Risk-neutral stock growth equals bank growth

Proved

FinancialDerivatives.FD02.riskNeutral_stock_martingale

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

arbitrageformal-verificationoption-pricing

The candidate risk-neutral probability is calibrated so the expected stock gross return equals the bank accumulation factor.

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_riskNeutralUp

Formal statement

namespace FinancialDerivatives.FD02

/-- Risk-neutral stock growth equals bank growth. -/
theorem riskNeutral_stock_martingale (m : TwoStateMarket) :
  riskNeutralUp m * m.up + (1 - riskNeutralUp m) * m.down = m.growth := by sorry

end FinancialDerivatives.FD02

Source

Derived from Wilmott, Howison and Dewynne (1995), §10.2, equation (10.4), p. 183. 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 $0\le D<U$ , defining $q=(R-D)/(U-D)$ , the algebraic identity $qU+(1-q)D=R$ holds. No assumption that $R$ is strictly between $D$ and $U$ , or that $q$ is nonnegative or at most one, is made.

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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