Black-Scholes Put Price
Theoretical price of a European put on a non-dividend-paying stock under the Black-Scholes-Merton model. Equivalent to the Black-Scholes call price connected by put-call parity.
When to use: Use to value European puts directly, or to back out the put price implied by an observed call via parity. For American puts the model is an approximation; the early-exercise premium is generally not negligible, especially for deep ITM puts.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| PutPrice | Put Price | Theoretical price of the European put | $ |
| S | Stock Price | Current price of the underlying | $ |
| Strike | Strike Price | Exercise price of the option contract | $ |
| Rf | Risk-Free Rate | Continuously compounded annual risk-free rate as a decimal | % |
| Sigma | Volatility | Annualized volatility of the underlying as a decimal (e.g. 0.25 for 25%) | % |
| T | Time to Expiration | Time to expiration in years (e.g. 0.25 for 3 months) | years |
Real-Life Examples
Example 1: ATM Put, 30% Vol, 90 Days
Stock trades at $100. A 90-day $100-strike put. Risk-free rate is 4%, volatility is 30%.
Given
Step-by-Step
The 90-day ATM put prices around $5.46 — slightly less than the equivalent call ($6.46) because of the positive risk-free rate. Parity confirms: C − P = 6.46 − 5.46 = 1.00 ≈ S − K·e^(−rT) = 100 − 99.005 = 0.995.
Frequently Asked Questions
When the risk-free rate is positive, a positive cost of carry tilts the model: holding stock has a positive expected drift under the risk-neutral measure. Calls benefit from upside drift; puts pay for it. Set r = 0 and ATM call and put prices match.
Less accurate than for non-dividend-paying calls. Deep ITM American puts are often worth more than the European Black-Scholes price because early exercise has positive value (you get the strike now and can earn interest on it). Pricing American puts properly requires a binomial tree or finite-difference method.
For a continuous dividend yield q, replace S with S·e^(−qT). Dividends raise put values (lower expected stock at expiration) and lower call values, so the impact is meaningful for dividend-paying names.