Calibrate Geometric Brownian Motion of trading volume time series

Let's say I'm modeling the trading volume of a stock price (e.g. Apple Inc.) to follow a Geometric Brownian Motion and I want to estimate the parameters (i.e. drift and volatility) using historical data. I am assuming that a GBM matches the time series pretty well. If I was modeling a stock price itself, I would calculate the drift and volatility by looking at the returns. However, applying this to the trading volume does not seem right to me since the trading volume, unlike a stock price, does not have any returns and if I am not looking at the returns the time series is non-stationary.

Q: How would I calculate the drift and volatility of such a historical time series to calibrate my GBM?

• If the dynamics of the trading volume do not match a GBM, why do not try a different process? Given the time-series of the trading volume, you could select an econometric model that fits the properties of you process. Jan 5 '21 at 16:38
• @alexbougias I didn't say that the time series does not match a GBM, in fact I assume that it matches a GBM pretty well. My question is only about the calculation of the drift and volatility based on historical data. To me it doesn't seem right to calculate the drift and volatility based on the returns since trading volume, unlike to a stock price, has no returns (you can't really invest in it and make a profit/loss). So again, I assume a GBM matches the time series, I just don't know how to calibrate it considering it is not a stock. Jan 6 '21 at 8:40
• If that is the case, then taking the returns would be ok. There is no economic rationale behind this, but a technical reason. You have to make your time series stationary first and then estimate the parameters. In econometrics, this is analogous to taking differences in the log-process to make the process I(1). Jan 6 '21 at 8:51
• @alexbougias That explanation makes sense. Just a follow up question since you mentioned the possibility that trading volume may not match a GBM, do you have a stochastic process in mind that I should look at? Note that it should be a stochastic process, an econometric model would go beyond the scope since the modeling of the trading volume is just a small piece of a rather complex model, so I don't want to develop a sub-model. Jan 6 '21 at 9:12
• I do not have a specific process, but my thoughts are that you should model the dependence between trading volume and stock returns simultaneously. There should be a work in the microstructure literature that explicitly models both variables. Jan 6 '21 at 12:18