Alternative (Black '76) formulation of the call and put prices

Proved

BlackScholesModel.price_eq_forward_formulation

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 $r\in\mathbb R$ , and put $\tau=T-t$ , $D=e^{-r\tau}$ (discount factor) and $F=e^{r\tau}S$ (forward price). With the forward-form parameters $d_\pm=\frac{1}{\sigma\sqrt\tau}\big[\ln\frac FK\pm\frac12\sigma^2\tau\big]$ , the Black–Scholes prices satisfy

C(S,t)=D\,[N(d_+)F-N(d_-)K],\qquad P(S,t)=D\,[N(-d_-)K-N(-d_+)F].

This is the special case of the Black '76 formula used in the Interpretation section, where $DN(d_+)F$ and $DN(d_-)K$ are the present values of an asset-or-nothing and a cash-or-nothing call.

Preamble

import Mathlib
import Definitions.Def_BlackScholesModel_Core

open Real MeasureTheory ProbabilityTheory Filter Topology
open scoped ContDiff

Formal statement

namespace BlackScholesModel

theorem price_eq_forward_formulation (K r σ T S t : ℝ) (hK : 0 < K) (hσ : 0 < σ) (hS : 0 < S) (ht : t < T) :
    callPrice K r σ T S t
        = callPriceFwd (Real.exp (-r * (T - t))) (Real.exp (r * (T - t)) * S) K σ (T - t) ∧
    putPrice K r σ T S t
        = putPriceFwd (Real.exp (-r * (T - t))) (Real.exp (r * (T - t)) * S) K σ (T - t) := 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; pp. 4–5 (Alternative formulation)

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 $P(S,t)=N(-d_-)Ke^{-r(T-t)}-N(-d_+)S$ ).

Let $D=e^{-r(T-t)}$ and $F=e^{r(T-t)}S$ . Let $\tilde C(D,F,K,\sigma,\tau)=D\big(N(\tilde d_+)F-N(\tilde d_-)K\big)$ and $\tilde P(D,F,K,\sigma,\tau)=D\big(N(-\tilde d_-)K-N(-\tilde d_+)F\big)$ with $\tilde d_+=\frac{\ln(F/K)+\sigma^2\tau/2}{\sigma\sqrt\tau}$ , $\tilde d_-=\tilde d_+-\sigma\sqrt\tau$ (definition file). The theorem asserts both

C(S,t)=\tilde C\big(D,F,K,\sigma,T-t\big)\quad\text{and}\quad P(S,t)=\tilde P\big(D,F,K,\sigma,T-t\big).

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).

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