Read a Greek as a local derivative
This calculator starts from your spot price S, strike K, remaining years T, annual volatility σ, continuously compounded rate r and continuous dividend yield q. It calculates European Black-Scholes prices and five sensitivities. Inputs such as 25% volatility are converted to 0.25 internally; days use T = days / 365 and months use T = months / 12. It does not fetch option quotes or solve for implied volatility.
Set d₁ = [ln(S/K) + (r − q + σ²/2)T] / (σ√T) and d₂ = d₁ − σ√T. Call delta is e⁻ᑫᵀN(d₁); put delta is e⁻ᑫᵀ[N(d₁) − 1]. Gamma is e⁻ᑫᵀφ(d₁)/(Sσ√T). N is the standard-normal cumulative distribution and φ its density. The implementation approximates N numerically; displayed decimals are not a claim of market precision.
Keep the sign and the unit
| Greek | Change represented | Interpretation |
|---|---|---|
| Delta | Spot increases $1 | Signed option-price change; put delta is negative in this model. |
| Gamma | Spot increases $1 | Change in delta, not a direct cash P&L number. |
| Theta | One calendar day passes | Signed time sensitivity; it can be positive for some long European options. |
| Vega | Volatility rises one percentage point | For example 25% to 26%, not a 1% relative change from 25%. |
| Rho | Annual rate rises one percentage point | Positive for calls and negative for puts under the stated model. |
For a hypothetical vega of $0.15 per share, a five-percentage-point IV increase implies approximately $0.75 per share locally. With one long contract whose verified multiplier is 100, that is $75 before other changes and costs. Treating $0.75 as the per-contract result understates exposure by a factor of 100. Portfolio Greeks across different underlyings cannot simply be added as if every underlying moves by the same dollar.
Delta is not a success probability
Call delta uses N(d₁) and the dividend discount factor. The model's risk-neutral probability of finishing above strike uses N(d₂). Those differ, and neither estimates a real-world probability of profit after paying the premium. A near-the-money label is not used to obscure this: the calculator calls spot exactly equal to strike ATM and otherwise uses the sign of the spot/strike difference.
Compare scenarios without assuming an execution result
- Choose call or put and record the source and timestamp of every assumption. Use the actual contract's exercise style and dividend treatment to decide whether this model is suitable.
- Click a Greek, then choose spot, remaining time or volatility. The curve samples one variable while holding the rest fixed. It includes your input within its range; the marker shows the exact-input calculation.
- For a proposed large change, reprice using the changed inputs rather than treating a local derivative as an exact gain. Gamma, theta and vega themselves change.
- Apply the verified contract multiplier and signed position quantity. Compare with executable quotes, spread, fees, financing and exercise constraints separately.
Model limitations and sources
Zero time and zero volatility are excluded rather than replaced with artificial small values. American early exercise, discrete dividend events and changing volatility surfaces require other methods. Model price can be below the spot exercise payoff for a European option, so their difference is shown with its sign. Options and hedges can lose money; a positive sensitivity is not a trading recommendation.
Reference: OIC on Black-Scholes, Delta, Theta, Vega and Rho. The formulas, input limits and sampling conventions above describe this page's implementation.