
Quite obviously, the typical financial mathematics & quantitative finance literature models equity prices with exponential Brownian motion.
But what about trading volume? I haven't found much papers on it. Wondering if anyone here ever tried to model trading volume as a stochastic process and/or statistical distribution.
I have a strong suspicion that trading volume is well suited for logNormal distribution much more so than modeling returns with logNormal. But what do I know.
Thanks! 



bullero


Total Posts: 62 
Joined: Feb 2018 

 
ronin


Total Posts: 591 
Joined: May 2006 


There is some reasonable understanding of how traded volume works.
Cumulative volume is strongly seasonal intraday. Google VWAP profile. It is also approximately linear in cumulative variance of the price process, i.e. the other way around.
Instantaneous volume is just not particularly interesting. If I am trying to do 100 lots, I can do it as one order for 100 lots, 100 orders for 1 lot each, and pretty much any other combination I can think of. And those orders can be filled in different ways depending on what is in the orderbook at the time. It does not make any difference to the economics of the transaction. The economics cares about how many lots get done, and at what price.
So the question "what is the distribution of traded quantitites" is irrelevant. The question "what is the process for cumulative traded quantity" is more interesting, but there is not a great deal to be said about it beyond the basics. And the basics are reasonably well understood. 
"There is a SIX am?"  Arthur 



Exactly!
It seems fairly straightforward (and almost obvious) that factors including recent price activity, other trading activity, cyclicality, and noncyclical seasonality (ie. what holidays are near coming up) tell you something about volume. I could see how such factors might even be used to predict future volume.
As you mentioned, I'm very much interested in the "underlying process" of trading volume. So far, I've only found 2 not so very wellknown papers that discuss this topic. Not sure if this is because modeling trading volume is not particularly a useful thing to do. 


bullero


Total Posts: 62 
Joined: Feb 2018 


There are two components that contribute to the "trade volume": (1) number of events (process), and (2) traded quantity per event (random variable) (...well, also (3) time). The intuition behind this is that traded volume may be large even if you have one large trade or a large number of smaller trades. The cumulative amount is a natural consequence of these two quantities plus time.
Since you are interested in the underlying (stochastic) process, the first order mathematical approximation could be a compound Poisson process. In compound Poisson the time interval between events (in your case, trades) is exponential rv. and the traded size is also some rv.
Now, in real life trading activity tends to exhibit self exciting property, clustering effect and long memory. More precisely, past activity tends to increase the trading activity in the future. In other words, the "Poisson" intensity in the compound Poisson that I mentioned previously is not a constant. However, it is not random either. Instead, it is sort of a decaying function of all the past events (trades). One possible way to model this phenomenon is to model the trade arrival process as a Hawkes process where the traded quantity is a random variable. Just Google it and you will find papers and code. Good luck. 



ronin


Total Posts: 591 
Joined: May 2006 


I have just googled "traded volume variance" and a bunch of papers come out. I have no idea if they are any good though.
But the basics are that cumulative volume is just a measure of time. If you count time in lots traded rather than physical time, a whole bunch of stuff simplifies a great deal.
Beyond that, what is there to say. It is a monotonically increasing process, highly correlated with variance, uncorrelated with the price. There is no particular reason why the increments would be lognormal. But they are always positive.

"There is a SIX am?"  Arthur 

nikol


Total Posts: 1143 
Joined: Jun 2005 


@ronin
"question "what is the distribution of traded quantitites" is irrelevant. The question "what is the process for cumulative traded quantity" is more interesting"
1 lot of size 100 is (local) monopoly for sure 100 lot of size 1 has some non zero probability of market diversity.
So, these convey different messages. 



ronin


Total Posts: 591 
Joined: May 2006 


> 1 lot of size 100 is (local) monopoly for sure
Eiher that, or a bookie aggregating 100 1lots. There is no information content, either way. 
"There is a SIX am?"  Arthur 
