FinanceDerivativesUniversity
AQAAPOntarioNSWCBSEGCE O-LevelMoECAPS

Black-Scholes (European Put Option Price) Calculator

Calculates the theoretical price of a European put option using the Black-Scholes model.

Use the free calculatorCheck the variablesOpen the advanced solver
Open Advanced Calculator
This is the free calculator preview. Advanced walkthroughs stay in the app.
Result
Ready
Put Option Price

Formula first

Overview

The Black-Scholes model is a fundamental tool in financial mathematics for pricing European-style options. For a put option, it estimates its fair value by considering five key inputs: the current stock price, the option's strike price, the time until expiration, the risk-free interest rate, and the volatility of the underlying asset. This model assumes a log-normal distribution for asset prices and continuous trading, providing a theoretical benchmark for option valuation.

Symbols

Variables

S_0 = Current Stock Price, X = Strike Price, T = Time to Expiration, r = Risk-Free Interest Rate, \sigma = Volatility

Current Stock Price
$
Strike Price
$
Time to Expiration
Risk-Free Interest Rate
Volatility
Cumulative Standard Normal Distribution Function
Put Option Price
$

Apply it well

When To Use

When to use: Use this formula when you need to determine the theoretical fair price of a European put option, given the current market conditions and the option's characteristics. It is applicable for options that can only be exercised at expiration and when the underlying asset does not pay dividends.

Why it matters: The Black-Scholes model revolutionized financial markets by providing a consistent framework for option pricing, enabling more efficient trading and risk management. It is crucial for traders, portfolio managers, and risk analysts to value options, hedge portfolios, and understand market implied volatility.

Avoid these traps

Common Mistakes

  • Using incorrect units for time (e.g., days instead of years).
  • Misinterpreting volatility (annualized standard deviation of returns).
  • Confusing put option formula with call option formula.
  • Incorrectly calculating N(d) values.

One free problem

Practice Problem

A European put option has a strike price of 105, the risk-free interest rate is 3% (annualized), and the stock's volatility is 25% (annualized). Calculate the theoretical price of this put option.

Current Stock Price105 $
Strike Price100 $
Time to Expiration0.5 years
Risk-Free Interest Rate0.03 decimal
Volatility0.25 decimal

Solve for:

Hint: Remember to calculate d1 and d2 first, then find N(-d1) and N(-d2) using a standard normal distribution table or calculator.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Hull, John C. Options, Futures, and Other Derivatives. 10th ed., Pearson, 2018.
  2. Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654.
  3. Wikipedia: Black-Scholes model
  4. Hull, J. C. (2018). Options, Futures, and Other Derivatives (10th ed.). Pearson.
  5. Hull, John C. Options, Futures, and Other Derivatives. Pearson.
  6. Black, Fischer, and Myron Scholes. 'The Pricing of Options and Corporate Liabilities.' Journal of Political Economy 81, no.