Constructed portfolio pays the down-state claim

Proved

FinancialDerivatives.FD02.replicatingTrade_down

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

arbitrageformal-verificationoption-pricing

The same fixed hedge reproduces the down-state payment exactly.

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

/-- Constructed portfolio pays the down-state claim. -/
theorem replicatingTrade_down (m : TwoStateMarket) (H : Bool → ℝ) :
  terminalPayoff m (replicatingTrade m H) false = H false := by sorry

end FinancialDerivatives.FD02

Source

Original derived theorem based on Wilmott, Howison and Dewynne (1995), §2.4, p. 30 and §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 any signed two-state claim with $H_+=H(\mathrm{true})$ and $H_-=H(\mathrm{false})$ , define $x=(H_+-H_-)/(S(U-D))$ and $c=(UH_--DH_+)/(R(U-D))$ . Then this trade's false-state terminal payoff is exactly the claim's false-state payoff: $xSD+cR=H_-$ . No no-arbitrage assumption or payoff sign restriction is needed.

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