On the origin of power law tails in price fluctuations

In a recent Nature paper, Gabaix et al. \cite{Gabaix03} presented a theory to explain the power law tail of price fluctuations. The main points of their theory are that volume fluctuations, which have a power law tail with exponent roughly -1.5, are modulated by the average market impact function, which describes the response of prices to transactions. They argue that the average market impact function follows a square root law, which gives power law tails for prices with exponent roughly -3. We demonstrate that the long-memory nature of order flow invalidates their statistical analysis of market impact, and present a more careful analysis that properly takes this into account. This makes it clear that the functional form of the average market impact function varies from market to market, and in some cases from stock to stock. In fact, for both the London Stock Exchange and the New York Stock Exchange the average market impact function grows much slower than a square root law; this implies that the exponent for price fluctuations predicted by modulations of volume fluctuations is much too big. We find that for LSE stocks the distribution of transaction volumes does not even have a power law tail. This makes it clear that volume fluctuations do not determine the power law tail of price returns.