How Options Greeks Explorer works

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

1. 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.
2. Continuous dividend yield: discrete dividends are approximated. For single names with large discrete dividends, use a dividend-adjusted model.
3. 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.
4. Log-normal prices: Black-Scholes assumes log-normal underlying. Observed equity returns have fatter tails. For deep OTM options, Black-Scholes under-prices tail.
5. 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.
6. 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.

Changelog

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