Put Delta: $\partial P/\partial S = -N(-d_+) = N(d_+) - 1$

Proved

BlackScholesModel.put_delta

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 Black–Scholes put price is differentiable in the spot price at $S$ , with

\frac{\partial P}{\partial S}(S,t)=-N(-d_+)=N(d_+)-1.

Preamble

import Mathlib
import Definitions.Def_BlackScholesModel_Core

open Real MeasureTheory ProbabilityTheory Filter Topology
open scoped ContDiff

Formal statement

namespace BlackScholesModel

theorem put_delta (K r σ T S t : ℝ) (hK : 0 < K) (hσ : 0 < σ) (hS : 0 < S) (ht : t < T) :
    HasDerivAt (fun S' => putPrice K r σ T S' t)
      (-stdNormalCDF (-dPlus S K r σ (T - t))) S ∧
    -stdNormalCDF (-dPlus S K r σ (T - t)) = stdNormalCDF (dPlus S K r σ (T - t)) - 1 := 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: Delta, Put)

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 two things: (i) the function $S'\mapsto P(S',t)$ is differentiable at $S$ with derivative $-N(-d_+)$ , where $d_+=d_+(S,K,r,\sigma,T-t)$ ; and (ii) $-N(-d_+)=N(d_+)-1$ for this value of $d_+$ .

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

Put Delta: ∂P/∂S=−N(−d+)=N(d+)−1\partial P/\partial S = -N(-d_+) = N(d_+) - 1∂P/∂S=−N(−d+​)=N(d+​)−1

Put Delta: $\partial P/\partial S = -N(-d_+) = N(d_+) - 1$