Black-Scholes Put Option Calculator
The Black-Scholes model values a European put option, the right to sell an asset at a set strike price at expiration, from the same five inputs used for a call: asset price, strike, time to expiration, risk-free rate, and volatility. This calculator returns the put price along with d1, d2, and the put delta. A put gains value as the underlying falls, which is why it is the classic instrument for hedging downside. The model assumes no dividends and constant volatility, so use it as a baseline.
Black-Scholes put formula
d1 = (ln(S / K) + (r + vol^2 / 2) * T) / (vol * sqrt(T))
d2 = d1 - vol * sqrt(T)
Put = K * e^(-r * T) * N(-d2) - S * N(-d1)
where N() is the standard normal cumulative distribution
N() is evaluated with a high-accuracy rational approximation. Put delta equals N(d1) minus 1 and is negative. Rate and volatility are entered as decimals (5% is 0.05).
Worked example
With S = 100, K = 100, T = 1 year, r = 0.05, volatility = 0.20: d1 = 0.35, d2 = 0.15. The put value is 100 * e^(-0.05) * N(-0.15) - 100 * N(-0.35) = 5.57 (rounded). The put delta is N(0.35) - 1 = -0.36.
Black-Scholes put: frequently asked questions
What does the Black-Scholes put formula compute?
It gives the theoretical fair value of a European put option, the right to sell an asset at a fixed strike price on the expiration date. As with the call, the price depends on the asset price, strike, time to expiration, the risk-free rate, and volatility. A put gains value when the underlying falls, so it acts as downside protection.
How does the put value relate to the call value?
Through put-call parity: call minus put equals the asset price minus the discounted strike. So once you know the call value for the same inputs, the put follows directly. This calculator computes the put from the model formula, and the parity relation provides an independent cross-check.
What are the model's main limitations?
The standard Black-Scholes model assumes European exercise, no dividends, a constant risk-free rate, constant volatility, and lognormal prices. Real markets violate these to varying degrees, so the output is a baseline. Traders adjust for implied volatility and dividends rather than treating the figure as an exact market price.
Sources
- The Black-Scholes-Merton option pricing model is a standard published result in financial mathematics (Black and Scholes, 1973; Merton, 1973). Options market structure background: U.S. Securities and Exchange Commission Investor.gov: Options.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.