Finding parameters of a function for optimal market making with real data

Question

I am reading this paper and trying to apply it with real data to do some simulations.

I will use realtime order book & market order data that I will receive from the exchange. This is a sample of the market order data(You can say execution data or tick data too).

side  price   size   execution_date
BUY   100     1.5    2018-08-06T03:24:29.023 

Using these data, I can't figure out how to find some parameters for the function below.

γ = Risk parameter, σ = Volatility, T = Terminal time, t = Current time, k = Trading intensity

I do not know how to compute parameter σ & k. I assume that σ would be the standard deviation by the previous few seconds of tick data.

And for k, using the total volume of previous few seconds of market orders and the current limit order book, we can know the temporary market impact. But I am not sure how to fit these information into parameter k.

I hope someone could give some help and advice how to find these parameters.

Thank you.

Answer 1

Parameter $k$ (I assume) is related to the model connected with control theory. You have to find the difference between transaction price and mid price - it can be called $\delta$. Then you have to fit your results to the intensity function: $\Lambda(\delta) = A \exp( - \delta k)$. These are your parameters. Spread does not depend on the parameter A but it is related to nonsymmetrical deltas.

I am going to answer you questions here:

1. Close to this. I would say that we would like to find the general pattern for the probability that your quotes will be filled using historical data from finite period of time.

2. Probably you can do it with several approaches. You should collect as many deltas as you can. If you have, let's say 10000 deltas, then we can start our estimation.

Let's define $\widehat{\Lambda} (\delta_n)$ as the number of transactions where difference between mid price and transaction price was less that $\delta_n$, where sequence $(\delta_n)_n$ can be define (in R language) as:

seq(0.1,5,0.1)

Then we have $\widehat{\Lambda} (\delta_n)$ and $\delta_n$ for every $n$ in your bucket. Then you can minimize the function:

$\sum_{n = 1}^{|D|} \left( \widehat{\Lambda} (\delta_n) - A e^{- k \delta_n} \right)^2$

where $|D|$ is the number of items in sequence $(\delta_n)_n$. Obviously we minimize with respect to $A$ and $k$.