The replicating quotation preserves strong no-arbitrage

Proved

FinancialDerivatives.FD02.fair_quote_noArbitrage

by WillR · Oct 27, 2026 · Mathlib 0df444a (Lean v4.33.1)

arbitrageformal-verificationoption-pricing

For any signed quantity of a claim at its hedge cost, replace that position by an equivalent stock-and-bank trade. The resulting zero-cost base-market strategy cannot yield positive free payoffs.

A proof must use the exact published definitions and be accepted by the authoritative Prove2Me compiler.

Preamble

import Mathlib.Data.Real.Basic
import Definitions.Def_fd02_TwoStateMarket
import Definitions.Def_fd02_Trade
import Definitions.Def_fd02_terminalPayoff
import Definitions.Def_fd02_Replicates
import Definitions.Def_fd02_replicatingTrade
import Definitions.Def_fd02_initialCost
import Definitions.Def_fd02_riskNeutralUp
import Definitions.Def_fd02_riskNeutralValue
import Definitions.Def_fd02_NoArbitrage
import Definitions.Def_fd02_ClaimNoArbitrage

Formal statement

namespace FinancialDerivatives.FD02

/-- The replicating quotation preserves strong no-arbitrage. -/
theorem fair_quote_noArbitrage (m : TwoStateMarket) (hNA : NoArbitrage m) (H : Bool → ℝ) :
  ClaimNoArbitrage m H (riskNeutralValue m H) := by sorry

end FinancialDerivatives.FD02

Source

Original derived theorem based on Wilmott, Howison and Dewynne (1995), §§3.2,10.3, pp. 33–34 and 187; https://doi.org/10.1017/CBO9780511812545.

Read-back

What the Lean code literally says, in plain math · GPT-6

For any market with real spot $S>0$ , bank growth $R>0$ , and stock factors $0\le D<U$ , assume the base-market no-arbitrage property that every signed real pair $(x,c)$ with $xS+c=0$ , $xSU+cR\ge0$ , and $xSD+cR\ge0$ has $xSU+cR=0$ and $xSD+cR=0$ . For any real claim $H$ with payoffs $H_+,H_-$ and $q=(R-D)/(U-D)$ and $V=(qH_++(1-q)H_-)/R$ , the claim quoted at price $p=V$ satisfies the stronger combined-market property: for every real signed share holding $x$ , cash holding $c$ , and claim quantity $a$ , if $xS+c+aV=0$ , $xSU+cR+aH_+\ge0$ , and $xSD+cR+aH_-\ge0$ , then $xSU+cR+aH_+=0$ and $xSD+cR+aH_-=0$ .

Human review

Endorsed by Shuze Chen · Oct 27, 2026
Confirmed by the moderator at approval.
Endorsed by WillR · Oct 27, 2026
Confirmed by the mission captain (proposal self-audit).

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