# 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

• Don't you know $\delta$? You have to estimate $A$, where $A \leq 1$? – quasi Apr 8 '14 at 5:57
• this looks like some type of poisson distribution to me - however it does not seem to be normalized to 1 ? – Probilitator Apr 8 '14 at 10:50
• @quasi heh yeah I meant A and k. I'll fix that. – Thomas Johnson Apr 8 '14 at 13:40
• @Probilitator Yeah Poisson is more correct than exponential, I'll change that, thanks – Thomas Johnson Apr 8 '14 at 15:26
• Maybe I'm thinking to simple but: why not use a standard MLE? – Bob Jansen Apr 8 '14 at 16:09

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.

• "This approach will output a parameterisation that minimizes the asolute distance in probability mass". I don't understand what this means. Sure you have an estimator but why not minimize the sum of squares instead? Can you get a confidence interval or at least a rate of convergence of this estimator? If you only had one size of ticks (a simple Poisson process) is that how you would estimate its intensity? I don't see what is the statistical reasoning here. – AFK Apr 9 '14 at 23:51
• @YBL your remark is completly valid. The choice of metric or "cost function" (as it is often called in optimisation) depends on your personal preference. One could also minimize the sum of squares. I personally am a fan of absolute difference in probability mass. The squared sum has the adventage of being analytically more tractable (differentiability etc.) – Probilitator Apr 10 '14 at 8:08
• The "density-fit" will obviously change depending on the dinstance-metric between empirical and analytical distribution. When I have to fit a density to a historical empirical distribution I check both - the absolute distance and the sum of squares. – Probilitator Apr 10 '14 at 8:11