title: Financial Model
author: Keith A. Lewis
institute: KALX, LLC
classoption: fleqn
fleqn: true
abstract: Univeral Financial Model
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\newcommand\RR{\bm{R}}
This note provides a mathematical model of finance.
Motivation: There are problems with mathematical finance.
Let $T$ denote the set of all times with the total order $t\le t'$ indicating $t$
occured before or at $t'$.
The set $E$ consists of all legal entities, i.e., all individuals and corporations.
The partial order $e\preceq e'$ indicates the entity $e$ is owned by entity $e'$.
Let $I$ denote the set of all instruments and let $I(t,e)$ the set of all instruments
that are available to entity $e$ at time $t$.
Amounts $A$ can be represented by a real number. Let
$A(t;i,e;i',e')\subseteq \bm{R}$ denote the set of amounts of instrument $i$ that entity $e$ can
exchange for instrument $i'$ with entity $e'$ at time $t$. This set is determined by
the seller $e'$. The buyer is $e$.
A holding $(a,i,e)$ is an amount, instrument, and entity.
A trade $(t;a,i,e;a',i',e')$ is an exchange of holdings between a buyer and a seller.
At time $t$ the buyer $e$ pays holding $(a,i,e)$ to the seller and receives holding $(a',i',e')$.
The price of the trade is $X = a/a'$. The buyer exchanges the holding $(a'X,i,e)$ for
the holding $(i',a',e')$ with the seller.
Instruments entail cash flows $C\colon T\times I\times I\to\RR$. Holding $(a,i,e)$
at time $t$ results in an additional holding $(aC(t;i,i'), i', e)$. The cash flow
is determined by the issuer of instrument $i$.
The seller determines the price $X\colon T\times I\times E\times A\times I\times E\to\bm{R}$
At time $t\in T$ the buyer $e$ can trade $a'X(t;i,e;a',i',e')$ of instrument $i$ for
amount $a'$ of instrument $i'$ with seller $e'$.
A trading strategy is a finite sequence of increasing times $(τ_j)$ and corresponding
amounts $(Γ_j)$. The trading times depend only on the information available to the
buyer. The amounts must satisfy $Г(t;i,e;i',e')$ belongs to $A(t;i,e;i',e')$.
The value
A relation on the cartesian product $A\times B = {(a,b):a\in A, b\in B}$
is a subset $R\subseteq A\times B$. We use $aRb$ to indicate $(a,b)\in R$.
The product of the relations $R\subseteq A\times B$ and $S\subseteq B\times C$
is the relation $RS = {(a,c):aRb, bSc\text{ for some }b\in B}\subseteq A\times C$.
The converse of a relation $R\subseteq A\times B$ is
$R' = {(b,a):(a,b)\in R} \subseteq B\times A$.
Let $∆_A = ∆ = {(a,a):a\in A$ be the diagonal relation on $A\times A$. A relation on
$R\subseteq A\times A$ is reflexive if $∆\subseteq R$, symmetric if $R = R'$,
antisymmetric if $R\cap R'\subseteq ∆$, and transitive if $R^2 = RR\subseteq R$.
Let $A^B = {f:B\to A}$ be the set of all functons from the set $B$ to
the set $A$.