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Lesson 1 of 6 · 12 min

Payoff at expiration and exercise value before it

Before expiration, an option's exercise value is what it would be worth if you could settle it now, comparing the spot price with the present value of the exercise price.

In short

At expiration a European call pays $\max(0, S_T - X)$ and a put pays $\max(0, X - S_T)$ . The buyer exercises only when the payoff is positive.
If an option expires unexercised, the buyer's loss is the premium paid.
Before expiration (t < T), the exercise value compares the spot price $S_t$ with the present value of the exercise price, $PV(X) = X(1+r)^{-(T-t)}$ .
Call exercise value = $\max(0, S_t - PV(X))$ . Put exercise value = $\max(0, PV(X) - S_t)$ . Neither can be negative.
With X set equal to the forward price, a call's exercise value matches a long forward's value whenever $S_t > PV(X)$ ; a put's exercise value matches a short forward only on the downside.

What happens at expiration

An option is a contingent claim: the buyer decides whether the trade happens. A European option can only be exercised at expiration, time T. The buyer pays a premium up front ( $c_0$ for a call, $p_0$ for a put) for this right.

At expiration the decision is simple. A call holder exercises only if buying at X is better than buying in the market, so if $S_T > X$ . A put holder exercises only if selling at X beats selling in the market, so if $S_T < X$ .

This gives the payoffs $c_T = \max(0, S_T - X)$ and $p_T = \max(0, X - S_T)$ . The max with zero is the whole point of an option: the holder can walk away, so the payoff is never negative. If the option expires unexercised, the buyer simply loses the premium.

Payoff at expiration for an option buyer (X = 100)
$S_T$	Call payoff $\max(0, S_T - X)$	Put payoff $\max(0, X - S_T)$
80	0 (not exercised)	20
100	0	0
115	15	0 (not exercised)
140	40	0 (not exercised)

Both payoffs are floored at zero and kink at the exercise price. The call gains one-for-one above 100, the put one-for-one below it; on the other side the holder simply walks away.

Exercise value before expiration

Between inception and expiration, traders want a quick gauge of what an option is worth right now. The exercise value (also called intrinsic value) answers: what would this option be worth if it could be settled today, at time t?

The curriculum builds the time value of money into this measure. The exercise price is paid (call) or received (put) only at T, so today it is worth its present value, $PV(X) = X(1+r)^{-(T-t)}$ , where $T - t$ is the time left in years.

So the call's exercise value is $\max(0, S_t - PV(X))$ : you own something worth $S_t$ today and owe something worth $PV(X)$ today. The put's exercise value is $\max(0, PV(X) - S_t)$ . As with the payoff, the floor at zero reflects the holder's right not to exercise.

At expiration $T - t = 0$ , so $PV(X) = X$ and the exercise value becomes the ordinary payoff. Before that, discounting makes $PV(X)$ smaller than X, which raises a call's exercise value and lowers a put's compared with simply using X.

CHF 80 call, six months left, r = 3%. Discounting moves the kink from 80 to 78.83, so the exercise value line sits 1.17 above the undiscounted one wherever the call is in the money.

Think of a call as 'buy the asset now, pay X later'. Paying later is cheaper in today's money, which is why the call's exercise value uses PV(X), not X.

Link to forward contracts

Suppose a call's exercise price equals the forward price, $X = F_0(T)$ . A long forward is worth $V_t = S_t - PV(F_0(T))$ at time t. Whenever $S_t > PV(X)$ , this is exactly the call's exercise value. The call and the long forward agree on the upside.

They differ on the downside. If $S_t < PV(X)$ , the long forward has a negative value, while the call's exercise value is floored at zero. The call holder is not obliged to buy.

A put with $X = F_0(T)$ mirrors a short forward in the same one-sided way. At maturity, if $S_T < X$ , both pay $X - S_T$ . If $S_T > X$ , the short forward must pay $S_T - F_0(T)$ to the counterparty, while the put simply expires unexercised.

Both comparisons ignore the premium. The option buyer paid $c_0$ or $p_0$ at t = 0 for the right to walk away; the forward cost nothing to enter.

Position (X = F₀(T))	Underlying rises	Underlying falls
Long call	Gains like a long forward	Value floored at zero
Long forward	Gains	Loses
Long put	Value floored at zero	Gains like a short forward
Short forward	Loses	Gains

With X equal to the EUR 50 forward price, the put and the short forward lie on the same line below 50. Above 50 the forward keeps losing while the put stops at zero; the premium is what pays for that cut-off.

Formulas

Payoffs at expiration: $c_T = \max(0,\, S_T - X) \qquad p_T = \max(0,\, X - S_T)$
S_T = underlying price at expiration, X = exercise price.
Exercise value at time t: $\text{Call: } \max\!\left(0,\, S_t - X(1+r)^{-(T-t)}\right) \qquad \text{Put: } \max\!\left(0,\, X(1+r)^{-(T-t)} - S_t\right)$
r = annual risk-free rate, T − t = years left to expiration. Assumes no income or cost from holding the underlying.
Buyer's profit at expiration (ignoring time value of money): $\Pi_{call} = \max(0,\, S_T - X) - c_0 \qquad \Pi_{put} = \max(0,\, X - S_T) - p_0$
Payoff minus the premium paid at t = 0.

Worked examples

Example 1 Exercise value of a call

A European call on a share has an exercise price of CHF 80 and six months left to expiration. The risk-free rate is 3% and the share trades at CHF 86. What is the call's exercise value? What would it be if the share traded at CHF 75?

Example 2 Exercise value of a put

A European put has an exercise price of GBP 120 and nine months to expiration. The risk-free rate is 2.5% and the underlying trades at GBP 111. What is the put's exercise value?

Example 3 Put versus short forward at maturity

An investor can either sell a stock forward at F₀(T) = EUR 50 or buy a put with X = EUR 50. Compare the two at maturity if S_T = EUR 42 and if S_T = EUR 57, ignoring the put premium.

On the exam

Expect a calculation of exercise value with PV(X), often paired with time value, and conceptual items comparing an option's payoff with a forward's. Wrong options usually use X instead of PV(X).

Using X instead of $PV(X)$ for the exercise value before expiration. This understates a call's and overstates a put's exercise value.
Discounting over the option's original life instead of the time remaining, T − t.
Reporting a negative exercise value. The minimum is zero, because the holder can choose not to exercise.
Forgetting that the forward comparison ignores the option premium paid at t = 0.

Check yourself

Q1A European call has an exercise price of EUR 60 and six months to expiration. The share trades at EUR 64 and the risk-free rate is 4%. The call's exercise value is closest to:
Q2A European put has an exercise price of JPY 2,000 and three months to expiration. The underlying trades at JPY 1,900 and the risk-free rate is 1%. The put's exercise value is closest to:
Q3A call with an exercise price of NOK 300 expires when the underlying trades at NOK 287. The call buyer's payoff at expiration is:
Q4A put's exercise price equals the forward price. Its payoff at maturity equals that of a short forward:
Q5Which statement about an option's exercise value before expiration is most accurate?

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