### How to get this limit order fill probability?

Posted:

**February 14th, 2022, 6:45 am**I find this power class limit order fill probability

$$h(\delta)=\frac{1}{1+(\kappa\delta)^{\gamma}}$$

where \(\delta\) is the distance to mid price;\(\kappa\) and \(\gamma\) need to be calibrated from data .

in https://tspace.library.utoronto.ca/bits ... thesis.pdf

My questions are (1) Is there any reference to justify this model? (2) And how to calibrate this model?

My primary understaning is that this model comes from:

(a) Market impact assumption: \( \Delta p=K(Q-c)^{\frac{1}{\gamma}} \), where \(\Delta p\) is the distance to mid price and \(Q\) is trading volume.

(b) Distribution of market order size (pdf): \(Q:x^{-2}\)

With this two assumptions,

$$h(\delta)=P(\Delta p>\delta)=P(K(Q-c)^{\frac{1}{\gamma}}>\delta)=P(Q>(\frac{\delta}{K})^{\gamma}+c)=\int^{\infty}_{\frac{\delta}{K}^{\gamma}+c} x^{-2}dx=\frac{1}{\frac{\delta}{K}^{\gamma}+c}=\frac{1}{c} \frac{1}{1+(\kappa\delta)^{\gamma}}$$

where\(\kappa=\frac{1}{c^{\frac{1}{\gamma}K}}\)

If this derivation is correct, then one need to get \(c\) and \(K\) from data to calculate \(\kappa\)?

From https://www.math.nyu.edu/~avellane/High ... rading.pdf, assumption (a) and (b) should be something like

(a') \( \Delta p=K(Q)^{\frac{1}{2}} \)

(b') \(Q:x^{-2.5}\)

With (a') and (b') is is possible to get the same formula of \(h(\delta)\)?

$$h(\delta)=\frac{1}{1+(\kappa\delta)^{\gamma}}$$

where \(\delta\) is the distance to mid price;\(\kappa\) and \(\gamma\) need to be calibrated from data .

in https://tspace.library.utoronto.ca/bits ... thesis.pdf

My questions are (1) Is there any reference to justify this model? (2) And how to calibrate this model?

My primary understaning is that this model comes from:

(a) Market impact assumption: \( \Delta p=K(Q-c)^{\frac{1}{\gamma}} \), where \(\Delta p\) is the distance to mid price and \(Q\) is trading volume.

(b) Distribution of market order size (pdf): \(Q:x^{-2}\)

With this two assumptions,

$$h(\delta)=P(\Delta p>\delta)=P(K(Q-c)^{\frac{1}{\gamma}}>\delta)=P(Q>(\frac{\delta}{K})^{\gamma}+c)=\int^{\infty}_{\frac{\delta}{K}^{\gamma}+c} x^{-2}dx=\frac{1}{\frac{\delta}{K}^{\gamma}+c}=\frac{1}{c} \frac{1}{1+(\kappa\delta)^{\gamma}}$$

where\(\kappa=\frac{1}{c^{\frac{1}{\gamma}K}}\)

If this derivation is correct, then one need to get \(c\) and \(K\) from data to calculate \(\kappa\)?

From https://www.math.nyu.edu/~avellane/High ... rading.pdf, assumption (a) and (b) should be something like

(a') \( \Delta p=K(Q)^{\frac{1}{2}} \)

(b') \(Q:x^{-2.5}\)

With (a') and (b') is is possible to get the same formula of \(h(\delta)\)?