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Optionsbeginner

Put Option

A put option gives the holder the right to sell the underlying asset at the strike price KK during the life of the contract. The payoff at expiry is max⁡(K−ST,0)\max(K - S_T, 0). Buyers are bearish on the underlying or are using the put for portfolio insurance. The writer accepts the obligation to buy the asset at KK if the put holder exercises. A long put's downside is capped at the premium PP, and the maximum payoff is KK, reached when the stock falls to zero.

Why it matters

A put is price insurance on something you own or might owe. A super fund holding ASX 200 stocks can buy XJO puts to protect against a market crash, much as a homeowner buys fire insurance. If the market falls, the put pays. If the market rises, the put expires worthless and the fund forfeits the premium, just as an unused insurance policy expires at year end. The premium is the price of peace of mind plus the chance of profit if conditions turn bad.

Formulas

Put payoff at expiry
Payoff=max⁡(K−ST,0)\text{Payoff} = \max(K - S_T, 0)
Zero whenever ST≥KS_T \geq K. Below the strike, payoff falls one-for-one as STS_T falls.
Long put profit at expiry
Profit=max⁡(K−ST,0)−P\text{Profit} = \max(K - S_T, 0) - P
PP is the premium paid. The position breaks even when ST=K−PS_T = K - P.

Worked examples

Scenario

An investor buys a put on Telstra with strike K=K = 4.50$ and pays P=P = 0.20$ per share. At expiry ST=S_T = 4.00$.

Solution

Payoff per share =max⁡(4.50−4.00,0)== \max(4.50 - 4.00, 0) = 0.50$. Profit per share =0.50−0.20== 0.50 - 0.20 = 0.30$. The investor exercises the right to sell at $4.50, locking in a gain above the lower spot price. Across one ASX contract of 100 shares, that is $30 of profit on $20 of premium.

Scenario

A self-managed super fund holds 1000 BHP shares at $45 and buys 10 protective put contracts (100 shares each) with K=K = 42$ at P=P = 0.80$.

Solution

Total insurance cost =1000×0.80== 1000 \times 0.80 = 800$. If BHP crashes to ST=S_T = 30,theputspay, the puts pay \max(42 - 30, 0) = $12$ per share, or $12000 total. The stock loss on the shares is $15 per share, or $15000. Net loss to the fund =15000−12000+800== 15000 - 12000 + 800 = 3800,versus, versus 15000 unhedged. The put has bought a floor at $42 below which losses stop accumulating.

Common mistakes

  • ✗You must own the underlying stock to buy a put. A speculative put requires no underlying holding. Hedge funds frequently buy puts to express a directional bearish view, with no shares to deliver if assigned.
  • ✗Put profits are unlimited like calls. Maximum put payoff is KK, achieved only if the stock falls to zero. The put writer's maximum loss is therefore K−PK - P, not infinity, in contrast to the uncapped loss on an uncovered short call.
  • ✗Buying a put means you are sure the stock will fall. Many puts are bought as hedges, not speculation. The buyer may hope the put expires worthless because that means the protected stock position rose in value.

Revision bullets

  • •Right to sell at strike KK
  • •Payoff =max⁡(K−ST,0)= \max(K - S_T, 0)
  • •Profit =max⁡(K−ST,0)−P= \max(K - S_T, 0) - P
  • •Bearish or used for portfolio insurance
  • •Max profit =K−P= K - P, reached only if ST=0S_T = 0

Quick check

A long put has positive payoff at expiry when:

An investor holding BHP shares buys put options on BHP. The motive is most likely:

Connected topics

MoneynessLong PutShort Put

In learning paths

Options Path

Sources

  1. Hull (2022), §10.1 to 10.4
    Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
    Standard textbook treatment of put payoff diagrams, protective puts, and the buyer-writer relationship.
  2. Merton (1973)
    Merton, Robert C. "Theory of Rational Option Pricing." Bell Journal of Economics and Management Science, vol. 4, no. 1, 1973, pp. 141 to 183.
    Foundational treatment of European and American put valuation and bounds, including the upper bound $P \leq K$.
  3. ASX Options Knowledge Hub
    Australian Securities Exchange. Types of Options. ASX Investor Education, accessed 2026.
    Local reference for put option specifications on ASX-listed equities and the use of XJO puts for index-level hedging.
  4. Cboe Options Institute Glossary
    Cboe Global Markets. Options Trading Glossary. Cboe Options Institute, accessed 2026.
    Industry definition of put option, exercise, and assignment in listed equity markets.
How to cite this page
Dr. Phil's Quant Lab. (2026). Put Option. Derivatives Atlas. https://phucnguyenvan.com/concept/put-option
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