Call Option
Put Option
Sensitivity to Volatility
| Volatility | Call Price | Put Price |
|---|
How Black-Scholes Works
The Black-Scholes model calculates the theoretical price of European call and put options by modeling the stock price as a geometric Brownian motion with constant drift and volatility. The model uses five key inputs to determine the fair value of an option.
Black-Scholes Formula
Call Option: C = S * N(d1) - K * e^(-rT) * N(d2)
Put Option: P = K * e^(-rT) * N(-d2) - S * N(-d1)
where:
d1 = [ln(S/K) + (r + sigma^2/2) * T] / (sigma * sqrt(T))
d2 = d1 - sigma * sqrt(T)
N(x) is the cumulative standard normal distribution function
Understanding the Inputs
Stock Price (S)
The current market price of the underlying asset. This is the most volatile input and directly affects option value. As stock price increases, call values rise and put values fall.
Strike Price (K)
The price at which the option can be exercised. The relationship between stock price and strike price determines the option's moneyness (in-the-money, at-the-money, or out-of-the-money).
Time to Expiration (T)
The time remaining until the option expires, measured in years. Longer time periods increase option values for both calls and puts due to greater uncertainty and more opportunity for favorable price movements.
Risk-Free Rate (r)
The theoretical return on a risk-free investment, typically based on government treasury yields. This represents the opportunity cost of capital and affects the present value calculation of the strike price.
Volatility (sigma)
The annualized standard deviation of the stock's returns, representing price uncertainty. Higher volatility increases both call and put option values because there's greater probability of large price movements in either direction.
Key Assumptions
The Black-Scholes model relies on several important assumptions that may not hold in real markets:
- European Exercise: The option can only be exercised at expiration, not before (unlike American options)
- No Dividends: The underlying stock does not pay dividends during the option's life
- Constant Volatility: Volatility remains constant over the option's life (in reality, volatility changes)
- Log-Normal Distribution: Stock prices follow a log-normal distribution with constant drift and volatility
- Efficient Markets: No transaction costs, no taxes, and assets are infinitely divisible
- Risk-Free Rate: The risk-free interest rate is constant and known
- No Arbitrage: There are no arbitrage opportunities in the market
Despite these limitations, Black-Scholes remains widely used as a benchmark for option pricing and understanding option behavior.
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Privacy & Limitations
- All calculations run entirely in your browser -- nothing is sent to any server.
- Results are estimates for planning purposes and should not replace professional financial advice.
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Black-Scholes Calculator FAQ
What is Black-Scholes Calculator?
Black-Scholes Calculator is a free finance & money tool that helps you Calculate option prices using the Black-Scholes model for calls and puts.
How do I use Black-Scholes Calculator?
Enter your input values, review the calculated output, and adjust inputs until you reach the result you need. The result updates in your browser.
Is Black-Scholes Calculator private?
Yes. Calculations run locally in your browser. Inputs are not uploaded to a server by default, and refreshing the page clears session data.
Does Black-Scholes Calculator require an account or installation?
No. You can use this tool directly in your browser without sign-up or software installation.
How accurate are results from Black-Scholes Calculator?
This tool applies standard formulas or deterministic processing logic for estimates. For medical, legal, tax, or investment decisions, verify with a qualified professional.
Can I save or share outputs from Black-Scholes Calculator?
You can bookmark this page and copy outputs manually. Results are not persisted in your account and are typically not embedded in the URL.