Controlling the speed and direction of trades becomes increasingly important to algorithmic trading desks. Brian Bulthuis, Julio Concha, Tim Leung and Brian Ward propose a new optimal execution algorithm with both limit and market orders. An optimal strategy is derived analytically and illustrated numerically.
High-frequency trading (HFT) is a type of algorithmic financial trading characterized by high speeds, high turnover rates, and high order-to-trade ratios that leverages high-frequency financial data and electronic trading tools. While there is no single definition of HFT, among its key attributes are highly sophisticated algorithms, co-location, and very short-term investment horizons. HFT can be viewed as a primary form of algorithmic trading in finance. Specifically, it is the use of sophisticated technological tools and computer algorithms to rapidly trade securities. HFT uses proprietary trading strategies carried out by computers to move in and out of positions in seconds or fractions of a second.
A substantial body of research argues that HFT and electronic trading pose new types of challenges to the financial system. Algorithmic and high-frequency traders were both found to have contributed to volatility in the Flash Crash of May 6, 2010, when high-frequency liquidity providers rapidly withdrew from the market. Several European countries have proposed curtailing or banning HFT due to concerns about volatility.
The rapid-fire computer-based HFT developed gradually since 1983 after NASDAQ introduced a purely electronic form of trading. At the turn of the 21st century, HFT trades had an execution time of several seconds, whereas by 2010 this had decreased to milli- and even microseconds. Until recently, high-frequency trading was a little-known topic outside the financial sector, with an article published by the New York Times in July 2009 being one of the first to bring the subject to the public's attention.
The common types of high-frequency trading include several types of market-making, event arbitrage, statistical arbitrage, and latency arbitrage. Most high-frequency trading strategies are not fraudulent, but instead exploit minute deviations from market equilibrium.
Some high-frequency trading firms use market making as their primary strategy. Automated Trading Desk (ATD), which was bought by Citigroup in July 2007, has been an active market maker, accounting for about 6% of total volume on both the NASDAQ and the New York Stock Exchange. In May 2016, Citadel LLC bought assets of ATD from Citigroup. Building up market making strategies typically involves precise modeling of the target market microstructure together with stochastic control techniques.
Quote stuffing is a form of abusive market manipulation that has been employed by high-frequency traders (HFT) and is subject to disciplinary action. It involves quickly entering and withdrawing a large number of orders in an attempt to flood the market creating confusion in the market and trading opportunities for high-frequency traders. 
Filter trading is one of the more primitive high-frequency trading strategies that involves monitoring large amounts of stocks for significant or unusual price changes or volume activity. This includes trading on announcements, news, or other event criteria. Software would then generate a buy or sell order depending on the nature of the event being looked for.
Another set of high-frequency trading strategies are strategies that exploit predictable temporary deviations from stable statistical relationships among securities. Statistical arbitrage at high frequencies is actively used in all liquid securities, including equities, bonds, futures, foreign exchange, etc. Such strategies may also involve classical arbitrage strategies, such as covered interest rate parity in the foreign exchange market, which gives a relationship between the prices of a domestic bond, a bond denominated in a foreign currency, the spot price of the currency, and the price of a forward contract on the currency. High-frequency trading allows similar arbitrages using models of greater complexity involving many more than four securities.
The TABB Group estimates that annual aggregate profits of high-frequency arbitrage strategies exceeded US$21 billion in 2009, although the Purdue study estimates the profits for all high frequency trading were US$5 billion in 2009.
A separate, "naïve" class of high-frequency trading strategies relies exclusively on ultra-low latency direct market access technology. In these strategies, computer scientists rely on speed to gain minuscule advantages in arbitraging price discrepancies in some particular security trading simultaneously on disparate markets.
High-frequency trading strategies may use properties derived from market data feeds to identify orders that are posted at sub-optimal prices. Such orders may offer a profit to their counterparties that high-frequency traders can try to obtain. Examples of these features include the age of an order or the sizes of displayed orders. Tracking important order properties may also allow trading strategies to have a more accurate prediction of the future price of a security.
The effects of algorithmic and high-frequency trading are the subject of ongoing research. High frequency trading causes regulatory concerns as a contributor to market fragility.Regulators claim these practices contributed to volatility in the May 6, 2010 Flash Crash and find that risk controls are much less stringent for faster trades.
In September 2011, market data vendor Nanex LLC published a report stating the contrary. They looked at the amount of quote traffic compared to the value of trade transactions over 4 and half years and saw a 10-fold decrease in efficiency. Nanex's owner is an outspoken detractor of high-frequency trading. Many discussions about HFT focus solely on the frequency aspect of the algorithms and not on their decision-making logic (which is typically kept secret by the companies that develop them). This makes it difficult for observers to pre-identify market scenarios where HFT will dampen or amplify price fluctuations. The growing quote traffic compared to trade value could indicate that more firms are trying to profit from cross-market arbitrage techniques that do not add significant value through increased liquidity when measured globally.
The brief but dramatic stock market crash of May 6, 2010 was initially thought to have been caused by high-frequency trading. The Dow Jones Industrial Average plunged to its largest intraday point loss, but not percentage loss, in history, only to recover much of those losses within minutes.
In the aftermath of the crash, several organizations argued that high-frequency trading was not to blame, and may even have been a major factor in minimizing and partially reversing the Flash Crash. CME Group, a large futures exchange, stated that, insofar as stock index futures traded on CME Group were concerned, its investigation had found no support for the notion that high-frequency trading was related to the crash, and actually stated it had a market stabilizing effect.
However, after almost five months of investigations, the U.S. Securities and Exchange Commission (SEC) and the Commodity Futures Trading Commission (CFTC) issued a joint report identifying the cause that set off the sequence of events leading to the Flash Crash and concluding that the actions of high-frequency trading firms contributed to volatility during the crash.
High-frequency trading comprises many different types of algorithms. Various studies reported that certain types of market-making high-frequency trading reduces volatility and does not pose a systemic risk, and lowers transaction costs for retail investors, without impacting long term investors. Other studies, summarized in Aldridge, Krawciw, 2017 find that high-frequency trading strategies known as "aggressive" erode liquidity and cause volatility.
In a September 22, 2010 speech, SEC chairperson Mary Schapiro signaled that US authorities were considering the introduction of regulations targeted at HFT. She said, "high frequency trading firms have a tremendous capacity to affect the stability and integrity of the equity markets. Currently, however, high frequency trading firms are subject to very little in the way of obligations either to protect that stability by promoting reasonable price continuity in tough times, or to refrain from exacerbating price volatility." She proposed regulation that would require high-frequency traders to stay active in volatile markets. A later SEC chair Mary Jo White pushed back against claims that high-frequency traders have an inherent benefit in the markets. SEC associate director Gregg Berman suggested that the current debate over HFT lacks perspective. In an April 2014 speech, Berman argued: "It's much more than just the automation of quotes and cancels, in spite of the seemingly exclusive fixation on this topic by much of the media and various outspoken market pundits. (...) I worry that it may be too narrowly focused and myopic."
In March 2012, regulators fined Octeg LLC, the equities market-making unit of high-frequency trading firm Getco LLC, for $450,000. Octeg violated Nasdaq rules and failed to maintain proper supervision over its stock trading activities. The fine resulted from a request by Nasdaq OMX for regulators to investigate the activity at Octeg LLC from the day after the May 6, 2010 Flash Crash through the following December. Nasdaq determined the Getco subsidiary lacked reasonable oversight of its algo-driven high-frequency trading.
In September 2014, HFT firm Latour Trading LLC agreed to pay a SEC penalty of $16 million. Latour is a subsidiary of New York-based high-frequency trader Tower Research Capital LLC. According to the SEC's order, for at least two years Latour underestimated the amount of risk it was taking on with its trading activities. By using faulty calculations, Latour managed to buy and sell stocks without holding enough capital. At times, the Tower Research Capital subsidiary accounted for 9% of all U.S. stock trading. The SEC noted the case is the largest penalty for a violation of the net capital rule. 2b1af7f3a8