Here is a list of Quant Books and Research Papers on Volume-time and Volatility
Is there an empirical or theoretical study, ideally across different asset classes, of the relationship between s_t, the volatility of a given security at (or around) time t, and v_t, the total $ volume of the given security traded in the market at (or around) time t?
Here are a few related studies:
• Ané, T., and H. Geman (2000): “Order Flow, Transaction Clock and Normality of Asset Returns”, Journal of Finance, 55: 2259–2284.
• Clark, P. K. (1970): “A Subordinated Stochastic Process Model of Cotton Futures Prices”, unpublished Ph.D. dissertation, Harvard University, May.
• Clark, P. K. (1973): “A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices”, Econometrica, 41 (1): 135-155.
• Mandelbrot, B., and M. Taylor (1967): “On the Distribution of Stock Price Differences”, Operations Research, 15 (6): 1057-1062.
• Mandelbrot, B. (1973): “Comments on ‘A subordinated stochastic process model with finite variance for speculative prices by Peter K. Clark’”, Econometrica, 41 (1): 157-159.
Here is how Mandelbrot and Taylor  saw it:
“Price changes over a fixed number of transactions may have a Gaussian distribution. Price changes over a fixed time period may follow a stable Paretian distribution, whose variance is infinite. Since the number of transactions in any time period is random, the above statements are not necessarily in disagreement. […] Basically, our point is this: the Gaussian random walk as applied to transactions is compatible with a symmetric stable Paretian random walk as applied to fixed time intervals.”
Hiemstra and Jones (Journal of Finance, 1994) did a generalized causality study, T. Andersen (Journal of Finance, 1996) used a microstructure framewok. More recently, Giot and Laurent (Journal of Empirical Finance, 2010) used a realized vol/jump component approach.
it’s not exactly on-point, but might be handy if you’re creating a framework, the author gave the paper at last year’s sofie conference in chicago:
Stochastic subordination and time change are classical theoretical machinery, per literature cited by Marcos.
Numerous time / clock formalisms have been proposed and empirical evaluated; for example: Mendelbrot with “trading time”; Dacorogna et al. (2001) with “upsilon time”; and Derman with “intrinsic time” (2002).
Literature on this topic continues to be active; for example, see references in McCulloch (2011).
Gallant Rossi Tauchen, RFS 1993, volatility and volume for the S&P500
Also: George Tauchen, Harold Zhangb, Ming Liua; Volume, volatility, and leverage: A dynamic analysis; Journal of Econometrics Volume 74, Issue 1, September 1996, Pages 177–208
The paper recommended by Thierry Michel [Trading activity, realized volatility and jumps (Giot, Laurent, Petitjean, 2010, Journal of Empirical Finance 17, 168-175] includes lieterature review and can be found here http://www.core.ucl.ac.be/~laurent/pdf/Petitjean.pdf
George Tauchen, Harold Zhangb, Ming Liua; Volume, volatility, and leverage: A dynamic analysis; Journal of Econometrics Volume 74, Issue 1, September 1996, Pages 177–208 can be found here:
It uses NONLINEAR, non-parameteric impulse response analysis to investigate the relationship between Volatility, Volume and the Leverage Effect. The nonlinearity is explored through perturbation analysis (so it is dealing with small nonlinearities).
the following are detailed citation for references in above comment:
Dacorogna, Michel et al. (2001), An Introduction to High-Frequency Finance. http://books.google.com/books?id=dobO95EBcqsC
Derman, Emanuel (2002), “The perception of time, risk and return during periods of speculation”. http://www.ederman.com/new/docs/qf-market_bubbles.pdf
Mandelbrot, B. and Taylor, H. (1967), “On the Distribution of Stock Price Diﬀerences”, Operations Research 15, 1057-1062. http://www.jstor.org/stable/168611
McCulloch, James (2011), “Fractal Market Time”. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1803888
I now post my TRADING ALERTS
into my personal FACEBOOK ACCOUNT
. Don't worry as I don't post stupid cat videos or what I eat!