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Lecture 21: Risk Neutral and Martingale Measure
Lecture 21: Risk Neutral and Martingale Measure

... 2. Modify to make sure that the discounted stock price process is a martingale - achieved by a change of measure 3. Other derivative prices (discounted) are also martingales: therefore a formula involving an expectation is obtained to price such a derivative. 4. This price is guaranteed to be arbitr ...
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Algorithmic trading

Algorithmic trading, also called algo trading and blackbox trading, encompasses trading systems that are heavily reliant on complex mathematical formulas and high-speed, computer programs to determine trading strategies. These strategies use electronic platforms to enter trading orders with an algorithm which executes pre-programmed trading instructions accounting for a variety of variables such as timing, price, and volume. Algorithmic trading is widely used by investment banks, pension funds, mutual funds, and other buy-side (investor-driven) institutional traders, to divide large trades into several smaller trades to manage market impact and risk.Algorithmic trading may be used in any investment strategy or trading strategy, including market making, inter-market spreading, arbitrage, or pure speculation (including trend following). The investment decision and implementation may be augmented at any stage with algorithmic support or may operate completely automatically.Many types of algorithmic or automated trading activities can be described as high-frequency trading (HFT), which is a specialized form of algorithmic trading characterized by high turnover and high order-to-trade ratios. As a result, in February 2012, the Commodity Futures Trading Commission (CFTC) formed a special working group that included academics and industry experts to advise the CFTC on how best to define HFT. HFT strategies utilize computers that make elaborate decisions to initiate orders based on information that is received electronically, before human traders are capable of processing the information they observe. Algorithmic trading and HFT have resulted in a dramatic change of the market microstructure, particularly in the way liquidity is provided.Profitability projections by the TABB Group, a financial services industry research firm, for the US equities HFT industry were US$1.3 billion before expenses for 2014, significantly down on the maximum of US$21 billion that the 300 securities firms and hedge funds that then specialized in this type of trading took in profits in 2008, which the authors had then called ""relatively small"" and ""surprisingly modest"" when compared to the market's overall trading volume. In March 2014, Virtu Financial, a high-frequency trading firm, reported that during five years the firm as a whole was profitable on 1,277 out of 1,278 trading days, losing money just one day, empirically demonstrating the law of large numbers benefit of trading thousands to millions of tiny, low-risk and low-edge trades every trading day.A third of all European Union and United States stock trades in 2006 were driven by automatic programs, or algorithms. As of 2009, studies suggested HFT firms accounted for 60-73% of all US equity trading volume, with that number falling to approximately 50% in 2012. In 2006, at the London Stock Exchange, over 40% of all orders were entered by algorithmic traders, with 60% predicted for 2007. American markets and European markets generally have a higher proportion of algorithmic trades than other markets, and estimates for 2008 range as high as an 80% proportion in some markets. Foreign exchange markets also have active algorithmic trading (about 25% of orders in 2006). Futures markets are considered fairly easy to integrate into algorithmic trading, with about 20% of options volume expected to be computer-generated by 2010. Bond markets are moving toward more access to algorithmic traders.Algorithmic trading and HFT have been the subject of much public debate since the U.S. Securities and Exchange Commission and the Commodity Futures Trading Commission said in reports that an algorithmic trade entered by a mutual fund company triggered a wave of selling that led to the 2010 Flash Crash. The same reports found HFT strategies may have contributed to subsequent volatility by rapidly pulling liquidity from the market. As a result of these events, the Dow Jones Industrial Average suffered its second largest intraday point swing ever to that date, though prices quickly recovered. (See List of largest daily changes in the Dow Jones Industrial Average.) A July, 2011 report by the International Organization of Securities Commissions (IOSCO), an international body of securities regulators, concluded that while ""algorithms and HFT technology have been used by market participants to manage their trading and risk, their usage was also clearly a contributing factor in the flash crash event of May 6, 2010."" However, other researchers have reached a different conclusion. One 2010 study found that HFT did not significantly alter trading inventory during the Flash Crash. Some algorithmic trading ahead of index fund rebalancing transfers profits from investors.
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