Black-Scholes Call Price
Theoretical price of a European call on a non-dividend-paying stock under the Black-Scholes-Merton model. Assumes log-normal stock returns, constant volatility, constant risk-free rate, and continuous trading.
When to use: Use to value European-style calls, sanity-check market quotes, or compute Greeks. For American options the model is an approximation (good for non-dividend calls, less so for puts and dividend-paying calls).
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| CallPrice | Call Price | Theoretical price of the European call | $ |
| 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 Call, 30% Vol, 90 Days
Stock trades at $100. A 90-day $100-strike call. Risk-free rate is 4% (continuous), volatility is 30%.
Given
Step-by-Step
A 90-day ATM call at 30% vol with a 4% risk-free rate prices around $6.46 — roughly 6.5% of the stock price, consistent with the heuristic that ATM premium ≈ 0.4 × σ × S × √T.
Frequently Asked Questions
European-style options can only be exercised at expiration, never earlier. American options allow early exercise. Most US equity options are American, but the early-exercise premium for non-dividend-paying calls is generally negligible, so Black-Scholes is widely used as an approximation.
N is the standard normal cumulative distribution function — the probability that a standard normal random variable is less than d. d1 and d2 are risk-neutral standardized measures of how far in or out of the money the call is, scaled by volatility and time.
The basic form ignores dividends. For a continuous dividend yield q, replace S with S·e^(−qT) in the formula. For discrete dividends, subtract the present value of expected dividends from S before applying the formula.
Constant volatility (real markets exhibit volatility smiles/skews and stochastic volatility), continuous trading (real markets jump and have transaction costs), log-normal returns (real returns are fat-tailed), and the assumption that the risk-free rate is known and constant.