Underpricing an attainable claim creates a static arbitrage

Proved

FinancialDerivatives.FD02.underpriced_claim_arbitrage

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

arbitrageformal-verificationoption-pricing

If the quote is below the hedge cost, buy the claim, short the hedge and invest the surplus. The resulting zero-cost payoff is strictly positive in both states.

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

/-- Underpricing an attainable claim creates a static arbitrage. -/
theorem underpriced_claim_arbitrage (m : TwoStateMarket) (H : Bool → ℝ) (p : ℝ) (hp : p < riskNeutralValue m H) :
  ¬ ClaimNoArbitrage m H p := by sorry

end FinancialDerivatives.FD02

Source

Original derived theorem based on Wilmott, Howison and Dewynne (1995), §3.2, pp. 33–34; 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 $0\le D<U$ , any real claim $H$ with payoffs $H_+,H_-$ , $q=(R-D)/(U-D)$ and $V=(qH_++(1-q)H_-)/R$ , and any real quotation $p$ strictly below $V$ (so $p<V$ ), the following combined-market no-arbitrage property is false: for every signed real stock holding $x$ , cash holding $c$ , and claim quantity $a$ , the conditions $xS+c+ap=0$ , $xSU+cR+aH_+\ge0$ , and $xSD+cR+aH_-\ge0$ imply $xSU+cR+aH_+=0$ and $xSD+cR+aH_-=0$ . Equivalently, signed real $x,c,a$ exist with combined initial cost zero and nonnegative combined terminal payoff in both Boolean states, with at least one payoff strictly positive. No base-market no-arbitrage hypothesis is required.

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).

View graph