28 — The microstructure of high frequency markets

This paper “microscopically” explores high-frequency trader agents in the financial market. In particular, the authors explore the models for a financial asset “traded electronically via a limit order book.”

The paper first focuses on the importance of superior information when studying a high-frequency trade system. This, the authors explain, when taken to an extreme, is insider-trading; but an agent with superior information is, simply, a better-informed agent.

A perfectly informed agent knows the fundamental price of an asset, and can thus base their trades on this information. (Of course, how the agent acts on this information can drastically affect their returns.)

A high-frequency trader, on the other hand, likely has no information on the fundamental value of an asset. (HFTs generally leverage the microstructure of the market to generate value.) But, like an inside-trader, a HFT’s profit is based on how this information is used, not in how accurately the information can be learned.

The paper contrasts the two traders: An insider trades always in the same direction; a HFT oscillates based on microstructure. An insider knows current price $p$ at time $n$ ($p_n$), and the price of the asset at time $p_N$. The HFT anticipates price $p_{n+1}$.1

By making some assumptions about a trader — namely, the trader can only sell assets in their inventory at time $n$, $L_n$, and holds $K_n$ in purchasing power — we can calculate wealth as:

\[X_n = L_n p_n + K_n\]

Now, defining the spread of an asset as $s_n$, we can derive the trading profits of an agent (eq 3.7 in the paper):

\[\Delta_n X = \texttt{op} \Big(L_n \Delta_n p, \frac{s_n}{2} \Big| \Delta_nL\Big| \Big) + \Delta_n L \Delta_n p\]

where $\Delta_n W$ is the change of $W$ between times $n$ and $n+1$, and $\texttt{op}$ is $+$ if limit-trading and $-$ when trading with market orders.

In other words, we can estimate the wealth-change of an agent by observing the right-hand-side variables.

This models the authors’ “self-financing” condition for HFTs.

The paper then explains some confounds to this formula: Options, dark pools, etc… and closes by demonstrating graphically and mathematically that the true wealth-flow and the wealth-flow of an agent modeled by this formula are indistinguishable in the presence of predicted trade-friction.