Options Pricing & Greeks Calculator
Model assumptions
Theoretical value and Greeks
Solve implied volatility from a market premium with these assumptions. The theoretical value shown here is not treated as a market quote, and this volatility input is not carried into the solver.
Test target-price probabilities with these distribution assumptions. Stock price, time, volatility, risk-free rate, and dividend yield carry forward; option type, strike, contract size, theoretical value, intrinsic value, and Greeks stay in Pricing & Greeks.
Build a strategy around this option contract. Option type, stock price, strike, time, volatility, risk-free rate, dividend yield, and contract size carry forward. Long/short side, entered premium, and contract quantity stay blank for you to choose; theoretical value and Greeks are not treated as strategy inputs.
Stock-price sensitivity
Only the stock price changes below. Time, volatility, rates, dividend yield, strike, and option type stay fixed, so the table shows how theoretical value and local price sensitivity change as moneyness changes.
| Stock price | Theoretical value / share | Delta | Gamma |
|---|---|---|---|
| $80.00 | $1.10 | +0.1400 | 0.0132 |
| $90.00 | $3.26 | +0.3012 | 0.0184 |
| $100.00 (current input) | $7.22 | +0.4917 | 0.0189 |
| $110.00 | $13.04 | +0.6669 | 0.0157 |
| $120.00 | $20.42 | +0.8008 | 0.0110 |
What the Greeks mean here
| Greek | Local model sensitivity |
|---|---|
| Delta | Approximate option-value change for a $1 increase in the stock price, holding the other model inputs fixed. |
| Gamma | Approximate change in Delta for a $1 increase in the stock price. |
| Theta | Local option-value change for one fewer calendar day to expiration, shown per share. |
| Vega | Approximate option-value change for a one percentage-point increase in annual volatility. |
| Rho | Approximate option-value change for a one percentage-point increase in the annual risk-free rate. |
Greeks are local derivatives of the pricing model, not promises about realized option-price changes. A large move in one or more inputs can make a simple Greek-based approximation materially different from a full repricing.
How the model works
d2 = d1 − σ√T
Call value = S e−qTN(d1) − K e−rTN(d2)
Put value = K e−rTN(−d2) − S e−qTN(−d1)
S is stock price, K is strike, T is calendar days divided by 365, σ is annual volatility, r is the continuously compounded annual risk-free rate, q is the continuous annual dividend yield, and N is the standard normal cumulative distribution function. The same inputs produce the displayed analytical Greeks.
Where this calculator stops
This is a Black-Scholes-Merton model for European-style exercise. Listed U.S. equity options are generally American-style, so they can be exercised before expiration. Early exercise, discrete dividends, borrow constraints, and other contract features can make an American option worth something different from this European model value.
The calculator assumes constant volatility, rates, and continuous dividend yield over the modeled life. It does not fetch option chains, solve implied volatility from a market premium, model volatility skew or term structure, estimate probability of profit or expected return, calculate margin or buying power, account for bid-ask spreads, commissions or taxes, or recommend a strike, expiration, strategy, or trade.
Use the Options Profit/Loss Calculator for single-leg expiration P&L from an entered premium, or the Options Strategy Builder for multi-leg expiration payoff analysis.
Pricing-model references
- Options Industry Council: Black-Scholes Formula, including the model inputs and the American-style equity-option limitation.
- Options Industry Council: Volatility & the Greeks, including Delta, Gamma, Theta, Vega, and Rho interpretation.
- Cboe Options Institute: Options Calculator, a current benchmark that reports theoretical option price and the same five primary Greeks.
- OCC: Equity Options Product Specifications, including standard equity-option contract-unit context and adjusted-contract exceptions.