October 3, 2025
Notes on Quantum Markets by Jack Sarkissian.
\tau – time between trades
s – price
x – log price
m – absolute value of order/trade
size
M – absolute value trade size
q – order size
Q – signed trade size
\rho – order size density s\,dm/ds \theta – order density flow \Delta – bid/ask spread
\epsilon – \Delta/s - relative spread
h – high minus low
I – execution imbalance
J – order imbalance
V – volume
v – order flow
L – price impact
A (legal) entitiy is a person or corporation that can own financial instruments.
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. Holdings are the atoms of finance.
An atomic transaction, or switch, is a triple \chi = (t, h, h') where t is the time the buyer holding h is exchanged with the seller holding h'. Both buyers and sellers are owners.
A switch 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\}. The amount of each instrument is switched between the buyer and seller. Sellers passively offer potential exchanges and buyers actively decide which to accept. Both buyers and sellers can buy/go long (amount is positive) or sell/go short (amount is negative).
A transaction is a collection of related switches involving, e.g., fees or taxes. Transactions are the molecules of finance.
The price of a switch 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 a switch. 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 switch \chi = (t,(X a', i, o), (a',i',o')) is available to a buyer wanting to acquire a' of seller’s 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.
The price a seller quotes can also depend on the buyer but in all cases the price recorded in books and records will be X = a/a'.
If i is assumed to be the preferred buyer currency, i' is a particular seller instrument, and we lump all buyers into o and all sellers into o' then market orders can be modeled as a sequence (t_j, a_j') of times and amounts of instrument i' acquired by the buyer. The buyer pays a_j = a_j' X_j in the preferred currency to the seller where X_j is the price of the switch.
An order book is a collection of limit orders. A limit order is a quintuple (t, l, a, i, o) indicating that owner o is willing to buy amount a > 0 or sell amount a < 0 of instrument i at price l any time after t. Limit orders can be cancelled by the owner.
The depth at time t for level l is the sum over the amount of every 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 is the lowest level with A(t, l) > 0. The bid is the highest level with A(t, l) < 0.
The order book can be represented by B(t, l) = \sum_{a_j > 0, t > t_j} a_j 1(l > l_j) + \sum_{a_j < 0, t > t_j} a_j 1(l < l_j) Note dB/dl = A and the inverse of B is the usual representation for order books.
TODO: explain better
Exchanges line up liquidity providers to seed the order book prior to market open with limit orders. Exchange customers can also place limit orders. Both liquidity providers and exchange customers can execute market orders during trading hours.
A market order specifies the holding (a, i, o) that buyer o would like to obtain at time t. If the amount a is positive the limit orders at the ask level are matched in priority of when they were placed and removed from the order book. This may result in a transaction with multiple switches having different limit owner orders.
If the depth at the ask is less than the amount a then the price will be the ask. If not, limit orders at the next highest level will be matched. If their depth is still less than the remaining amount the process repeats until the entire buy is filled. This results in a transaction containing switches for each level that was filled.
If the amount a is negative the limit orders at the bid level are matched in priority of when they were placed and removed from the order book. If the depth at the bid is less than the amount a then the price will be the bid. If not, limit orders at the next lowest level will be matched. If their depth is still less than the remaining amount the process repeats until the entire sell is filled.
The usual representation is the inverse of B with level a function of amount.
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 not unlike the Heisenberg Uncertainty Principle.
The ask for a order book is the lowest level with positive depth. The bid for a order book 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.
Now that we have established rigorous definitions of order books and market orders we will revisit the notation from the first section.
Let (t_j, a_j') be the market orders where the times are increasing. Clearly \Delta t_j = t_{j+1} - t_j is the time between trade j and trade j + 1. Given two times u < v the average time between trades in [u, v] is \tau(u,v) = (\sum_{j=m}^n \Delta t_j)/(n - m + 1) where the sum is over \Delta t_j\subseteq [u,v].
The last price to trade at s_t is X_j = a_j/a_j' where j = j(t) = \max \{i \mid t_i < t\}. We write s_t^+ if traded at the ask and s_t^- if traded at the bid. The mid is s_t = (s_t^+ + s_t^-)/2.
m, M, q, Q?