Black-Scholes Options Price Calculator
Calculate the theoretical price of a European call or put option using the Black-Scholes model. Enter stock price, strike, volatility, risk-free rate, and time to expiration.
Implied volatility as a percent (e.g., 25).
Annual risk-free rate as a percent (e.g., 5).
Days to expiration (e.g., 30 DTE).
Optional — dividend yield as a percent (e.g., 2).
Enter option parameters above and click Calculate price to see the Black-Scholes theoretical price.
Black-Scholes Formulas
d1 = [ln(S/K) + (r - q + 0.5σ²)T] / (σ√T)
d2 = d1 - σ√T
Call = Se^(-qT)N(d1) - Ke^(-rT)N(d2)
Put = Ke^(-rT)N(-d2) - Se^(-qT)N(-d1)
Worked Examples
Example 1: Pricing an at-the-money call option
A stock trades at $100.00. You want to price a $100 call with 30 days to expiration. Volatility is 25%, the risk-free rate is 5%, and there is no dividend.
- T = 30/365 = 0.0822 years
- σ√T = 0.25 × √0.0822 = 0.0717
- d1 = [ln(100/100) + (0.05 + 0.5 × 0.0625) × 0.0822] / 0.0717
- d1 = [0 + 0.006678] / 0.0717 = 0.0932
- d2 = 0.0932 − 0.0717 = 0.0215
- N(d1) = 0.5371, N(d2) = 0.5086
- Call = 100 × 0.5371 − 100 × e^(−0.05 × 0.0822) × 0.5086
- Call = 53.71 − 99.59 × 0.5086 = 53.71 − 50.67 = $3.04
The theoretical price of this ATM call is about $3.04. Since it is at the money, the entire premium is extrinsic (time value).
Example 2: Pricing a deep out-of-the-money put
A stock trades at $150.00. You want to price a $130 put with 60 days to expiration. Volatility is 30%, the risk-free rate is 4.5%, and dividend yield is 1.5%.
- T = 60/365 = 0.1644 years
- σ√T = 0.30 × √0.1644 = 0.1216
- d1 = [ln(150/130) + (0.045 − 0.015 + 0.045) × 0.1644] / 0.1216
- d1 = [0.14310 + 0.01233] / 0.1216 = 1.2793
- d2 = 1.2793 − 0.1216 = 1.1577
- N(−d1) = 0.1004, N(−d2) = 0.1235
- Put = 130 × e^(−0.045 × 0.1644) × 0.1235 − 150 × e^(−0.015 × 0.1644) × 0.1004
- Put = 129.04 × 0.1235 − 149.63 × 0.1004 = 15.94 − 15.02 = $0.92
This deep OTM put is priced at roughly $0.92. The low price reflects the low probability of the stock dropping from $150 to below $130 in 60 days.
How to Use This Calculator
- Select option type — choose Call or Put depending on the option you want to price.
- Enter the underlying price (S) — the current market price of the stock or ETF.
- Enter the strike price (K) — the strike of the option you are pricing.
- Enter volatility (σ) — the implied volatility as a percentage. Find this on your broker's option chain.
- Enter the risk-free rate (r) — typically the current Treasury bill rate. A common default is 4–5%.
- Enter time to expiration (T) — select Days or Years, then enter the value. For 30 DTE, enter 30 with Days selected.
- Optionally enter dividend yield (q) — if the underlying pays dividends, enter the annualized yield. Leave blank for non-dividend stocks.
- Click Calculate price — the calculator returns the theoretical option price, d1, d2, and a summary of your inputs.
Frequently Asked Questions
- What is the Black-Scholes model?
- The Black-Scholes model is a mathematical formula for pricing European-style options. Developed by Fischer Black, Myron Scholes, and Robert Merton in 1973, it calculates the theoretical fair value of an option based on five inputs: stock price, strike price, time to expiration, risk-free rate, and volatility. It remains the foundation of modern options pricing.
- What are d1 and d2?
- d1 and d2 are intermediate values in the Black-Scholes formula. N(d1) represents the option's delta (sensitivity to the underlying price), while N(d2) approximates the probability that the option expires in the money. Together, they determine how much of the stock price and strike price contribute to the option value.
- Does Black-Scholes work for American options?
- Black-Scholes is designed for European options, which can only be exercised at expiration. American options can be exercised early, so their prices may differ — particularly for deep ITM puts or calls on dividend-paying stocks. For most purposes, the Black-Scholes price is a close approximation, but it may undervalue American options in certain cases.
- Why does my calculated price differ from the market price?
- Market prices reflect supply and demand, early exercise premium (for American options), and the volatility smile/skew. Black-Scholes assumes constant volatility across all strikes, which does not match real markets. The difference between the model price and market price is one way traders identify potentially mispriced options.
- How does dividend yield affect the option price?
- Dividends reduce the expected stock price at expiration because the stock drops by approximately the dividend amount on the ex-date. This decreases call values and increases put values. If you leave dividend yield blank (0%), the calculator assumes no dividends. For stocks with significant yields (2%+), including the dividend improves accuracy.