Two-state no-arbitrage exactly means strict return separation

Proved

FinancialDerivatives.FD02.noArbitrage_iff_growth_between

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

arbitrageformal-verificationoption-pricing

Strong no-arbitrage holds exactly when d<R<u, with d<u and positive stock/bank prices already included in the market structure.

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

/-- Two-state no-arbitrage exactly means strict return separation. -/
theorem noArbitrage_iff_growth_between (m : TwoStateMarket) :
  NoArbitrage m ↔ m.down < m.growth ∧ m.growth < m.up := by sorry

end FinancialDerivatives.FD02

Source

Derived from 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 $0\le D<U$ , the following two statements are equivalent: (i) every signed real pair $(x,c)$ with $xS+c=0$ , $xSU+cR\ge0$ , and $xSD+cR\ge0$ has $xSU+cR=0$ and $xSD+cR=0$ ; and (ii) both strict inequalities $D<R$ and $R<U$ hold. Neither equality at an endpoint is allowed by (ii).

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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