Initial capital is additive across trades

Proved

FinancialDerivatives.FD02.initialCost_add

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

arbitrageformal-verificationoption-pricing

For any two stock-and-cash holdings, the cost of holding their sum equals the sum of their separate costs.

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

Formal statement

namespace FinancialDerivatives.FD02

/-- Initial capital is additive across trades. -/
theorem initialCost_add (m : TwoStateMarket) (a b : Trade) :
  initialCost m ({ shares := a.shares + b.shares, cash := a.cash + b.cash } : Trade) = initialCost m a + initialCost m b := 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 $0\le D<U$ and any two signed trades $a=(x_1,c_1)$ and $b=(x_2,c_2)$ , the initial cost of the componentwise sum $(x_1+x_2,c_1+c_2)$ equals the sum of their initial costs: $(x_1+x_2)S+(c_1+c_2)=(x_1S+c_1)+(x_2S+c_2)$ .

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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