Monday, March 26, 2018

Classic Jacod (1994) Paper

J. Financial Econometrics will soon publish Jean Jacod's brilliant and beautiful 1994 paper, "Limit of Random Measures Associated with the Increments of a Brownian Semimartingale", which I just had the pleasure of reading for the first time. (Ungated version here.) Along with several others, I was asked to supply some comments for the issue's introduction. What follows is adapted from those comments, providing some historical background. (Except that it's not really historical background -- keep reading...)

Jacod's paper effectively lays the foundation for the vast subsequent econometric "realized volatility" (empirical quadratic variation) literature of the past twenty years.  Reading it leads me to recall my early realized volatility work with Torben Andersen and Tim Bollerslev in the late 1990's and early 2000's. It started in the mid-1990's at a meeting of the NBER Asset Pricing Program, where I was the discussant for a paper of theirs, eventually published as Andersen and Bollerslev (1998). They were using realized volatility as the "realization" in a study of GARCH volatility forecast accuracy, and my discussion was along the lines of, "That's interesting, but I think you've struck gold without realizing it -- why not skip the GARCH and instead simply characterize, model, and forecast realized volatility directly?".

So we decided to explore realized volatility directly. Things really took off with Andersen et al. (2001) and Andersen et al. (2003). The research program was primarily empirical, but of course we also wanted to advance the theoretical foundations. We knew some relevant stochastic integration theory, and we made progress culminating in Theorem 2 of Andersen et al. (2003). Around the same time, Ole Bardorff-Nielsen and Neil Shephard were also producing penetrating and closely-related results (most notably Barndorff-Nielsen and Shephard, 2002). Very exciting early times.

Now let's return to Jacod's 1994 paper, and consider it against the above historical background of early econometric realized volatility papers. Doing so reveals not only its elegance and generality, but also its prescience: It was written well before the "historical background"!! One wonders how it went unknown and unpublished for so long.

References

Andersen, T. G. and T. Bollerslev (1998), "Answering the Skeptics: Yes, Standard Volatility Models do Provide Accurate Forecasts," International Economic Review, 39, 885-905.

Andersen, T.G., T. Bollerslev, F.X. Diebold, and P. Labys (2001), "The Distribution of Realized Exchange Rate Volatility," Journal of the American Statistical Association, 96, 42-55.

Andersen, T.G., T. Bollerslev, F.X. Diebold, and P. Labys (2003), "Modeling and Forecasting Realized Volatility," Econometrica, 71, 579-625.

Barndorff-Nielsen, O. and N. Shephard (2002), "Econometric Analysis of Realized Volatility and its Use in Estimating Stochastic Volatility Models," Journal of the Royal Statistical Society, 64,
253-280.

Jacod, J. (1994), "Limit of Random Measures Associated with the Increments of a Brownian Semimartingale," Manuscript, Institute de Mathematiques de Jussieu, Universite Pierre et Marie Curie, Paris.