putoption

A put option is a financial derivative that gives the holder the right, but not the obligation, to sell a specified quantity of an underlying asset at a predetermined price, known as the strike price, on or before a specified expiration date. The buyer pays a premium to the seller for this right. If the market price of the underlying remains above the strike price, the put may expire worthless; if it falls below the strike, the option is in the money and can be exercised.

The payoff at expiration for a long put is max(strike price minus the final underlying price, 0). Depending on the contract, settlement can be physical (delivery of the underlying) or cash-settled. Put options can be traded on exchanges or over-the-counter and may be American-style (exercisable any time before expiry) or European-style (exercisable only at expiry). Some contracts are Bermudan or have other exercise features.

Intrinsic value and time value determine an option’s price. For a put, intrinsic value equals max(strike price minus current price, 0). The total premium reflects intrinsic value plus time value, which depends on factors such as time to expiry, volatility, interest rates, and dividends. The option’s price also has sensitivity measures known as Greeks; puts generally have negative delta, with other Greeks capturing time decay and sensitivity to volatility and rates.

Common uses include hedging and risk management through protective puts, or speculative bets on declines in the underlying. For the writer (short put), the seller has the obligation to buy the underlying at the strike price if exercised. Put-call parity links put prices to call prices and other market parameters in arbitrage-free pricing.