A simple theory of the ‘ ‘ cubic ’ ’ laws of stock market activity ∗

We show empirically a series of sharp patterns in stock market fluctuations, trading activity and their contemporaneous relationships. We link together and explain the following facts: (i) the cubic law of returns: returns follow a power law distribution with exponent 3. This “cubic” law seems to hold both across time and internationally. Stock market “crashes” (e.g. the 1929 and 1987 crashes) are not outliers to this law; (ii) the half cubic law of volumes: volumes follow a power law distribution with exponent 3/2; (iii) the square root law of price impact (the price impact of a volume V is proportional to V ); (iv) Zipf’s law for mutual funds: mutual funds size follow a power law distribution with exponent 1. The model also makes predictions about the crossconditional relationships between various trading variables. They all appear to be verified empirically. The model makes a series of other, out of sample, testable predictions. Finally, it shows that a Tobin tax, or circuit breakers, do not affect that size of extreme fluctuations. However, a tax that increases with the size of the transactions does reduce the magnitude of those very large fluctuations. ∗[email protected], [email protected], [email protected], [email protected]. We thank Alex Weisgerber, Tal Fishman, and especially Kirk Doran for outstanding research assistance, and Tobias Adrian, Olivier Blanchard, Joel Hasbrouck, Augustin Landier, Thomas Phillipon, Jon Reuter, Gideon Saar, Andrei Shleifer, Dimitri Vayanos, Jessica Wachter, Jiang Wang, seminar participants at Delta, John Hopkins University, MIT, Princeton for helpful comments.