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Kch


Total Posts: 3
Joined: Aug 2017
 
Posted: 2017-09-20 20:59
In market making models derived originally from Avellaneda-Stoikov, there is a function lambda that represents the arrival rate of orders. In its prodigy, there are different representations of lambda, some linear, exponential, etc. But, all of these are functions of the difference between the maker's quote and the reference price. How do you calibrate it on initialization?

Also, in a Gueant-Lehalle-Tapia model, how are A and K practically calibrated?

mrdivorce


Total Posts: 4
Joined: Jan 2017
 
Posted: 2017-10-17 01:08
I'd also be pretty interested in finding out how to estimate these from data (if anyone's just passing through and is curious, one of the papers where this is used is https://people.orie.cornell.edu/sfs33/LimitOrderBook.pdf; the arrival rate is defined on page 7)

I was glancing through 'Optimal market making' by Gueant and he mentions that the A and k for the credit indices under consideration were 'estimated with classical likelihood maximization techniques using real quotes posted by the bank and the trades occurring between the bank and other market participant'. So I guess you apply some magic and use your observed quotes vs mid and trades done to find the MLE of A and k in the arrival rate formula (A * exp(-k * d)). Not totally sure though, sorry I couldn't be of more help!

mrdivorce


Total Posts: 4
Joined: Jan 2017
 
Posted: 2017-10-18 23:52
I tried my own advice and didn't get anywhere with MLE, though my maths is rubbish. I think you could use good ol' least squares on a log-transformed version though - if for a distance d_i you have orders o_i then:

o_i = A * exp(-K * d_i) -> ln(o_i) = ln(A) - K * d_i

and then calculate the gradient/intercept as usual to find A and K?

Kch


Total Posts: 3
Joined: Aug 2017
 
Posted: 2017-10-19 17:45
I used nonlinear optimization to solve A and k by capturing limit orders over a reference time, spreading them to reference, then scaling them to the tick time. In my case, the arrival of the new limit orders was a Poisson process and could be modelled exponentially.
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