Fundamental Finance

In Physics, a theory that does not agree with experiment is recognized as deficient and serious scientists get to work on improving the theory. Mathematical Finance seems to ignore fundamental shortcomings. Continuous time hedging is impossible. Traders need to know when, and by how much, they should adjust their hedges. This short note takes a page out of the physicsts playbook in an attempt to identify the fundamental components of real-world trading and define a vocabulary for a rigorous mathematical framework to reason about them.

A market consists of instruments. Stocks and bonds come into existence by companies wanting to raise capital. Commodities are instruments that involve growing crops, raising livestock, and digging things out of the ground. Currencies are issued by nation states to facilitate trustworth transactions. Exchanges provide a means for buyers and sellers to transact instruments.

Every instrument has a price at which it can be bought or sold and holding an instrument entails cash flows to its owner. Stocks pay dividends, or may incur borrow costs when shorted. Bonds pay coupons. Futures always have price zero and have daily margin adjustements. Currencies never have cash flows.

Instruments do not trade in a vacuum. Companies bring them into the market by issuing them and legal entities, companies or individuals, trade them. Price is typically quoted in terms of the amount that is converted to a native currency required to exchange ownership. Stocks trade in shares. Bonds have notionals. Futures have contract sizes. Exchanging one currency for another is the foreign exchange rate.

The elementary particles of finance are holdings, (a,i,e), indicating amount a of instrument i is owned by entity e. Holdings interact via transactions, (t;a,i,e;a',i',e'), indicating the buyer e transferred ownership of amount a of instrument i to seller e' in exchange for amount a' of instrument i'. This corresponds to a measurement indicating the price for this particular transaction of instruments between the entities is a'/a.

Transactions are not symmetric. The buyer decides when and how much of an instrument they hold to bid. The seller offers a price X = X(t;a,i,e;i',e') that the buyer can lift for the transaction (t;a,i,e;aX,i'e'), so a' = aX and X = a'/a is the price.