Hey Max,

Thanks for this cool way to follow along with Das's paper!

I was with you right up until updating the bid-ask spread, and I was actually a bit confused with Das's explanation (what does he mean by "cycling down"?):

The fixed point equations 2.10 and 2.11 are approximately solved by using the result from Glosten and Milgrom that Pb ≤ E[V ] ≤ Pa and then, to find the bid price, for example, cycling from E[V] downwards until the difference between the left and right hand sides of the equation stops decreasing. The fixed point real-valued solution must then be closest to the integral value at which the distance between the two sides of the equation is minimized.

Reading eq. 2.10 the first time, I assumed that the P_b on the left hand side of the equation was the posterior whereas the P_b used in the Σ expression was the prior.

But after puzzling over his explanation of the solution, I think that P_b is the same on both sides of the equation, and that his proposed approximate solution is to

1. iterate over potential P_b values,
2. plug each into the equation, and compute the difference between (1) the value that you plugged in and (2) the result from the equation, and
3. choose the bid price that minimizes the magnitude of that difference.

So you'd go up and down a few ticks from the last bid price (down if your bid was hit, up if your quote was lifted) until minimizing that difference. I think what your implementation is doing is setting the bid price equal to the mean of how the probability distribution would be updated if the bid were hit again, and vice versa for the ask.

Does that make sense? I guess I'm not completely sure that I'm interpreting that correctly, but it did end up producing more realistic values for me.

A couple other things that I don't totally get:

• Why use volatility as σ instead of the standard deviation between ticks? This was confusing to work through with some real data because it happened in my case that annualized volatility of returns are in the same ballpark as the standard deviation between ticks! I tried checking for like three different instruments; no heuristic solution to this one for me haha
• After updateDensity I think the distribution needs to be "re-normalized" by making a second pass and diving the probability for each tick by the new integral for the curve, but it's totally possible that I'm missing something.

Btw his 2005 paper explores the same topic, and even includes whole sections verbatim from the 2003 paper, so for anybody else that may end up reading this, it's worth checking out as well.

Cheers!