Calibrating Exponential Order Arrival Rates from Market Data

Summary

The note proposes fitting an exponential order-arrival model, in which execution intensity declines with distance from the mid-price. Its two parameters control the baseline arrival level and how quickly the rate falls as quote distance increases. The intended use is to estimate these parameters from observed market data for a particular contract.

The suggested procedure is to construct empirical arrival rates across quote distances, compare them with rates implied by candidate parameter values, and optimize the parameters to reduce the summed absolute differences over a selected range. The range could theoretically extend indefinitely, though the answer suggests a finite range may be adequate. This is a simple calibration proposal rather than a fully specified statistical procedure: it does not detail how to estimate arrival rates from event data, choose the fitting range, weight observations, or quantify uncertainty. Since it fits historical observations, the resulting parameters are backward-looking and may not remain representative as market conditions change.

Key ideas

• The model assumes order arrival intensity falls exponentially as quote distance from mid-price increases.
• The baseline level and distance sensitivity can be fitted by minimizing differences from empirical arrival rates.
• A finite set of quote distances can be used for the fitting objective.
• Historical calibration describes past order flow and may not capture future market conditions.

Tags

# How do you calibrate a poisson arrival rate process?


# How do you calibrate a poisson arrival rate process?












Many papers in the microstructure literature assume an order arrival rate of the form

$\lambda^a(\delta) = \lambda^b(\delta) = Ae^{-k\delta}$

That is, an order that's placed $\delta$ away from the mid-price is likely to be executed with probability $Ae^{-k\delta}$. How would you choose k and A given data from a real contract?

In particular, this is used in the seminal paper by Avellanda and Stoikov (http://www.math.nyu.edu/faculty/avellane/HighFrequencyTrading.pdf) in section 2.5

## Answer by Probilitator (score 2, accepted)

https://quant.stackexchange.com/a/10871

Here is how I would approach such a calibration.

Assuming we have the necessary market data one can easily construct the emprical distribution of the arrival rate.

Let $\lambda_{emp}(\delta)$ be the empirical distribution. Then one can define a metric by

$$ m(k,A,N)=\sum_{i=1}^N |\lambda_{emp}(i)-\lambda^a(i)| $$ After you have decided upon a suitable $N$ (it would be formally correct to set $N=\infty$ but I don't think this is necessary to get a decent calibration result)

One can now run an optimisation routine on $m(k,A,N)$ to determine parameters $k,A$.

This approach will output a parameterisation that minimizes the asolute distance in probability mass.

Note however that this would give you a "backward looking" calibration for one will be fitting to historical data.

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