Black–Scholes

Black & Scholes 1973, Journal of Political Economy§II — Valuation of options

Closed-form European-option pricing under geometric Brownian motion of the underlying with constant volatility, constant risk-free rate, and log-normal terminal distribution. The crate consumes the closed-form pricer plus a bisection IV solver; the analytic layer composes against the kernel rather than re-implementing it.

Reference →

Authoritative references

• Black, F. & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81(3), 637–654.
  • DOI: https://doi.org/10.1086/260062.
• Hull, J. C. Options, Futures, and Other Derivatives (11th ed.), §15 (Black–Scholes–Merton) and §19 (Greek letters). Pearson, 2021.

Consumed by

• Greeks — closed-form Black–Scholes Greeks and IV bisection.
• Gex — per-strike Black–Scholes gamma.
• IvChange — IV deltas computed against the same bisection result.

Closed-form formulas

For a European call on a non-dividend underlying $S$, strike $K$, time-to-expiration $T$, risk-free rate $r$, and volatility $\sigma$:

$$C=S\cdot N(d_1)-K{e}^{-rT}\cdot N(d_2)$$$$d_1=\frac{\ln (S/K)+(r+{\sigma }^2/2)T}{\sigma \sqrt{T}},d_2=d_1-\sigma \sqrt{T}$$

A continuous-dividend yield $q$ enters by replacing $S$ with $S\cdot e^{-qT}$. Puts follow by put-call parity.

The Greek letters used by the crate:

Greek	Definition	Notation
Delta	$\partial V/\partial S$	$\mathrm{\Delta }$
Gamma	${\partial }^{2}V/\partial S^2$	$\mathrm{\Gamma }$
Vega	$\partial V/\partial \sigma$ (per 1.0 vol, not per 1 %)	$\nu$
Theta	$\partial V/\partial t$ (per year)	$\theta$
Rho	$\partial V/\partial r$	$\rho$

See Hull §19 for the closed-form derivations.

Implementation notes

• Implied volatility is solved by bisection against the closed-form pricer. Maximum 30 iterations; the residual at the final bracket is surfaced on every GreekTick via the iv_error field. Consumers reading non-zero iv_error should treat the emission as a tightened-but-not-converged solve and gate downstream consumption per their own tolerance.
• All Greeks are re-evaluated at the solved IV so that the emitted tick is internally consistent.
• The pricing kernel sits inside the analytic; it does not re-implement the closed form separately.