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### How to calibrate an exponential distribution with scale?

Posted: **May 27th, 2019, 3:46 am**

by **EdisonCruise**

I want to fit an exponential arrival rate from my data with below model, a more detail is given in my next post:

$$\lambda(\delta)=Aexp(-\kappa\delta) $$

MATLAB can fit an exponential distribution with the following pdf:

$$Y(\delta)=\frac{1}\mu exp(-\frac{\delta}\mu) $$

But is there any way to convert the MATLAB fitting for the first equation? I need to get \(A\) and \(\kappa\).

Thank you for any suggetions.

### Re: How to calibrate an exponential distribution with scale?

Posted: **May 27th, 2019, 4:27 pm**

by **Alan**

pdf's have norm 1

### Re: How to calibrate an exponential distribution with scale?

Posted: **May 27th, 2019, 9:14 pm**

by **bearish**

Even less cryptical: the exponential distribution is fully characterized by a single parameter.

### Re: How to calibrate an exponential distribution with scale?

Posted: **May 28th, 2019, 1:32 am**

by **EdisonCruise**

Thank you for your reminding. I confused something here. \(\lambda(\delta)=Aexp(-\kappa\delta) \) is not a pdf.This model is from the paper "High-frequency trading in a limit order book "by MARCO AVELLANEDA and SASHA STOIKOV.

\(\lambda(\delta)=Aexp(-\kappa\delta) \) is acutally the exponential arriaval rates of market order, where \( \delta \) is the distance from mid price. I am finding a way to calibrate this model in practice.

In the 2006 version of this paper, the authors say

"Once we have chosen our time step \(dt\),e we obtain the value for \(A\) through the relation \(\lambda(M/2)=Aexp(\kappa M/2)dt\), which says that if we post a limit order at a distance \(M/2\) from the mid-price, this limit order is essentially a market order that will get executed with probability one in a \(dt\) time interval.

"

I am not clear about

(1) how to select the time interval \(dt\)

(2)how to calibrate \(A\) and \(\kappa\) from \(\lambda(\delta)=Aexp(-\kappa\delta) \) and \(\lambda(M/2)=Aexp(\kappa M/2)dt\). It seems to me that these two equations may not be consistent,because by counting the market order the former equation can be calibrated alone to get \(A\) and \(\kappa\).

### Re: How to calibrate an exponential distribution with scale?

Posted: **May 28th, 2019, 10:09 pm**

by **ISayMoo**

A high-frequency trading model which assumes continuous prices is worthless.

### Re: How to calibrate an exponential distribution with scale?

Posted: **May 29th, 2019, 12:34 am**

by **EdisonCruise**

I think the model should work for small-tick-size assets. But for most of other asset types, this model is not applicable directly.

### Re: How to calibrate an exponential distribution with scale?

Posted: **May 29th, 2019, 11:40 pm**

by **ISayMoo**

Yeah, I was thinking about equities mostly.