black scholes calculator - Aaron Graves, PhDude Replica

Current Stock Price (S)

Strike Price (K)

Time to Expiration in Years (T)

Risk-Free Interest Rate % (r)

Volatility % (σ)

Dividend Yield % (q)

Enter rates and volatility as percentages (example: 5 means 5%).

Enter your inputs and click calculate.

What This Black-Scholes Calculator Does

This tool estimates the theoretical value of European call and put options using the Black-Scholes model. Given the stock price, strike, time to expiration, risk-free rate, volatility, and dividend yield, the calculator returns both call and put prices instantly.

The model is a cornerstone of modern options pricing and gives a consistent framework for comparing market option prices against a mathematically derived fair value.

Inputs Explained

Current Stock Price (S): The underlying asset price today.
Strike Price (K): The price at which the option can be exercised.
Time to Expiration (T): Time left in years (e.g., 0.5 for six months).
Risk-Free Rate (r): Annualized continuously compounded risk-free rate.
Volatility (σ): Annualized standard deviation of returns.
Dividend Yield (q): Annualized continuous dividend yield of the underlying.

Black-Scholes Formula

For a dividend-paying stock, the formulas are:

Call = S·e^(-qT)·N(d1) - K·e^(-rT)·N(d2) Put = K·e^(-rT)·N(-d2) - S·e^(-qT)·N(-d1) d1 = [ln(S/K) + (r - q + 0.5σ²)T] / (σ√T) d2 = d1 - σ√T

Here, N(x) is the cumulative distribution function of the standard normal distribution. In practical terms, it converts distance-from-mean values into probabilities under a normal curve.

How to Use This Calculator

Step-by-step

Enter the underlying stock price and strike.
Set time to expiration in years.
Enter an annual risk-free rate and implied/expected volatility.
Optionally include dividend yield if the stock pays dividends.
Click Calculate Option Prices.

The result panel shows call price, put price, plus intermediate values (d1 and d2) to help you audit and understand the output.

Important Assumptions and Limitations

Applies best to European options (exercise only at expiration).
Assumes volatility and interest rates are constant over the option life.
Assumes lognormal price behavior and frictionless markets.
May be less accurate for deep in/out-of-the-money contracts or event-driven markets.

Practical Interpretation

If the market price of an option is significantly above the Black-Scholes value, the contract may be expensive relative to your assumptions (especially volatility). If significantly below, it may appear cheap. In real trading, model output should be one input among many, not a standalone signal.

Quick Example

With S=100, K=100, T=1, r=5%, σ=20%, q=0%, Black-Scholes returns a call price around 10.45 and a put price around 5.57 (values vary slightly by numerical approximation). Try these defaults in the calculator to verify.

Educational use only. This page does not provide financial advice. Option trading involves substantial risk, including possible loss of principal.