How to use this calculator
- Enter the share price and the strike you want to price.
- Set days to expiration and the implied volatility of that expiration.
- Set the risk-free rate (a short-term Treasury yield is the usual proxy).
- Read the result: the theoretical call and put value, then delta, gamma, theta, vega, and rho - all per share, so multiply by 100 for one contract.
What it tells you: the theoretical fair value of a call and put and how each Greek will move that price as the inputs change.
How this calculator works
Black-Scholes-Merton prices a European option by assuming the stock drifts and diffuses as a lognormal random walk with constant volatility. The whole model turns on two intermediate terms: d1 = [ln(S/K) + (r + σ²/2)·T] ÷ (σ·√T) and d2 = d1 − σ·√T, where S is the share price, K the strike, r the risk-free rate, σ the implied volatility and T the time to expiration in years (we use calendar days ÷ 365).
The call value is S·N(d1) − K·e−rT·N(d2) and the put value is K·e−rT·N(−d2) − S·N(−d1), where N(·) is the standard-normal cumulative distribution. The two are tied together by put-call parity, so this calculator always returns prices that satisfy it exactly. Every Greek is the partial derivative of those formulas with respect to one input.
The Greeks, in plain English
- Delta — how much the option moves per $1 move in the stock. A 0.50 delta call gains about $0.50 (×100 = $50 a contract) on a $1 rise. It also approximates the chance of finishing in the money.
- Gamma — how fast delta itself changes. High near the money and near expiration; it is why short options get dangerous late in their life.
- Theta — the value bled per calendar day from time decay. It is the seller's friend and the buyer's rent, and it accelerates as expiration approaches.
- Vega — sensitivity to implied volatility, per 1 percentage point. A vega of 0.11 means the option gains about $0.11 if IV rises one point. Long options are long vega; sellers are short it.
- Rho — sensitivity to interest rates, per 1 percentage point. The smallest Greek for short-dated retail trades, but it grows with time to expiration.
Worked example
A fixed, hypothetical illustration — not live market data.
Take an at-the-money option: a stock at $100, a $100 strike, 30 days out, a 4% risk-free rate and 25% implied volatility — the calculator's defaults.
- Call value ≈ $3.02, put value ≈ $2.69 (≈ $302 and $269 per contract).
- Delta: call ≈ 0.53, put ≈ −0.47 — roughly even money to finish in the money, as expected at the money.
- Gamma ≈ 0.055 and vega ≈ 0.114, shared by both legs.
- Theta: call ≈ −$0.053/day, put ≈ −$0.042/day — the daily rent of holding.
- Rho: call ≈ 0.041, put ≈ −0.041 per 1% rate — small, as it should be at 30 days.
Assumptions and limits
- European exercise. The model assumes exercise only at expiration. American puts and options on dividend payers can carry an early-exercise premium it misses.
- No dividends. This version sets the dividend yield to zero. A dividend lowers calls and raises puts versus the figures here.
- Constant volatility. One IV across all prices. Real markets price a volatility smile, so deep in- and out-of-the-money strikes differ from the single-σ value.
- Frictionless and before costs. No commissions, no bid-ask spread, continuous trading. Treat the output as a theoretical mid-price.
Common mistakes
- Reading the price as a quote. It is the model's fair value, not what the market will fill — check the live bid-ask.
- Forgetting the ×100 multiplier. Every figure is per share; one contract controls 100 shares.
- Using a stale IV. The output is only as good as the volatility you feed it; use the IV of the expiration you are trading.
- Trusting it through earnings or on dividend payers. Events and dividends break the model's assumptions.
Frequently asked questions
What is the Black-Scholes model?
Black-Scholes-Merton is the classic closed-form formula for the fair value of a European option. It takes the share price, strike, time to expiration, risk-free rate and implied volatility, assumes the stock follows a lognormal random walk with constant volatility, and returns a theoretical price plus the option’s sensitivities — the Greeks. It earned Scholes and Merton the 1997 Nobel Prize and still anchors how options are quoted.
What do delta, gamma, theta, vega and rho mean?
Delta is how much the option price moves per $1 move in the stock (calls 0 to 1, puts −1 to 0). Gamma is how much delta itself changes per $1 move. Theta is the value lost per day from time decay. Vega is the change in value per one-point change in implied volatility. Rho is the change per one-point change in interest rates. Together they describe how the price reacts as each input shifts.
What units are the Greeks shown in?
Everything is per share — multiply by 100 for one contract. Delta and gamma are per $1 move in the underlying. Theta is per calendar day (we divide the annual figure by 365). Vega is per 1 percentage-point change in implied volatility, and rho is per 1 percentage-point change in the risk-free rate. These match the conventions on a typical broker option chain.
Does this handle American options or early assignment?
No — it prices European options, which can only be exercised at expiration. For non-dividend-paying stocks an American call is worth essentially the same as a European one, so the price is a close guide. American puts and any option on a dividend payer can carry an early-exercise premium this formula does not capture. We never model early assignment.
Why does my broker’s price differ slightly?
Several reasons: dividends (this calculator assumes none), American versus European exercise, the fact that real markets price each strike with its own implied volatility (the volatility smile) rather than one number, the bid-ask spread, and small differences in the assumed interest rate. Black-Scholes gives the theoretical mid-price under its assumptions, not the live quote.
Is implied volatility an input or an output here?
It is an input. This tool prices an option from a volatility you supply. The reverse problem — backing out the implied volatility that matches a market price — is solved numerically and is what your broker’s “IV” column shows. Use the IV Rank calculator to judge whether that implied volatility is historically high or low.
Related tools and guides
Turn delta into a probability with the Probability Calculator, frame the move with the Expected Move Calculator, model the P&L of actually buying that call or put in the Long Call & Long Put Calculator, and judge whether the IV you typed is rich with the IV Rank Calculator.
New to the Greeks? The Greeks for option sellers guide explains each one, or look up any term in the options glossary.
Educational tool only. Nothing here is financial advice. Black-Scholes gives a theoretical value under idealized assumptions; real option prices differ, and options trading carries the risk of significant loss. Verify every number before you trade on it.