Choosing the Time Scale in Brownian Market-Making Models

Summary

The document addresses what the time increment means when a market-making model treats its reference price as an arithmetic Brownian motion. It explains that Brownian motion is self-similar, so the model’s mathematical form can be considered across time scales, while estimates derived from data can differ substantially depending on the sampling interval.

For a high-frequency market maker, the relevant interval depends on the market and the shortest time step that can be observed or used in practice. The cited work’s continuous-time formulation treats time as continuous in theory; an implementation may approximate it with discrete observations. The discussion offers general interpretation rather than a specific sampling rule, and it does not establish that estimates remain interchangeable across intervals.

Key ideas

• The model’s time increment depends on the market context and the data available.
• Brownian motion’s self-similarity supports formulating the model at different time scales.
• Estimated parameters may change materially when the sampling interval changes.
• A continuous-time model can be approximated with discrete observations in application.

Tags

# Arithmetic Brownian Motion in Market Making papers


# Arithmetic Brownian Motion in Market Making papers












We often consider high-frequency market maker and suppose that the reference price is the arithmetic Brownian Motion:

$dS_{t} = \sigma d W_t$

What is the difference $t_n - t_{n-1}$ in this case? Is is one day or one second? Estimation in those two cases based on datasets would be different, so what is the case here?

My question is based on the paper: Dealing with inventory risk - a solution to the marker making model by Gueant, Lehalle and Tapia.

## Answer by sfmiller940 (score 4)

https://quant.stackexchange.com/a/49101

The time step typically depends on the context. Due to the self-similarity of Brownian motion the mathematics should work similarly on any time scale, although the resultant estimates might vary greatly (as you mention).

Since the cited article assumes a "high-frequency market maker," the implied time step seems to be the shortest time step available or attainable in a given market.

Edit: Also the cited article references a paper by Avellaneda and Stoikov. Towards the end of section 2.1 this paper states that it's using a "continuous-time model." So the time variable is continuous in theory, while discrete approximations are most likely used in application.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.