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EdisonCruise
Topic Author
Posts: 105
Joined: September 15th, 2012, 4:22 am

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

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.
Last edited by EdisonCruise on May 28th, 2019, 3:25 am, edited 1 time in total.

Alan
Posts: 9929
Joined: December 19th, 2001, 4:01 am
Location: California
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### Re: How to calibrate an exponential distribution with scale?

pdf's have norm 1

bearish
Posts: 5283
Joined: February 3rd, 2011, 2:19 pm

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

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

EdisonCruise
Topic Author
Posts: 105
Joined: September 15th, 2012, 4:22 am

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

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 $\kappa$.

ISayMoo
Posts: 2250
Joined: September 30th, 2015, 8:30 pm

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

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

EdisonCruise
Topic Author
Posts: 105
Joined: September 15th, 2012, 4:22 am

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

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

ISayMoo
Posts: 2250
Joined: September 30th, 2015, 8:30 pm

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

Yeah, I was thinking about equities mostly.