Finance-Theory · Unit 12 · Video 2 · Interactive Practice

A Put Is Insurance on a Stock: It Keeps the Upside, and Fear Sets the Premium

IKey Formulas

Formula	Name	What it means
$P_T = \max[0,\ K - S_T]$	Put payoff at maturity	The right to sell at the strike $K$ pays only when the stock ends below it
$S_T + P_T = \max[S_T,\ K]$	Protective put (share + put)	A floor at $K$ under the downside; above $K$ the holder keeps every dollar of the rise
$\text{Deductible} = S_t - K$	Loss before coverage starts	Today's price minus the strike: with the stock at $20 and $K$ at $18, the first $2 of a fall per share is the holder's
$S_T + (F - S_T) = F$	Share + short futures at $F$	Locks in $F$ (ignoring carrying costs): the futures loss on a rise gives the gain back

Key Insight: A put's payoff can fall to zero but never below it, so it adds a floor without a ceiling; a short futures payoff goes negative as the price rises, and that loss is the ceiling. The protection costs the put price $P_t$ , paid up front.

IIThe Insurance Policy Inside a Put

A fall in the stock splits into the deductible the holder bears and the claim the put pays.

Today's price S_t $20 Strike K $18 Price at maturity S_T $10

Insurance	Put option	Value
Asset insured	The stock	one GE share
Current value	$S_t$	$20 a share
Term of policy	Time to maturity $\tau$	a few months
Maximum coverage	Strike $K$	$18 a share
Deductible	$S_t - K$	$2 a share
Premium	Put price $P_t$	paid today
Claim at maturity	$P_T = \max[0,\ K - S_T]$	$8

IIIA Floor Without a Ceiling: Protective Put vs Short Futures

Both hedges protect the share on a fall, but only the put leaves the rise to the holder.

Price at maturity S_T $10 Put price P_t (illustrative) $0

💡 The premium is priced by the swings the market expects. The VIX, the implied volatility in S&P 500 option prices, closed at 80.86 on 20 November 2008 and at 17.71 on 16 September 2026, and in autumn 2008 one month of protection against a 10% fall in the index cost roughly ten times what it had a year earlier.

IVQuiz Questions

Problem 1 · The Floor at the Strike

Given: A stock trades at $36 today. You hold one share and a put on it with strike $32. At maturity the stock is at $19. Find the put's payoff and the value of share plus put (setting aside the price paid for the put).

What does the put pay?

What are the share and the put worth together?

✅ Correct! The put pays $13 and the share is worth $19, so together they reach exactly the $32 floor.

❌ That is the call's formula.

\max[0,\ S_T - K]

is a call's payoff; a put pays

\max[0,\ K - S_T]

, which is positive here because the stock ended below the strike.

❌ That is the share's value, not the put's payoff. The put pays the strike minus the stock price.

❌ The payoff is a difference, not a sum. $51 adds the strike and the stock price; the put pays

K - S_T

, the gap between them.

❌ That is the put's payoff alone. The share is still worth $19 at maturity; add it to the put's $13.

❌ That is the share alone. The put pays $13 on top of the share's $19.

❌ Not quite. Use

P_T = \max[0,\ K - S_T]

with

K

= 32 and

S_T

= 19.

❌ Not quite. Add the share's value at maturity to the put's payoff.

Show solution

Put payoff at maturity, in dollars per share:

P_T = \max[0,\ K - S_T] = \max[0,\ 32 - 19] = 13

Share plus put: $19 + $13 = $32. Whenever $S_T$ is below the strike, $S_T + P_T = S_T + (K - S_T) = K$ , so the pair is worth exactly $32 however far the stock falls. The first $4 of the fall, from $36 to $32, is the deductible, and it is not covered.

Problem 2 · Does the Put Give Up the Gains?

Given: A stock trades at $45. Investor A holds one share plus a put with strike $40; Investor B holds one share plus a short futures position at $45 (ignore carrying costs). At maturity the stock is at $58. Find each position's value at maturity, setting aside the price of the put.

Investor A (share + put)?

Investor B (share + short futures)?

✅ Correct! The put expires worthless and A keeps all $58; B's futures lose $13 and B is back to the locked-in $45.

❌ The strike is a floor, not a ceiling. Above $40 the put is simply not exercised, and the share's full value stays with the holder.

❌ That is the futures hedge's outcome. Only the futures position locks the value in; the put can fall to zero but never below it.

❌ Check the put's payoff. $76 adds $18 as if the put paid

S_T - K

; a put pays

\max[0,\ K - S_T]

, which is zero here.

❌ The futures position has a payoff too. A short futures at $45 loses when the price rises: $45 − $58 = −$13.

❌ Check the sign. A short futures gains when the price falls and loses when it rises; at $58 it loses $13.

❌ $40 is the strike of A's put. B holds no put; the short futures locks in its futures price, $45, whatever the price at maturity.

❌ Not quite. Share plus put is worth

\max[S_T,\ K]

❌ Not quite. Share plus short futures is worth

S_T + (F - S_T) = F

Show solution

Investor A. The put pays $\max[0,\ 40 - 58] = 0$ , so share plus put is worth $58 + $0 = $58. The put's value fell to zero as the stock rose, but it cannot go negative, so none of the gain is given up.

Investor B. The short futures pays $F - S_T$ : $45 − $58 = −$13. Share plus futures: $58 − $13 = $45. The futures loss is what caps the hedge.

A's advantage is not free: A paid the put's price up front, which this comparison sets aside.

Problem 3 · Deductible Plus Premium

Given: A stock trades at $54. You buy one share and a put with strike $49 for a premium of $1.25 (ignore interest on the premium). Find the largest possible loss per share by maturity, and the prices at maturity at which this protected position beats a share hedged with a short futures position at $54 (ignore carrying costs).

Largest possible loss per share, premium included?

The protected position ends ahead of the futures hedge when the price at maturity is:

✅ Correct! The worst case is the $5 deductible plus the $1.25 premium; and the put hedge must beat the futures' locked $54 by enough to recover its premium.

❌ That is the deductible alone. The premium is also gone whatever happens, so it adds to the worst-case loss.

❌ The put covers only falls below the strike. The first $5 of a fall, from $54 to $49, is the deductible, and it comes on top of the premium.

❌ The premium is a cost, not a credit. It adds to the deductible rather than offsetting it.

❌ Compare with the futures hedge, not the strike. The futures hedge is worth $54 at every price, so the put position must exceed $54 after its premium.

❌ The premium was forgotten. Above the strike the protected position is worth

S_T

minus the $1.25 already paid.

❌ The premium moves the break-even up, not down. The put position is worth $1.25 less than the share alone, so it needs a higher price.

❌ Not quite. Find the worst value of share plus put at maturity, subtract the premium, and compare with the $54 paid for the share.

❌ Not quite. Set the put position's net value above the strike,

S_T - 1.25

, against the futures hedge's $54.

Show solution

Worst case. Share plus put is worth $\max[S_T,\ 49]$ dollars, never less than $49. Net of the $1.25 premium, the worst value is $49 − $1.25 = $47.75, a loss of $54 − $47.75 = $6.25 per share: the deductible ($54 − $49 = $5) plus the premium.

Against the futures hedge. The share plus a short futures at $54 is worth $54 at every price. Below the strike the put position is worth $47.75, which is less. Above the strike it is worth $S_T - 1.25$ dollars, and

S_T - 1.25 > 54 \iff S_T > 55.25

So the protected position ends ahead only when the stock finishes above $55.25: the price of keeping the upside is the premium plus the deductible on the downside.

Problem 4 · Insuring an Index

Given: An investor holds a portfolio that tracks the S&P 500, with the index at 1,250 (illustrative numbers). She buys a one-month European put on the index with strike 1,125, which can be exercised only at expiry. The index then falls to 950. Find the percentage fall in her protected position (portfolio plus put, before the premium), and decide what the put was worth on the day she bought it.

Percentage fall of portfolio plus put, before the premium?

On the day she bought it, the European put was worth:

✅ Correct! The strike sits 10% below the starting level, so the protected position falls 10% however far the index drops; and the right to choose at expiry is real protection, so it has a price.

❌ That is the unprotected fall. From 1,250 to 950 is 24%, but the put pays 1,125 − 950 = 175 points on top.

❌ The put does not simply subtract 10 points of percentage. Below the strike, portfolio plus put is worth exactly the strike, 1,125.

❌ The strike is below today's level. The first 125 points of the fall are the deductible, and they are the investor's.

❌ The exercise date is not the difference. A futures contract commits both sides on its date; the put holder never has to exercise, so she holds protection and must pay for it.

❌ Early exercise is not the source of the value. The payoff at expiry,

\max[0,\ K - S_T]

, is never negative and is positive whenever the index ends below the strike.

❌ A European put also gives protection. Its single exercise date is still a choice, not an obligation.

❌ Not quite. Below the strike, portfolio plus put is worth the strike; compare 1,125 with 1,250.

❌ Not quite. Ask whether the holder can ever be made worse off at expiry by holding the put.

Show solution

Protected position. At 950 the put pays $\max[0,\ 1125 - 950] = 175$ points, so portfolio plus put is worth $950 + 175 = 1125$ points. The fall is

\frac{1250 - 1125}{1250} = \frac{125}{1250} = 10\%

which is the deductible, $S_t - K$ = 125 points, as a share of today's level. The unprotected portfolio fell $\tfrac{1250 - 950}{1250} = 24\%$ .

Value on day one. A futures contract is worth zero when struck because both sides are committed to transact. The European put's holder has one date on which she may exercise; she exercises only if the index is below the strike, so the payoff is never negative and is sometimes positive. The seller provides real protection, so the put has a positive price, paid up front.

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