- Options: General Properties 169
Proof
Suppose thatCA(T′)>CA(T′′). We write and sell one option expiring at
timeT′and buy one with the same strike price but expiring at timeT′′,invest-
ing the balance without risk. If the written option is exercised at timet≤T′,
we can exercise the other option immediately to cover our liability. The posi-
tive balanceCA(T′)−CA(T′′)>0 invested without risk will be our arbitrage
profit.
The argument is the same for puts.
7.5 Time Value of Options
The following convenient terminology is often used. We say that at timeta
call option with strike priceXis
- in the moneyifS(t)>X,
- at the moneyifS(t)=X,
- out of the moneyifS(t)<X.
Similarly, for a put option we say that it is
- in the moneyifS(t)<X,
- at the moneyifS(t)=X,
- out of the moneyifS(t)>X.
Also convenient, though less precise, are the termsdeep in the moneyanddeep
out of the money, which mean that the difference between the two sides in the
respective inequalities is considerable.
An American option in the money will bring a positive payoff if exercised
immediately, whereas an option out of the money will not. We use the same
terms for European options, though their meaning is different: Even if the
option is currently in the money, it may no longer be so on the exercise date,
when the payoff may well turn out to be zero. A European option in the money
is no more than a promising asset.
Definition 7.1
At timet≤Ttheintrinsic valueof a call option with strike priceXis equal
to (S(t)−X)+. The intrinsic value of a put option with the same strike price
is (X−S(t))+.
We can see that the intrinsic value is zero for options out of the money or at
the money. Options in the money have positive intrinsic value. The price of an