Black-Scholes Put Price

Options

Updated Apr 2026 • Has calculator

The theoretical fair value of a European put option derived from the Black-Scholes model.

What is BS Put?

The Black-Scholes put price gives the theoretical fair value of a European put option. A put grants the right to sell shares at the strike price before expiration. The formula is derived from put-call parity applied to the call price, and requires the same five inputs: stock price, strike, time to expiry, risk-free rate, and volatility. Put value rises when the stock price falls or volatility increases.

Formula

P = K·e^(−rT)·N(−d₂) − S·N(−d₁)

Worked Example

Representative Q1 2024 market conditions

Step 1 Stock price (S): $185.00

Step 2 Strike price (K): $185.00 (at-the-money)

Step 3 Risk-free rate (r): 5.25%, Time to expiry (T): 0.082 yrs

Step 4 Implied volatility (σ): 28%

Step 5 d₁ = 0.114, d₂ = 0.034 (same as call)

Step 6 P = 185×e^(−0.0043)×N(−0.034) − 185×N(−0.114)

Step 7 P ≈ $4.66 per share ($466 per contract)

Step 8 → Verify via put-call parity: P = C − S + K·e^(−rT) = 6.11 − 185 + 184.20 ≈ $5.31

Source: Black & Scholes (1973) — Journal of Political Economy (2024-01-15)

Calculate BS Put

Stock Price

Current market price of the underlying stock

Strike Price

Price at which the option can be exercised

Risk-Free Rate

Annual risk-free rate (e.g. 3-month T-Bill yield)

Time to Expiry

Time to expiration in years (e.g. 0.25 = 3 months)

Implied Volatility

Annualised implied volatility of the underlying stock

Put Premium

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Not investment advice.

How to Interpret BS Put

< 0

Pricing error — put price cannot be negative

0 – 1

Low premium — far OTM or short-dated option

1 – 10

Typical ATM premium — near-term option

> 10

Deep ITM or long-dated — significant downside hedge

📚 Advanced Options — Complete the path

Sources