# Finding parameters of an utility function in a market making strategy to apply it in practice

I am reading this paper below about optimal bid-ask spread in a market making strategy. It finds an approximation for optimal solution, but I cannot understand how it's practice to set the parameters for a sample stock (eg. AAPL). Assuming, I have this stock below, how I can find all the parameters for the optimal bid-ask spread?

-> How to set $A$ and $k$, for my example stock? Which is the parameter for the tick size?

Example:

Stock: AAPL

$\sigma = 0.2$

$\mu = 0.01$

$S_t$= 169.23 USD

Tick=0.01 USD

Utility function to maxime

Intensity function

How fast my order (bid/ask) will be filled respect to the mid-price in the market at t

page 13

Paper source: Dealing with the Inventory Risk. A solution to the market making problem https://arxiv.org/abs/1105.3115

• Sophie Laruelle did it in this working paper: events.chairefdd.org/… unfortunately it is in French, but the sensity of Figures and formula should make it readable by non French-speaking people – lehalle Jan 4 '18 at 14:48

All the parameters of the solution need to be estimated for your specific stock. Stochastic process-specific parameters, i.e. $\mu$, $\sigma$, have to be estimated by some classical method (e.g. MLE, minimum contrast, etc.). No parameter of tick size is incorporated in the model, you will have to decide at the end whether the in-between quote shall be assigned to the higher or lower price value. And parameters relating to the order arrival need to be estimated similarly as derived in the original paper of Avellaneda & Stoikov. There you may find the derivation from three separate formulas:

1. Constant frequency $\Lambda$ of market buy/sell orders estimated by dividing the total volume traded over a day by the average size of market orders on that day.

2. The distribution parameter of the size of market orders $$f^{Q}(x) \propto x^{-1-\alpha}$$

where $\alpha$ will be overtaken from the literature (presented in the original study), or else needs to be calculated from your specific dataset.

1. Temporary impact of a large market order:

$$\Delta p \propto ln(Q),$$

also either overtaken from the literature or estimated on the historical values of your specific stock.

By putting all these three formulas together, you obtain the original result:

$$\lambda(\delta) = Aexp(-k\delta),$$

where $A = \Lambda / \alpha$ and $k = \alpha K$, $K$ being a scaling parameter for the temporary impact of a market order.

Source: Avellaneda, Marco, and Sasha Stoikov. "High-frequency trading in a limit order book." Quantitative Finance 8.3 (2008): pp. 220. https://www.math.nyu.edu/faculty/avellane/HighFrequencyTrading.pdf