Strict return separation precludes zero-cost arbitrage

Proved

FinancialDerivatives.FD02.noArbitrage_of_growth_between

by WillR · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

arbitrageformal-verificationoption-pricing

If the bank growth lies strictly between the down and up factors, any nonzero zero-cost stock position creates opposite-signed state payoffs. Hence no arbitrage exists.

This is an original derived theorem target; a remote compiler must validate its declaration and proof.

Preamble

import Mathlib.Data.Real.Basic
import Definitions.Def_fd02_TwoStateMarket
import Definitions.Def_fd02_Trade
import Definitions.Def_fd02_initialCost
import Definitions.Def_fd02_terminalPayoff
import Definitions.Def_fd02_NoArbitrage

Formal statement

namespace FinancialDerivatives.FD02

/-- Strict return separation precludes zero-cost arbitrage. -/
theorem noArbitrage_of_growth_between (m : TwoStateMarket) (hd : m.down < m.growth) (hu : m.growth < m.up) :
  NoArbitrage m := by sorry

end FinancialDerivatives.FD02

Source

Derived from Wilmott, Howison and Dewynne (1995), §§2.4,3.2, pp. 30–34. 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 $0\le D<U$ with the additional strict inequalities $D<R$ and $R<U$ , the following holds: every signed real pair $(x,c)$ with $xS+c=0$ , $xSU+cR\ge0$ , and $xSD+cR\ge0$ has $xSU+cR=0$ and $xSD+cR=0$ . Thus every signed zero-cost trade with nonnegative payoffs in both states has both payoffs exactly zero.

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