Order Book Normalization Questions in Avellaneda–Cont Models

Summary

The document raises questions about how two order book models normalize initial bid and ask sizes. It compares the median-size normalization associated with an Avellaneda model to a square-root-of-count normalization in a Cont and de Larrard treatment, and asks how either relates to decile values used in analytical probability calculations.

It also asks how to interpret a proposition involving an infinite sum and Bessel functions, including which kind of Bessel function is intended and whether the series can be evaluated numerically. These are unresolved questions in the supplied text: it contains no derivation, answer, or numerical evidence. The material is useful as a map of technical issues in applying and reconciling the models, but does not establish that the normalizations are equivalent or provide implementation guidance.

Key ideas

• The document compares alternative normalizations of initial bid and ask queue sizes in two order book models.
• It asks whether median-based and square-root-based normalizations produce equivalent formulations.
• It distinguishes these normalizations from decile values used in analytical probability calculations.
• It raises questions about interpreting Bessel functions and evaluating an infinite series, without resolving them.

Tags

# Avellaneda/Cont model Order Book Model


# Avellaneda/Cont model Order Book Model












The model given in the following paper by Avellaneda et al

http://people.stern.nyu.edu/jreed/Papers/limitorder.pdf

On page 7 he explains that the initial Bid and Ask size should be normalised by their median size. This leads to equation 3.8 on p10.

The same equation was also derived by Cont and de LARRARD eqn 36 p30

http://arxiv.org/pdf/1202.6412v1.pdf

However when he normalises the initial Bid and Ask sizes he uses the sqrt of n see eqn 12 on p20?

How are these normalisations equivalent? They seem quite different yet lead to the same solution?

More confusingly when Avellaneda et al calculate the analytical probs on p14 only the decile values are used (e.g 0.1) which is different again from the two normalisations mentioned above?

Q1. Are the three normalisations equivalent?

Lastly what is the correct interpretation of the eqn shown in Proposition 2 on p27 of Cont and de LARRARD?

Q2.) Are the Bessel functions of the first or second kind? (First I think).

Q3.) Is the infinite sum shown an example of an infinite Bessel series?

If the answer to Q3 is yes can these be evaluated numerically for example in Matlab?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.