Vega: $\partial V/\partial\sigma = SN'(d_+)\sqrt{T-t}$ for calls and puts

Proved

BlackScholesModel.vega

by Lucas · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

black-scholesmathematical-financeoption-pricing

Let $K>0$ , $\sigma>0$ , $S>0$ , $t<T$ and $r\in\mathbb R$ . Then the call and put prices are differentiable in the volatility at $\sigma$ , and

\frac{\partial C}{\partial\sigma}=\frac{\partial P}{\partial\sigma}=SN'(d_+)\sqrt{T-t}.

Preamble

import Mathlib
import Definitions.Def_BlackScholesModel_Core

open Real MeasureTheory ProbabilityTheory Filter Topology
open scoped ContDiff

Formal statement

namespace BlackScholesModel

theorem vega (K r σ T S t : ℝ) (hK : 0 < K) (hσ : 0 < σ) (hS : 0 < S) (ht : t < T) :
    HasDerivAt (fun σ' => callPrice K r σ' T S t)
      (S * stdNormalPDF (dPlus S K r σ (T - t)) * Real.sqrt (T - t)) σ ∧
    HasDerivAt (fun σ' => putPrice K r σ' T S t)
      (S * stdNormalPDF (dPlus S K r σ (T - t)) * Real.sqrt (T - t)) σ := by sorry

end BlackScholesModel

Source

Wikipedia, "Black–Scholes model" (snapshot supplied as Black–Scholes_model.pdf, 22 pp.), https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model; p. 7 (The Options Greeks, table: Vega)

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic)

Non-blind read-back. This read-back was written by the same agent that drafted the Lean statement (at the explicit instruction of the proposal owner), not by an independent blind auditor. It is not independent testimony; reviewers should compare it against the Lean code themselves.

Here $K,r,\sigma,T,S,t$ are arbitrary real numbers subject to the hypotheses $K>0$ , $\sigma>0$ , $S>0$ and $t<T$ (so $\tau:=T-t>0$ ); $r$ is unrestricted (it may be zero or negative). $N$ , $N'$ , $d_\pm$ , $C$ , $P$ are the functions of the companion definition file (normal distribution function given by its integral, normal density in closed form, $d_\pm=d_\pm(S,K,r,\sigma,T-t)$ , call price $C(S,t)=N(d_+)S-N(d_-)Ke^{-r(T-t)}$ , put price $P(S,t)=N(-d_-)Ke^{-r(T-t)}-N(-d_+)S$ ).

The theorem asserts that the function $\sigma'\mapsto C(S,t)$ computed with volatility $\sigma'$ in place of $\sigma$ (all of $K,r,T,S,t$ fixed) is differentiable at $\sigma'=\sigma$ with derivative $S\,N'(d_+)\sqrt{T-t}$ , where $d_+=d_+(S,K,r,\sigma,T-t)$ and $N'(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ ; and the same for the put price $P$ with the same derivative.

Human review

Endorsed by Shuze Chen · Oct 10, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 10, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

Vega: ∂V/∂σ=SN′(d+)T−t\partial V/\partial\sigma = SN'(d_+)\sqrt{T-t}∂V/∂σ=SN′(d+​)T−t​ for calls and puts

Vega: $\partial V/\partial\sigma = SN'(d_+)\sqrt{T-t}$ for calls and puts