What does the Options Greeks Explorer methodology page document?

Generalized Black-Scholes formulas, Greeks definitions, assumptions, and limitations. Source citations, assumption deltas, and as-of dates included. It states the formulas, assumptions, data sources, limitations, and reproducibility steps behind the Options Greeks Explorer, in the Finance category.

When was the Options Greeks Explorer methodology last reviewed?

This methodology was last reviewed on 2026-04-20. The matching tool is at https://aifinhub.io/options-greeks-explorer/.

Are the Options Greeks Explorer numbers reproducible?

Yes. This page embeds a worked example whose output is the verbatim result of running the shipped options-greeks-explorer engine on a fixed input; the embedded JSON is recomputed and diffed against the engine in CI, so the numbers cannot drift from the code.

Methodology: Options Greeks Explorer

Model

Generalized Black-Scholes with continuous dividend yield q:

d1 = [ln(S/K) + (r − q + σ²/2)·T] / (σ·√T)
d2 = d1 − σ·√T

call = S·e^(−qT)·Φ(d1) − K·e^(−rT)·Φ(d2)
put  = K·e^(−rT)·Φ(−d2) − S·e^(−qT)·Φ(−d1)

Greeks

Delta: ∂Price/∂S. Call: e^(−qT)·Φ(d1). Put: −e^(−qT)·Φ(−d1).
Gamma: ∂²Price/∂S² = e^(−qT)·φ(d1) / (S·σ·√T). Same for call + put.
Theta (per day): the per-day change in price. Scaled as ann_theta / 365.
Vega (per 1% vol point): S·e^(−qT)·φ(d1)·√T / 100.
Rho (per 1% rate): call K·T·e^(−rT)·Φ(d2) / 100. Put negative.

Assumptions + limitations

European exercise: no early exercise. For American options (most single-name equity options), Black-Scholes is an approximation. For index options (SPX, RUT) it's exact in structure.
Continuous dividend yield: discrete dividends are approximated. For single names with large discrete dividends, use a dividend-adjusted model.
Constant volatility + rate: in reality, both have term structure. For multi-leg strategies where term structure matters, use a multi-factor or local-vol model.
Log-normal prices: Black-Scholes assumes log-normal underlying. Observed equity returns have fatter tails. For deep OTM options, Black-Scholes under-prices tail.
No transaction costs, no bid-ask: model price is the theoretical fair value, not a quote. Live bid-ask can be 2–20% of theoretical on low-liquidity strikes.
Friction-free hedging: the Greeks are instantaneous sensitivities. Actual delta-hedging faces gamma scalping + transaction costs.

Scope

This tool is for intuition-building and structure-checking — seeing how delta rotates with spot, how theta accelerates near expiry, how vega concentrates near-the-money. For production pricing + risk management, use a dedicated library (e.g. QuantLib) with proper American exercise + discrete dividends + implied-vol surfaces.

References

Black, F., & Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81(3).
Merton, R. C. (1973). "Theory of Rational Option Pricing." Bell Journal of Economics 4(1).
Hull, J. C. (2017). Options, Futures, and Other Derivatives, 10th ed.

External resources

Changelog

2026-04-20 — Initial release with generalized Black-Scholes (continuous dividend yield).

Worked example

Frequently asked questions

How Options Greeks Explorer works