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Is there a consensus on a formula for measuring the market depth of a book at a given point in time? Or a possible proxy for this measurement?

I see so many articles / people discussing market depth but no description on how they calculate it or measure it.

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  • $\begingroup$ The theoreticians use "Kyle's Lambda" (a single number), the BTC traders use the Market Depth Diagram (a 2-dimensional representation) bitcoincharts.com/charts/bitstampUSD/… . The (absolute value of) the slope of the two sides of the chart is essentially $\frac{1}{\lambda}$ $\endgroup$
    – nbbo2
    Jun 9, 2017 at 14:14
  • $\begingroup$ I will look into Kyle's lambda as I am looking for a quantitative way to compare one time to another (rather than a heuristic diagram) $\endgroup$
    – guy
    Jun 9, 2017 at 14:15
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    $\begingroup$ I don't know what you're trying to do, but you might also search for liquidity proxies. Eg. this paper has some discussion. $\endgroup$
    – Matthew Gunn
    Jun 9, 2017 at 19:56
  • $\begingroup$ I want to compare a trade of size X from one moment in time to another in terms of market impact so i thought normalizing the volume trading at a given time by market depth would be a good way to do this. $\endgroup$
    – guy
    Jun 10, 2017 at 1:00

1 Answer 1

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There is no official definition of market depth (this is only a qualitative concept), only the cost of a roundtrip for a given number of shares of contracts. Take $V$ shares, on average, knowing the shape of the book at time $t$, what is the cost of buying and selling them immediately? You obtain a cost $C(V,t)$. Then you need to average or to choose an adequate $V$ or an adequate $t$.

  • for $t$ there is no reason to choose a specific time, but you can either average on seconds ${\cal T}=(t_1, \ldots, t_K)$ or according to the intraday volume seasonality ${\cal B}=(\tau_1, \ldots,\tau_N)$ (more volume the morning and at the end of the day)
  • for $V$ you can take only a typical volume of interest for you (if you know you will always split your metaorder in chunks of a given size $V^*$)
  • otherwise you can use the average trade size of a day $\bar V$,
  • or you can reuse a typical distribution of trade volumes of the day ${\cal V}=(v_1,\ldots,v_L)$. Be careful when I speak about trades, I have in mind market orders you need to aggregate several trades.

You end up with different measures of market depth, like $$\mathbb{E}(C(V,t)|V\in {\cal V}, t\in {\cal T}),$$ or $$\mathbb{E}(C(V^*,t)|t\in {\cal B}).$$

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    $\begingroup$ "There is not market depth" ? What do you mean? This is what I am referring to. $\endgroup$
    – guy
    Jun 10, 2017 at 22:27
  • $\begingroup$ I came across this paper that proposes the measure of the Bid-Ask volume ratio at a particular price level as $ W = \log \frac{\sum \exp(V^b)}{\sum \exp(V^a)}$ where $V^b$ is the volume on the bid side and $V^a$ is the volume on the ask side at the time instant just before a trade comes in. $\endgroup$
    – guy
    Jun 11, 2017 at 2:41
  • $\begingroup$ @tbone I mean "there is no official measure of market depth", this is a qualitative definition (I edited my answer). The closest measure is the one I give (the roundtrip), and I add in my answer this idea of averaging. $\endgroup$
    – lehalle
    Jun 11, 2017 at 8:09
  • $\begingroup$ @tbone the ZMA paper you talk about is your second comment is about the imbalance that is different from depth. See for instance this one too: arxiv.org/abs/1610.00261 $\endgroup$
    – lehalle
    Jun 11, 2017 at 8:12
  • $\begingroup$ @lehalle Would you please clarify what you mean when taking an expectation here? Which part is random? It seems that if I placed an order at time $t$, with volume $V$, the outcome is completely deterministic, the amount that it cost at that time with that volume. I'm also confused about what looks like conditional probability here. For example, is $V$ random and you're defining the event $V\in {\cal V}$ for the conditional probability?. I come from an engineering background and am having trouble understanding either the notation, concepts or a bit of both. $\endgroup$
    – Colin Hicks
    Mar 8, 2021 at 23:23

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