af.md

title	author	classoption
Atomic Finance	Keith A. Lewis	fleqn

\newcommand\ZZ{\bm{Z}}

There is an inevitable trajectory in the financial world toward more data, faster calculations, and increasing complexity.

The atoms of finance are holdings: an amount, instrument, and the owner of the holding. Holding an instrument often entails auxiliary cash flows for the owner. A buyer can exchange the amount and instrument of their holding with that of a seller. The price of the exchange is the ratio of buyer amount to seller amount. A portfolio is a collection of holdings. Portfolios evolve over time via exchanges. A transaction is a collection of associated exchanges.

Holding

Let $I$ be the set of all market instruments and $O$ the set of all (potential) instrument owners. A holding is a triple $$ h = (a, i, o) $$ where $i\in I$ is an instrument, $o\in O$ is the instrument owner, and $a\in\ZZ$ is the amount as an integer multiple of the minimal trading increment of the instrument. Holdings can entail cash flows such as, stock dividends, bond coupons, futures margin adjustments, broker fees, and taxes.

Exchange

Let $T$ be the set of all trading times. Holdings interact via an exchange: a triple $$ \chi = (t, h, h') $$ where $t\in T$, $h = (a,i,o)$ is the buyer holding, and $h'=(a',i',o')$ is the seller holding. The price of an exchange is $X = X_{i'}^i = a/a'$. Sellers quote a price $X$ and buyers decide if they want to trade at time $t$ amount $a'X$ of instrument $i$ they hold for amount $a'$ of instrument $i'$ the seller holds. An instrument is fungible if $(a, i, o)$ can be split into $(b, i, o)$ and $(a - b, i, o)$ when $0 < b < a$. Cash is fungible. An instrument is a numeraire if it has no cash flows. Cash is a numeraire.

For example, if a buyer holds 8 dollars and a seller holds 2 shares of Ford stock and quotes a price of 4 dollars per share then the buyer can acquire two shares of F for $8$ dollars. If the buyer wants to short two shares of Ford the seller will most like quote a price higher than $4$ per share.

Portfolio

A portfolio is a collection of holdings. Exchanges act on portfolios. A portfolio ${{\ldots,(a,i,o),\ldots,(a',i',o'), \ldots}}$ at time $t$ becomes ${{\ldots,(a',i',o),\ldots,(a,i,o'), \ldots}}$ after the exchange $\chi = (t,h,h')$ at time $t$ settles. $$ {\ldots,(a,i,o),\ldots,(a',i',o'), \ldots}\overset{\chi}{\to}{\ldots,(a',i',o),\ldots,(a,i,o'), \ldots} $$

A transaction is a set of related exchanges, for example an exchange may involve a broker fee or tax payment.

Value

Given a portfolio the value of instrument $i$ held by owner $o$ is $$ N(o, i) = \sum_h {a\mid h = (a, i, o)} $$ for $h$ in the portfolio.

The mark-to-market of a position in a cash numeraire $i_0$ requires specifying putative prices $X^i_{i_0}$ for $i\in I$. make imaginary exchanges of $(a'X^i_{i_0}, i, o)$ for $(a', i', o')$

Remarks

Mathematics allows you to think big. We can let $I$ be all market instruments that exist now, in the past, or potentially in the future.

It is quite common for owners to have different best guesses of $X_t^i_{i_0}$. Accountants might use "book," "market," "liquidation," or "going concern" values. This model does not suggest which is the most appropriate in any given situation, it only makes that choice explicit. may involve a broker fee or tax obligation.

Implementation

See Domain Driven Design

Holdings are stored in a relational table with columns Id, Amount, Instrument, and Owner. The Id is a unique identifier that includes the time the holding was entered. (E.g., UUID Version-4 or Time-First.) The Amount is an signed integer value indicating the multiple of the minimum trading increment of the Instrument. The Owner can be more refined than, e.g., a Legal Entity Identifier. Legal entities have subgroups and even individual employees associated with a holdings.

Id

A unique identifier including the time a holding was inserted into the Holdings table.

Amount

Integral multiple of smalled instrument trading unit. 64-bit signed integer. 32-bit is too small.

Instrument

This can be more refined than a stock ticker or CUSIP. For example, a stock might trade on multiple exchanges. CUSIPs can be reused so this should be taken into account.

Entity

LEI is too coarse. Need to subdivide into groups or even employee. This requires a slowly time-varying auxiliary table.

Remarks

If it is complicated, it is wrong.

Can we make it a function? (Easier to reason about.)