Treating the Bid/Ask spread as a Normal Distribution...

Treating the Bid/Ask spread as a Normal Distribution and
computing the probability for getting a fill for the own offer price
(that lies between the last/current Bid and Ask in the orderbook).

Obviously the probability for getting a fill for the MidPrice has to be p = 0.5 (ie. 50%).
And in case of a Long:
p(Bid) = 0.0, p(Ask) = 1.0

And in case of a Short:
p(Bid) = 1.0, p(Ask) = 0.0

The task is of course to compute this:
p(OwnOfferPrice) = ...

Bid and Ask are the ultimate limits, ie. the whole address space lies inbetween.

So, the question is: how to do this mathematically as we don't have a StdDev?
Can we arbitrarily use 34.1% for the StdDev?
Any other solution?

 

probability of getting filled at mid is not 50%. Also, would need to look at size. Is this something like hitting times?

 

KISS.

Your bid will be filled at the market's ask... and vice versa for your sale. Unless a MM decides to play your order between the bid/ask spread, your limit bid gets filled when the market moves down so that your bid has become the market's ask.

(I understand... you want to buy at the bid and sell at the ask. Unfortunately, doesn't usually work that way.)

 

In my case it's enough to assume that own qty is <= qty in the top of the book.
So, in this case we can ignore the size.
And: I don't have Level-2 data, rather just the top-of-the-book, ie. Level-1 quotes.

 

My story behind this ideá:
I first used DistancePercent of own Price from the Ask (or Bid in case of short), but then decided that this new idea here is much better when programatically scanning many (hundreds) trade candidates.

 

Why don’t you just run a study by analyzing a historical data set of bid/ask and comparing to tas data?

 

I need to apply this in a scanner. I only have the current Level-1 data.

 

Came across a great interview by Lex Fridman of Neal Stephenson (Snow Crash, Cryptonomicon,..)

He so happened to also be the first employee of Blue Origin and tasked with designing a new propulsion system for space travel. A lasting comment from that interview was as he spent his mental capital "designing" later he discovered someone had already invented it and done all the math. He pivoted and started to just extensively research and leveraged the prior works of others.




Estimating the Components of the Bid/Ask Spread by L. Harris et al
https://www8.gsb.columbia.edu/researcharchive/articles/1547

 

Found a Q&D solution:
An example for Long, Bid=80, Ask=120 using a const StdDev=34.134475 (= (0.5 - z2p(-1.0)) * 100). and zMin=-5.0, zMax=5.0 :

Code:

Sx=80.00 : z=-5.00000 p=0.00000 pPct=0.000 
Sx=81.00 : z=-4.75000 p=0.00000 pPct=0.000 
Sx=82.00 : z=-4.50000 p=0.00000 pPct=0.000 
Sx=83.00 : z=-4.25000 p=0.00001 pPct=0.001 
Sx=84.00 : z=-4.00000 p=0.00003 pPct=0.003 
Sx=85.00 : z=-3.75000 p=0.00009 pPct=0.009 
Sx=86.00 : z=-3.50000 p=0.00023 pPct=0.023 
Sx=87.00 : z=-3.25000 p=0.00058 pPct=0.058 
Sx=88.00 : z=-3.00000 p=0.00135 pPct=0.135 
Sx=89.00 : z=-2.75000 p=0.00298 pPct=0.298 
Sx=90.00 : z=-2.50000 p=0.00621 pPct=0.621 
Sx=91.00 : z=-2.25000 p=0.01222 pPct=1.222 
Sx=92.00 : z=-2.00000 p=0.02275 pPct=2.275 
Sx=93.00 : z=-1.75000 p=0.04006 pPct=4.006 
Sx=94.00 : z=-1.50000 p=0.06681 pPct=6.681 
Sx=95.00 : z=-1.25000 p=0.10565 pPct=10.565 
Sx=96.00 : z=-1.00000 p=0.15866 pPct=15.866 
Sx=97.00 : z=-0.75000 p=0.22663 pPct=22.663 
Sx=98.00 : z=-0.50000 p=0.30854 pPct=30.854 
Sx=99.00 : z=-0.25000 p=0.40129 pPct=40.129 
Sx=100.00: z=+0.00000 p=0.50000 pPct=50.000 
Sx=101.00: z=+0.25000 p=0.59871 pPct=59.871 
Sx=102.00: z=+0.50000 p=0.69146 pPct=69.146 
Sx=103.00: z=+0.75000 p=0.77337 pPct=77.337 
Sx=104.00: z=+1.00000 p=0.84134 pPct=84.134 
Sx=105.00: z=+1.25000 p=0.89435 pPct=89.435 
Sx=106.00: z=+1.50000 p=0.93319 pPct=93.319 
Sx=107.00: z=+1.75000 p=0.95994 pPct=95.994 
Sx=108.00: z=+2.00000 p=0.97725 pPct=97.725 
Sx=109.00: z=+2.25000 p=0.98778 pPct=98.778 
Sx=110.00: z=+2.50000 p=0.99379 pPct=99.379 
Sx=111.00: z=+2.75000 p=0.99702 pPct=99.702 
Sx=112.00: z=+3.00000 p=0.99865 pPct=99.865 
Sx=113.00: z=+3.25000 p=0.99942 pPct=99.942 
Sx=114.00: z=+3.50000 p=0.99977 pPct=99.977 
Sx=115.00: z=+3.75000 p=0.99991 pPct=99.991 
Sx=116.00: z=+4.00000 p=0.99997 pPct=99.997 
Sx=117.00: z=+4.25000 p=0.99999 pPct=99.999 
Sx=118.00: z=+4.50000 p=1.00000 pPct=100.000
Sx=119.00: z=+4.75000 p=1.00000 pPct=100.000
Sx=120.00: z=+5.00000 p=1.00000 pPct=100.000

 

This would be difficult to do if there is only one or two price steps between the bid and ask. Maybe you could do this for less liquid instruments, where the gap between bid and ask is a (large) multiple of the minimum price step size.