January 26, 2025
The financial world is a big, messy affair. No mathematical theory or software implementation can accurately model reality. The best a theory can do is to identify features relevant to practitioners that can be practically implemented on a computer.
The atoms of finance are holdings. A holding is an amount, an instrument, and an entity. We write \alpha = (a, i, e) to indicate entity e holds amount a\in\bm{Z} of instrument i. The amount is an integral multiple of the minimum trading quantity of the instrument. Holdings are element of the cartesian product A\times I\times E where A = \bm{Z} is the set of integers, I is the set of market instruments, and E is the set of entities. The set I can be the set of all instruments ever traded and E can be all past, current, and future possible entities. Math let’s you think big.
Holdings can entail cash flows. Stocks have dividends, bonds have coupons, futures have daily margin adjustments. (Taxes?)
A position is a (multi)set of holdings \pi\subseteq A\times I\times E. The total amount of instrument i held by entity e is A(\pi, i, e) = \sum_{\alpha\in\pi}\{\alpha.a\mid \alpha.i = i, \alpha.e = e\}
This can be implemented as a positions table in a
database with columns amount, instrument, and
entity. The above calculation corresponds to the SQL
query
SELECT SUM(amount) AS A
FROM positions
WHERE instrument = i AND entity = eAn exchange occurs when two entities swap the amounts and instruments of their holdings. Given holdings {\alpha = (a, i, e)} and {\alpha' = (a', i', e')} then, after the exchange settles, the entities hold {(a', i', e)} and {(a, i, e')} respectively.
Often entity e is a buyer and entity e' is a seller. A buyer decides whether or not to execute the exchange based on the price offered by the seller. The post facto price of the exchange is {X = a'/a}. A seller quotes a pre facto price X for instrument i' in terms of a currency i. If the buyer wants to acquire n shares of i' they must give the seller nX units of currency i. If n is large, or negative, the seller will adjust their quoted price. They might also adjust the price based on the buyer.
Financial exchanges sign up liquidity providers to supply limit orders. Each provider offers to buy or sell a fixed amount of some instrument if there are any takers. Exchanges make money by charging a fraction of the amount exchanged. They only care about the volume of trades.
Exchange customers open an account by depositing a margin.
The order book shows the total amounts available to buy or sell at each price level. They do not make the identity of liquidity providers available.
The “price” of a market order executed by a customer of the exchange is determined by the existing limit orders. The amount of the market order is matched against limit orders. If the amount of level one orders are less than the amount of the market order than the trade is matched against the next level.
We write {\chi = (t, \alpha, \alpha')} to indicate the exchange occurred at time t.
A transaction is a collection of related exchanges. A broker might charge the buyer, the seller, or both for facilitating an exchange. If an entity holds an instrument they get all associated cash flows proportional to the amount held at the time of the cash flow. These are related exchanges but usually between the buyer and the instrument issuer.
How much is a position worth? In order to determine that you must find putative prices for instruments you hold in terms of the currency used for reporting. Accountants might use “book,” “market,” “liquidation,” or “going concern” values.
Mathematics is agnostic to that important problem. Given a base instrument (the reporting currency) i_0, a price function X\colon I\to\bm{R} determines the value of converting instrument i\in I to i_0.