August 22, 2025
Notes on Quantum Markets by Jack Sarkissian.
\tau – time between trades
\rho – order size density s\,dm/ds \theta – order density flow \Delta – bid/ask spread
s – price
x – log price
\epsilon – \Delta/s - relative spread
h – high minus low
I – execution imbalance
J – order imbalance
m – order/trade imbalance
q – order size
M – absolute value trade size
Q – signed trade size
V – volume
v – order flow
L – price impact
A holding is a triple h = (a, i, o) indicating amount a of instrument i. is held by owner o. A portfolio is a collection of holdings.
An exchange is a triple \chi = (t, h, h') where t is the time of the exchange, h is a holding of the buyer, and h' is a holding of the seller. An exchange results in the portfolio of the buyer o changing from \{\ldots,(a,i,o),\ldots\} to \{\ldots,(a',i',o),\ldots\} and the portfolio of the seller o' changing from \{\ldots,(a',i',o'),\ldots\} to \{\ldots,(a,i,o'),\ldots\}.
A transaction is a collection of related exchanges. Transactions often involve exchanges such as broker fees or taxes. Savvy buyers should take these into account. Sellers have already done so.
Sellers passively offer exchanges as limit orders and buyers actively decide which to accept/lift using market orders.
Exchanges are the atoms of finance. Transactions are the molecules.
The price of an exchange is X = a/a' where h = (a, i, o) and h' = (a', i', o'). If a buyer pays (8, \$) for (2, \text{F}) shares of Ford stock then the price is 8/2 = 4 dollars per share. Typically i is the preferred currency of the buyer.
There is no uncertainty in the price of an exchange, it is simply a number entered into the books and records of each owner.
If a seller offers a price X at time t for buyers to exchange instrument i' for instrument i then it is approximately correct that the transaction \chi = (t,(Xa', i, o),(a',i',o')) is available to a buyer wanting to acquire a' of instrument i'. If the buyer wants to buy a' > 0 of i' then the seller price X will be the ask. As a' becomes a larger positive number, the ask will increase. If the buyer want to sell a' < 0 of i' then the seller price X will be the bid. As a' becomes a larger negative number, the bid will decrease.
An order book is a collection of limit orders. A limit order is a quintuple (t, l, a, i, o) indicating that seller o is willing to buy amount a > 0 or sell mount a < 0 of instrument i at price l any time after t. Limit orders can be cancelled by the seller.
A market order specifies the holding (t, a, i, o) that buyer o would like to obtain at time t. A limit order placed at time t will eventually have price l, but there is no guarantee when, or if, it will get matched by a market orders. A market order is immediately matched with existing limit orders, but its price is uncertain. This is similar to the Heisenberg Uncertainty Principle.
The depth at time t for level l is the sum over limit orders placed prior to t having level l A(t, l) = \sum_{(t_j,l_j,a_j,i_j,o_j)} \{a_j\mid t_j < t, l_j = l\}.
The ask for a order book at t is the lowest level with positive depth. The bid for a order book at t is the highest level with negative depth. The mid is the average of the bid and ask. The high over an interval is the largest trade price over the interval. The low over an interval is the smallest trade price over the interval. The open is the price of the first trade of the session and the sign of the amount. The close is the price of the last trade of the session and the sign of the amount.