This payoff amount is also the value of the option at

This payoff amount is also the value of the option at expiration. It represents value because it is
what the option is worth at that point. If the holder of the option sells it to someone else an
instant before expiration, it should sell for that amount because the new owner would exercise it
and capture that amount. To the seller, the value of the option at that point is – Max (0, S
T
– X ),
which is negative to the seller if the option is in the money and zero otherwise.
Using the payoff value and the price paid for the option, we can determine the profit from the
strategy, which is denoted with the Greek symbol Π. Let us say the buyer paid c
0
for the option
at time 0. Then the profit is
Π = Max (0, S
T
– X ) – c
0
(profit to the call buyer),
To the seller, who received the premium at the start, the payoff is
– c
T
= – Max (0, S
T
– X ) (payoff to the call seller),
The profit is
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Π = – Max (0, S
T
– X ) + c
0
(profit to the call seller),
Exhibit 3
illustrates the payoffs and profits to the call buyer and seller as graphical
representations of these equations, with the payoff or value at expiration indicated by the dark
line and the profit indicated by the light line. Note in Panel A that the buyer has no upper limit
on the profit and has a fixed downside loss limit equal to the premium paid for the option. Such
a condition, with limited loss and unlimited gain, is a temptation to many unsuspecting
investors, but keep in mind that the graph does not indicate the frequency with which gains and
losses will occur. Panel B is the mirror image of Panel A and shows that the seller has unlimited
losses and limited gains. One might suspect that selling a call is, therefore, the worst investment
strategy possible. Indeed, it is a risky strategy, but at this point these are only simple strategies.
Other strategies can be added to mitigate the seller’s risk to a substantial degree.
Exhibit 3. Payoff and Profit from a Call Option
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Now let us consider put options. Recall that a put option allows its holder to sell the underlying
asset at the exercise price. Thus, the holder should exercise the put at expiration if the
underlying asset is worth less than the exercise price ( S
T
< X ). In that case, the put is said to be
in the money. If the underlying asset is worth the same as the exercise price ( S
T
= X ), meaning
the put is at the money, or more than the exercise price ( S
T
> X ), meaning the put is out of the
money, the option holder would not exercise it and it would expire with zero value. Thus, the
payoff to the put holder is
p
T
= Max (0, X – S
T
) (payoff to the put buyer),
If the put buyer paid p
0
for the put at time 0, the profit is
Π = Max (0, X – S
T
) – p
0
(profit to the put buyer),
And for the seller, the payoff is
– p
T
= – Max (0, X – S
T
) (payoff to the put seller),
And the profit is
Π = – Max (0, X – S
T
) + p
0
(profit to the put seller),
Exhibit 4 illustrates the payoffs and profits to the buyer and seller of a put.