What are the parameters’ units in the Avellaneda and Stoikov model?

Question

I'm studying a draft of the paper “Dealing with the Inventory Risk: A solution to the market making problem” by Guéant et al from July 2012.

According to the paper, the closed form solution to the optimal control problem is:

$ \delta_{\infty}^{b *}(q) \simeq\frac{1}{\gamma} \ln \left(1+\frac{\gamma}{k}\right)+\frac{2 q+1}{2} \sqrt{\frac{\sigma^2 \gamma}{2 k A}\left(1+\frac{\gamma}{k}\right)^{1+\frac{k}{\gamma}}} $

and

$ \delta_{\infty}^{a *}(q) \simeq \frac{1}{\gamma} \ln \left(1+\frac{\gamma}{k}\right)-\frac{2 q-1}{2} \sqrt{\frac{\sigma^2 \gamma}{2 k A}\left(1+\frac{\gamma}{k}\right)^{1+\frac{k}{\gamma}}}. $

Now assume:

$ A = 0.9, k = 0.3, \sigma = 0.3 , \gamma = 0.01, q = 0. $

as also shown in the screenshot below from the paper:

I understand the above process but what are the units of the parameters in the solution and $\delta$?

If the parameters’ units are based on the price, we have the same sizes of $\delta^{a,b}$ regardless of the stock price being large or small and that is weird. And yet, if the units are based on tick, Brownian motion, $dS_t = \sigma dW_t$, can't be calculated well.

Answer 1

A good way to understand the parameters is to look where they first appear in the model:

1. $dS=\sigma\,dW$ means that $S$ is a price and $\sigma$ is the volatility in the same unit than $dt$ (ie dollar per unit of time). You are right, it is well known that the high frequency volatility is difficult to estimate. Note that this volatility is used measure the risk of holding an inventory of $N$ shares, it may help you choose the adequate estimator.
2. $\delta^b=s-s^b$ and $\delta^a=s^a-s$ says that the $s$ are prices (i.e. in dollars), $\delta$ is in dollars too. You are right the question of the tick size is not addressed in the paper (neither is the position in the queues, at least it is not done explicitly: all is ``embedded'' in the intensities $\lambda$).
3. $\lambda(\delta)=A\exp -k\delta$ means that $k$ is mapping the distance to the ``fair price'' (that you can take as being the mid price) to a quantity without unit (hence it is in $1/$dollar), and $A$ is an intensity, i.e. in number of trades per unit of time ($dt$ again).
4. the utility function being $\mathbb{E}_t -\exp-\gamma\cdot(X_T+q_TS_T)$, it says that $\gamma$ is a risk aversion parameter. It is always difficult to choose, trials and error can help, based on common sense applied to the result.

You link to the paper is broken, here is the one on the arxiv version of it.