# 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.

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.

• I see your point but the problem is that all of the parameters depend on the time frame. In the formulas related to bid and ask price we consider transaction intensity and sigma. Transaction intensity is invariant under the change of the time frame but the sigma is not. If we consider parameters estimation under the one second or two seconds timeframe, the results will be extremely different. I think that the shortest time step available is not satisfactory answer because it depends not only on the asset but also on the technology within the exchange/company etc.
– ltrd
Oct 8, 2019 at 15:52
• @Lipa_FNTE After a closer look I think t is a continuous time variable. Your paper begins with "Market makers continuously set bid and ask quotes." Your paper also references a paper by Avellaneda and Stoikov that integrates and differentiates relative to time. In section 3.1 they use a PDE that's continuous in t. All of this strongly implies that t is continuous... I imagine that practical applications with discrete time steps are an approximation of the continuous theory (as is fairly usual). Oct 8, 2019 at 18:41
• @Lipa_FNTE Also in that Avellaneda and Stoikov paper they mention their "continuous-time model" towards the end of section 2.1... Anyway for your question about $t_n - t_{n-1}$ I think $dS_t$ is the derivative in theory, so in an approximation you would use something like $\frac{S_{t_n} - S_{t_{n-1}}}{t_n - t_{n-1}}$. Of course these estimates will vary depending on the underlying data. Oct 8, 2019 at 18:55