Black-Scholes Option Pricing Calculator
Calculate theoretical European call and put option prices.
Understanding the Black-Scholes Model
The Black-Scholes model, also known as the Black-Scholes-Merton (BSM) model, is a mathematical framework used to estimate the theoretical value of European-style options. Developed in 1973 by Fischer Black, Myron Scholes, and Robert Merton, it revolutionized modern finance by providing a systematic way to price derivative contracts.
Key Inputs Explained
- Stock Price (S): The current market price of the underlying asset.
- Strike Price (K): The pre-determined price at which the option holder can buy (Call) or sell (Put) the underlying asset.
- Time to Expiry (T): The time remaining until the option contract expires, usually expressed as a fraction of a year.
- Volatility (σ): A measure of how much the stock price is expected to fluctuate. This is often the most critical and difficult input to estimate.
- Risk-Free Rate (r): The theoretical rate of return on an investment with zero risk, typically based on government bond yields (like the US Treasury).
The Mathematical Logic
The model assumes that stock prices follow a geometric Brownian motion with constant drift and volatility. It uses a partial differential equation to derive the pricing formulas for calls and puts. The core of the calculation involves finding 'd1' and 'd2', which represent probabilities related to the option finishing "in the money."
Example Calculation
Suppose a stock is trading at 100, and you are looking at a call option with a strike price of 100 that expires in 30 days. If the volatility is 20% and the risk-free rate is 5%, the Black-Scholes model would output a theoretical call price of approximately 2.45 and a put price of approximately 2.04.
Limitations of the Model
While powerful, the Black-Scholes model has limitations: it assumes constant volatility, no transaction costs, and it is strictly designed for European options (which cannot be exercised before the expiration date). American options, which allow early exercise, often require more complex models like the Binomial Options Pricing Model.