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

Posted: May 27th, 2019, 3:46 am
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
pdf's have norm 1

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

Posted: May 27th, 2019, 9:14 pm
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
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 priceI 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.
"
(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
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
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
Yeah, I was thinking about equities mostly.