other anonymous reviewers for helpful comments. Ian Bandeen's many and detailed comments on the paper, coupled with lengthy discussions with the author, were particularly useful, and resulted in more substantive changes to the paper than can be adequately acknowledged in a short thank-you. 1 HF traders do not hold stock or other assets in which they trade for long; the duration of the average round trip -that is, the combined buy/sell or sell/buy transaction -is measured in seconds or milliseconds. 2 The proportion of HF trades as a proportion of all trades is sensitive to the market under scrutiny. HF traders participate, but differentially so, in both "lit" public markets and dark pools. In addition, HF traders such as Getco, ATD, Knight, and Citadel are significant "internalizers," purchasing order flow from retail brokers and internally matching buy-and-sell orders, rather than executing them over a stock exchange or other external trading venue. Although internalization is not technically HFT, it is an important artifact of market structure. In calculating the percentage of trades in which HF traders participate, one should take care to exclude crosses; these are big block trades that are negotiated outside of the lit market and merely reported on a public market. 3 See also "A head start for HFT; or a downward spiral?" FTSE Global Markets 66 (November 2012), available online at http://www.ftseglobalmarkets.com/issues/issue-66-november-2012/a-headstart-for-hft-or-a-downward-spiral.html. 4 Those most likely to be HF traders had two particular attributes: they were "fast," and they traded using direct market access; see IIROC (2011, 26).High frequency trading (HFT) -the use of extremely high speed computers and automated trading algorithms to trade high volumes of stock at lightning speed 1 -has transformed world capital markets, nowhere more completely, or more rapidly, than in the United States.