Put–call parity: $P = Ke^{-r(T-t)} - S + C$

Proved

BlackScholesModel.put_call_parity

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

black-scholesmathematical-financeoption-pricing

For all real $K,r,\sigma,T,S,t$ , the Black–Scholes put and call prices satisfy

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

Equivalently, the two expressions $Ke^{-r(T-t)}-S+C(S,t)$ and $N(-d_-)Ke^{-r(T-t)}-N(-d_+)S$ given for the put price agree. Put–call parity is the basic no-arbitrage relation between European calls and puts.

Formalization Note The identity holds for all real parameter values (it only uses $N(x)+N(-x)=1$ ), so no positivity hypotheses are imposed.

Preamble

import Mathlib
import Definitions.Def_BlackScholesModel_Core

open Real MeasureTheory ProbabilityTheory Filter Topology
open scoped ContDiff

Formal statement

namespace BlackScholesModel

theorem put_call_parity (K r σ T S t : ℝ) :
    putPrice K r σ T S t = K * Real.exp (-r * (T - t)) - S + callPrice K r σ T S 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; p. 4 (put price based on put–call parity)

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.

For all real numbers $K,r,\sigma,T,S,t$ , with no hypotheses at all,

P(S,t)=K\,e^{-r(T-t)}-S+C(S,t),

where $C(S,t)=N(d_+)S-N(d_-)Ke^{-r(T-t)}$ and $P(S,t)=N(-d_-)Ke^{-r(T-t)}-N(-d_+)S$ are the call and put functions of the definition file, $d_\pm=d_\pm(S,K,r,\sigma,T-t)$ computed with the formal system's conventions (division by zero is $0$ , $\sqrt{x}=0$ for $x\le0$ , $\ln$ extended by $\ln0=0$ , $\ln x=\ln|x|$ ), and $N$ is the normal distribution function defined by its integral.

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).
Endorsed by WillR · Oct 27, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

Put–call parity: P=Ke−r(T−t)−S+CP = Ke^{-r(T-t)} - S + CP=Ke−r(T−t)−S+C

Put–call parity: $P = Ke^{-r(T-t)} - S + C$