April 25, 2024
Most mathematical finance models lack some important real-world features. It is customary to assume there is only one entity, the trader, involved in a trading strategy. The trader/buyer decides what hedging should be done, but the issuer/seller sometimes also has a say. For example, a callable bond allows the issuer to revoke ownership from the buyer by providing an early principal payment. The initial transaction also typically involves a third entity, the broker/dealer or an exchange, that receives a cash flow for providing liquidity. It is rare to find a model that accounts for taxes that must be paid due to trading activity. These can amount to a significant fraction of the cash flows involved and depend on the entities invovled.
We outline a treatment that allows more realistic modeling. It is essential to identify who owns what amount in each instrument. Holding an instrument often involves cash flows that depend on the amount held and the entity holding it. Holdings are transacted over time at a price. Each entity decides when and how much of each instrument they wish to trade. A complete model must allow for trading strategies to be specified by each entity.
Issuers create instruments. A company can issue stocks or bonds to raise capital, exchanges create contracts to connect liquidity providers with their customers. Commodities originate from producers of the physical product and currencies are issued by governments of countries.
Some instruments have cash flows. Stocks may have dividends, bonds may have coupons, futures have periodic margin adjustments. Commodities may involve holding costs. Futures always have price zero and only pay margin adjustment cash flows. Currencies do not have cash flows. The issuer of an instrument determines when and how much to give per unit of amount to instrument holders.
A holding is an amount of an instrument held by a legal entity. Amounts have units based on the instrument: stocks have shares, bonds have notional, futures have contract size. Commodities have physical quantities and currencies have denominations. Currencies never have cash flows. The price of a futures is always zero.
Holdings are transacted over time. Sellers determine what holdings they are willing to exchange with buyers. Buyers decide when to transact an available exchange of holdings with a seller. A transaction involves a trade date, settlement date, and an exchange of holdings between the buyer and seller. The price of the transaction is the quotient of the buyer amount and the seller amount. Ownership of holdings is transferred on the settlement date. The position of an entity is their set of holdings at any given time.
The sum of the amounts held of each instrument in a position is the net position in that instrument. Given prices of instruments in some currency, the net positions can be converted to the value, or mark-to-market, of the position in that currency. Determining the “price” of an illiquid instrument is problematic. The difference of the net value of positions at the beginning and end of a period is the profit and loss over the period.
Profit and loss does not capture the dynamics over the period. Cash flows and transactions cause changes to positions. Instrument holders receive cash flows based on the amount held and transactions involve an exchange of holdings between the buyer and seller. These show up in the trade blotter account. Other measures such as drawdown can be used to mangage trading risk and determine trading strategy.
Let E be the set of market entities.
Let T indicate the set of possible trading times.
Let I be the set of market instruments.
Let A = 𝑹, be the set of real numbers indicating the amounts that can be traded in each instrument.
A holding is a triple (a, i, e)\in A\times I\times E indicating entity e hold amount a of instrument i.
A transaction is an exchange of two holdings between a buyer and a seller at a time t. Sellers offer potential transactions to buyers who decide whether or not to accept them. This corresponds to buy-side and sell-side firms.
The market at time t is the set of all holdings \mathcal{H}_t = \{(a,i,e)\}. If a buyer holding (a,i,e) accepts a seller’s offer of (a',i',e') then the market holdings are changed. The holdings (a,i,e) and (a',i',e') become (a',i',e) and (a,i,e') when the transaction settles. The price of the transaction is X = a'/a.
The market gets updated by cash flows even if there are no transactions. If the holding (a,i,e) recieves a cash flow (b,',e) at time t that is added to \mathcal{H}_t.
The net amount entity e holds in instrument i is N_t(i,e) = \sum \{a' \mid (a',i',e')\in\mathcal{H}_t, i' = i, e' = e\}. The profit and loss over the period (t, u] is P_{t,u}(i, e) = N_u(i,e) - N_t(i,e).
The price offered by a seller to a buyer depends on the amount of the instrument being purchased from the seller and the amount the buyer is asking for.
X_t(i,e;a',i',e')