Black-Scholes Put Option Price Calculator
Price a European put option with the Black-Scholes-Merton formula. Get the fair value, d1, d2, and time value in one click.
📉 What is the Black-Scholes Put Option Price Calculator?
The Black-Scholes put option price calculator computes the theoretical fair value of a European-style put option using the Black-Scholes-Merton model, the same 1973 no-arbitrage pricing framework used for calls. A put option gives its holder the right, but not the obligation, to sell an underlying asset at a fixed strike price on or before expiry, making it the mirror image of a call: puts gain value as the underlying falls, while calls gain value as it rises.
Options traders use this calculator to price protective puts (insurance against a stock decline), to evaluate cash-secured put writing strategies, and to sanity-check a quoted market premium before trading. Portfolio managers use it to mark hedge positions to a model price. Finance students verify textbook put-pricing problems with it, and it pairs naturally with put-call parity checks against the Black-Scholes Call Option Price Calculator.
A frequent point of confusion is that European and American puts can have meaningfully different values even on the exact same stock and strike. Because early exercise of a deep-in-the-money put can be optimal (locking in and reinvesting the strike proceeds today has time value of its own), an American put is worth at least as much as the European price this calculator produces, and often strictly more. For calls on non-dividend stocks this early-exercise premium is zero, but for puts it never fully disappears, which is why serious put pricing on American-style contracts uses a binomial tree instead of the closed-form formula.
This calculator is useful because it isolates the pure European put value, breaks it into intrinsic and time value, and shows the intermediate N(-d1) and N(-d2) terms, alongside a chart of how the price responds to changes in the underlying's spot price.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — At-the-Money Stock Option
A stock trades at $100, with a $100 strike, 1 year to expiry, a 5% risk-free rate, and 20% volatility.
Example 2 — Out-of-the-Money Put
A stock trades at $150, with a $140 strike, 6 months to expiry, a 4% risk-free rate, and 25% volatility.
Example 3 — In-the-Money Put with Dividends
A dividend-paying stock trades at $50, with a $55 strike, 3 months to expiry, a 3% risk-free rate, 30% volatility, and a 1% dividend yield.
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Black-Scholes formula for a put option?
P = K x e^(-rT) x N(-d2) - S x e^(-qT) x N(-d1), where d1 = [ln(S/K) + (r - q + sigma^2/2)T] / (sigma x sqrt(T)) and d2 = d1 - sigma x sqrt(T). S is spot price, K is strike, T is time to expiry in years, r is the risk-free rate, q is the dividend yield, and sigma is volatility.
What do N(-d1) and N(-d2) mean in the put formula?
N(-d1) and N(-d2) are the standard normal CDF evaluated at the negatives of d1 and d2. N(-d2) approximates the risk-neutral probability the option expires in the money (stock below strike). N(-d1) is used to derive put Delta, which equals -e^(-qT) x N(-d1).
Why does a higher strike price increase the put price?
A put lets you sell at the strike, so a higher strike means a more valuable right to sell, all else equal. Put price increases monotonically with strike price K, holding spot, time, rate, and volatility fixed.
How is the put price related to the call price?
Put-call parity states C - P = S x e^(-qT) - K x e^(-rT) for European options with the same strike and expiry. Use this to cross-check a put price against the matching call price from the Black-Scholes Call Option Price Calculator.
Does this calculator price American or European puts?
This calculator prices European-style puts, exercisable only at expiry, using the closed-form Black-Scholes-Merton formula. American puts can be exercised early and are typically worth slightly more; use the Binomial Options Tree Calculator (CRR) to value those.
Why can an American put be worth more than a European put?
If a put is deep in the money, exercising immediately locks in the strike proceeds today rather than waiting for expiry, and reinvesting those proceeds at the risk-free rate has value. European puts cannot capture this early-exercise value, so American puts on the same stock are worth at least as much.
What happens to the put price as the stock price falls?
The put price rises, approaching K x e^(-rT) for deep out-of-the-money-for-the-call (deep in-the-money-for-the-put) stock prices as the probability of exercise approaches 1. This calculator's chart plots exactly this relationship.
Does dividend yield increase or decrease a put's price?
A higher dividend yield increases a put's price. Dividends reduce the stock's expected future price, since it typically drops by roughly the dividend amount on the ex-dividend date, which raises the expected payoff of holding a put.
Can the Black-Scholes put price ever exceed the strike price?
No. A put's maximum possible value is the present value of the strike, K x e^(-rT), reached only in the limit as the stock price approaches zero. The Black-Scholes formula respects this upper bound by construction.
Why is my calculated put price different from the market price?
Market option prices reflect implied volatility, the market's forward-looking volatility estimate, which can differ from any historical volatility figure you enter. Use the Implied Volatility Calculator on this site to back out the volatility the market is pricing in.
What is the difference between intrinsic value and time value for a put?
Intrinsic value is max(K - S, 0), what the put would be worth if exercised right now. Time value is the remainder of the premium, and it decays to zero as expiry approaches, the same time-decay pattern seen in calls.