Finance-Theory · Unit 12 · Video 1 · Interactive Practice

Calls and Puts: The Right Without the Obligation

IKey Formulas

Formula	Name	What it says
$C_T = \text{Max}\left[\,0,\ S_T - K\,\right]$	Call payoff at maturity	The holder buys at $K$ only when the share is worth more; with $K$ at $70 and $S_T$ at $80 the call pays $10
$P_T = \text{Max}\left[\,0,\ K - S_T\,\right]$	Put payoff at maturity	The holder sells at $K$ only when the share is worth less; with $K$ at $60 and $S_T$ at $40 the put pays $20
$S_T - K$	Long futures to buy at $K$	The same gain as the call above the strike, but no floor below it: buying at $60 a share worth $40 is a $20 loss the holder cannot avoid
$S_t,\ K,\ C_t,\ P_t,\ \tau$	Notation	Stock price, strike, call price, put price, time to maturity; $K$ has no time subscript because it is fixed in the contract

Key Insight: The holder exercises only when exercising gains something, so an option's payoff at maturity is never negative. That floor at zero, which a futures position does not have, is the asymmetry behind everything else in this unit.

IIExercise or Walk Away

The holder exercises only when it gains, so the payoff is the larger of zero and the exercise gain.

Contract Strike K

IIIWhy a Call Above Today's Price Is Worth Something

A call struck at $70 on a $60 stock has value because some price paths end above $70.

Paths 40

💡 These paths are illustrative random draws, not a forecast. Turning the chance of such payoffs into a price today is the pricing problem later in this unit.

IVOptions Net to Zero; Warrants Do Not

Every option held long was written by another investor; a warrant is written by the company itself.

VQuiz Questions

Problem 1 · A Call at Maturity

Given (illustrative numbers): a call with strike price $50 on a stock that closes at $62 on the maturity date. Find the call's payoff at maturity, per share.

✅ Correct! The holder buys at $50 a share worth $62 and gains $12.

❌ Not quite. The payoff is zero only when the stock ends at or below the strike; here

S_T > K

, so the holder exercises.

❌ Check the formula.

K - S_T

is the put's exercise gain. A call pays

S_T - K

, and its payoff is never negative anyway.

❌ Not quite. Exercising means paying the strike. The holder receives a share worth $62 but pays $50 for it.

❌ Not quite. Use

C_T = \text{Max}[\,0,\ S_T - K\,]

with

S_T = 62

and

K = 50

Show solution

With $S_T = 62$ and $K = 50$ (dollars per share):

C_T = \text{Max}\left[\,0,\ S_T - K\,\right] = \text{Max}\left[\,0,\ 62 - 50\,\right] = \text{Max}\left[\,0,\ 12\,\right] = 12

The stock ends above the strike, so the holder exercises, buys at $50 and holds a share worth $62: a payoff of $12 per share.

Problem 2 · A Put That Ends Above Its Strike

Given (illustrative numbers): a put with strike price $45 on a stock that closes at $52 on the maturity date. Find the put's payoff at maturity, per share.

✅ Correct! Selling at $45 a share worth $52 would lose $7, so the holder lets the put expire.

❌ Close, but…

K - S_T = -7

is what exercising would give. The holder is not obliged to exercise, so the payoff is the larger of that and zero.

❌ Not quite.

S_T - K = 7

is a call's exercise gain. A put pays when the stock ends below the strike.

❌ Not quite. Exercising a put means handing over a share worth $52 in return for $45; the strike is not a payoff by itself.

❌ Not quite. Use

P_T = \text{Max}[\,0,\ K - S_T\,]

Show solution

With $K = 45$ and $S_T = 52$ :

P_T = \text{Max}\left[\,0,\ K - S_T\,\right] = \text{Max}\left[\,0,\ 45 - 52\,\right] = \text{Max}\left[\,0,\ -7\,\right] = 0

The put gives the right to sell at $45, but the market pays $52, so the holder does not exercise and the put expires worthless: a payoff of $0, never −$7.

Problem 3 · Obligation Versus Right

Given (illustrative numbers): the stock ends at $52 on the maturity date. Trader A holds a long futures position obligating her to buy one share at $60 on that date; trader B holds a call with strike price $60 maturing on the same date. Find each trader's payoff at maturity.

Trader A (futures, must buy at the agreed price)

Trader B (call, strike $60)

✅ Correct! A must buy at $60 a share worth $52 and loses $8; B simply walks away with $0. The $8 gap is exactly the loss the right without the obligation avoids.

❌ Check the sign. A pays $60 for something worth $52: that is a loss,

S_T - K = 52 - 60

❌ Close, but… A zero floor belongs to options. A futures contract obliges A to buy, so she takes the loss.

❌ Check trader A. A futures payoff at maturity is

S_T - K

with no floor.

❌ Not quite. A does not lose the whole share price: she pays $60 and receives a share worth $52, so the payoff is the difference,

S_T - K

❌ Check the direction.

K - S_T = 8

is how far the stock ended below the strike, which is what a put would pay. A call pays only when the stock ends above the strike.

❌ Not quite. The strike is the price B would pay on exercise, not a payoff. Paying $60 for a share worth $52 would lose money, so B does not exercise.

❌ Close, but… That is what exercising would give. B is not obliged to exercise and will not.

❌ Check trader B. Use

C_T = \text{Max}[\,0,\ S_T - K\,]

Show solution

Trader A is obligated to buy at $60:

S_T - K = 52 - 60 = -8

a loss of $8 per share that she cannot decline.

Trader B has the right, not the obligation:

C_T = \text{Max}\left[\,0,\ 52 - 60\,\right] = \text{Max}\left[\,0,\ -8\,\right] = 0

B lets the call expire, with a payoff of $0. Above $60 the two traders would gain the same amount; below $60 only the futures holder loses.

Problem 4 · A European Put

Given (illustrative numbers): a European put with strike price $55 and one month to expiration. The stock trades at $50 today.

Can the holder exercise today?

The stock ends at $58 on the expiration date. What is the put's payoff?

✅ Correct! Only the expiration date counts for a European put, and on that date the stock is above the strike, so the put expires worthless.

❌ Not quite. Exercising at any time up to expiration is the American rule. "European" names the exercise rule: expiration date only.

❌ Not quite. Being below the strike makes exercising attractive, but a European option can be exercised only on the expiration date, wherever the stock trades before then.

❌ Check the direction.

S_T - K = 3

is a call's exercise gain. A put pays when the stock ends below the strike; at $58 it ends above $55.

❌ Check the exercise rule. A European option may be exercised only on the expiration date, wherever the stock trades before then.

❌ Not quite. $5 is what exercising today would give, but a European put cannot be exercised today. Its payoff depends on

S_T

only.

❌ Close, but…

K - S_T = -3

is what exercising would give; the holder declines, so the payoff is zero.

❌ Check the payoff. Use

P_T = \text{Max}[\,0,\ K - S_T\,]

with the price on the expiration date.

Show solution

Exercise rule. An American option may be exercised at any time up to and including expiration; a European option only on the expiration date. The $5 gap between the strike and today's price cannot be collected today.

Payoff. On the expiration date $S_T = 58$ and $K = 55$ :

P_T = \text{Max}\left[\,0,\ 55 - 58\,\right] = \text{Max}\left[\,0,\ -3\,\right] = 0

Selling at $55 a share worth $58 would lose $3, so the put expires with a payoff of $0.

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