The two-state stock-and-bank market is complete

Proved

FinancialDerivatives.FD02.everyClaim_uniquely_replicable

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

arbitrageformal-verificationoption-pricing

Every real-valued claim on the two terminal states has exactly one portfolio of shares and cash reproducing it.

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

Formal statement

namespace FinancialDerivatives.FD02

/-- The two-state stock-and-bank market is complete. -/
theorem everyClaim_uniquely_replicable (m : TwoStateMarket) (H : Bool → ℝ) :
  ∃! t : Trade, Replicates m t H := by sorry

end FinancialDerivatives.FD02

Source

Original derived theorem based on Wilmott, Howison and Dewynne (1995), §10.3, p. 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 $0\le D<U$ and every real-valued payoff function $H$ on the two Boolean states, there exists exactly one signed pair of real holdings $(x,c)$ satisfying both replication equations $xSU+cR=H(\mathrm{true})$ and $xSD+cR=H(\mathrm{false})$ . Uniqueness is of the whole pair of holdings, and no sign or no-arbitrage hypothesis is added.

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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