rate;
P = the current market value of the put;
S= the current market value of the share
It is derived purely using arbitrage arguments. It applies only to
European options, since a possibility of early exercise could cause a
divergence in the present values of the two portfolios. With regard to
your example at the money means X=S
Thus C=PV(S) = P+S
As PV(S)<S (we divide by 1+r typically with r the suitable risk free rate)
C-P = S-PV(S) as said above S-PV(S) is positive thus
C-P must be positive and then C>P.
This of course only hold if there are time to expiration left,
otherwise S=PV(S) and C-P =0 and in this case C=P, which if at the
money will mean C=P=0. But this is a special rare case.
Intuitively this hold as well ? keep in mind the value of an option is
linked to the probability or chance that it will end in the money and
by how much this value is (X-S for a put and S-X for a call). For a
put this value can be max X, thus if share falls to $0, unlikely but
then we did have a dot com boom! On the call side the value is maximum
? S-X but the share p[rice can go in theory to infinity. For example
warren Buffets Berkshire Hathaway was listed I think for $18 some 35y
ago. But now is worth some $90?000. Thus a 35y call strike $18 would
now be worth ($90 000 -$18) = $89?982. While if it totally blew up and
went bust to zero the put will only be worth $18-0 = $18. The chance
that the call gives you to make insane amounts of money while the
profit of a put is capped, implies that a call must be worth more than
a put everything else being equal. |