Methods and systems related to securities trading

[0001] The present application claims the benefit of U.S. Provisional Patent Application No. 61/355,398, filed Jun. 16, 2010, and is a CIP of U.S. patent application Ser. No. 13/083,711, filed Apr. 11, 2011, which claims priority to U.S. Provisional Patent Application No. 61/322,637, filed Apr. 9, 2010. The entire contents of the above-listed applications are incorporated by reference in their entirety into the present disclosure.

[0002] As an increasing number of traders turn to post-trade transaction cost analysis (TCA) to measure the quality of their executions, there has been an explosion in the number of TCA product offerings within the financial services sector. However, to date, all of these TCA products have focused on an “after-the-fact” analysis of various measurements of the trading costs that can be attributed to a particular trade, group of trades, a trader, a firm, etc. In addition, historical TCA offerings have lacked sophistication, having been largely limited to benchmark comparisons with large groups of trades based on fairly generic criteria (for example, breaking down averages into buckets by trade size or listing). While these broad and generic comparisons are very common, more often than not they are at best not helpful and at worst counter-productive, as illustrated by the following examples:

• • [0003] The variation in implementation shortfall (IS) performance of different traders on a desk is dominated by differences in their order flow. In contrast, the VWAP benchmark creates arbitrary incentives: it encourages risk-averse traders to spread smaller trades over the day regardless of the urgency of each trade, and for large trades creates an incentive to front-load the execution profile or use buying power to defend price levels to “game” the benchmark. Both practices increase average shortfalls.
  • [0004] Evaluating algorithms based on IS favors algorithms that tend to be used with a tight limit (and therefore can only execute if the market is favorable); paradoxically, the use of tight limits is most common for less-trusted, aggressive algorithms where the trader feels the need for the limit as a safety protection. Vice-versa, the best and most trusted algorithms that traders prefer to use for difficult non-discretionary market orders will never end up at the top of an IS ranking in universe comparisons.
  • [0005] Negotiated block crossing networks have zero shortfalls by definition but leave the trade unfilled when there is no natural contra; opportunistic algorithms and “aggressive in the money” strategies benefit from a similar selection bias. The practice of selectively executing the trades for which a natural contra is available is a great way to win a place at the top of broker rankings.
  • [0006] Universe comparisons of institutional managers promotes the practice of canceling orders in the most difficult trades where the stock is running away, or increasing the size of easier trades. In cases where trade-day performance is correlated to long-term residual alpha, this practice damages the fund's information ratio.

[0007] And while TCA based on these ineffective and generic comparisons has become the norm, it is fundamentally limited in its scope because at its foundation it is a static, “backwards-looking,” and often highly generic, not to mention one-dimensional, assessment of cost. As a result, even though traders are constantly required to make numerous decisions and weigh countless variables, all of which will have a dramatic impact on the quality and cost of an execution, this very basic and generic foundation of traditional TCA has neither accounted for all of these variables and decisions nor has it offered any tools that allow traders to better assess the impact of their choices or to understand the effect of their decisions in different situations.

[0008] For example, on a daily basis, traders need to choose between aggressive and opportunistic (less aggressive) trading strategies. Using the right limit price in relation to the information in a trade can enhance execution quality by selecting execution price points that are more attractive to a given target. Using more patient trading strategies can help reduce impact costs. But if the limit price or speed selection is too passive, it will delay the execution and result in substantial delays and opportunity costs. Other important variables include but are not limited to the optimal choice of algorithm, trading speed, trading venue, order size, time of order entry, time in force etc. Yet while the calculation behind traditional TCA products may penalize a trader for making the wrong choices in any of these selections, they offer no method for understanding the impact of a given choice or suggesting what would have been a better choice.

[0009] Furthermore, as explained in greater detail below, because traditional TCA products have neither leveraged the use of predictive analytics nor taken into account an analysis of what the market in a stock would have looked like had the trade or trades in question not occurred (e.g., whether the observed price movements were due to the trading activity associated with the trade in question or to exogenous market events), these static TCA offerings based on generic benchmark comparisons are often unhelpful if not counter-productive.

[0010] More specifically, the compromise between impact and opportunity costs requires an understanding of the urgency of the trade, or “short-term alpha.” Post-trade analysis too often defines short-term alpha as the average realized returns from the start of trading, ignoring the fact that a large part of this return is caused by market impact from the trade in question. A trader who believes his orders have high urgency will tend to trade aggressively, which causes more impact and therefore reinforces the perception of short-term alpha. The urgency of a trade depends ultimately on the stock's expected performance without executing the trade—or the impact free price estimation.

[0011] What has been lacking in the prior art are one or more TCA tools that can help traders understand the costs associated with their trades in a way that decomposes the various components responsible for trade execution outcome, included but not limited to estimating the components of implementation shortfall, which includes but is not necessarily limited to alpha loss, an algorithm's impact, adverse selection and opportunistic savings; as well as the trade-offs associated with the speed of execution, participation rates and limit prices. At least some of such tools would preferably also be capable of adjusting (and reflecting this adjustment) the realized returns for the estimated impact of the execution in question. This adjustment is preferred in order to identify the correct compromise between impact and opportunity, because making the correct inferences entails being able to identify how the price of the stock would have behaved if no trade had taken place. Although not always required, if this step is not taken, the results of the analysis may be spurious, making one believe that the orders require more urgent execution than they actually do. One or more of these tools may also offer the ability to accurately estimate the hypothetical cost that would have been incurred had a different strategy been chosen.

[0012] By way of a specific example that shows how one or more exemplary embodiments provide improvements over the prior art, FIG. 36 shows an example of how to decide the optimal trading speed: is it the 20% participation rate that the customer has chosen or an alternative 10% participation rate? The y axis of FIG. 36 is Profit/(Loss) in basis points. The x axis is time. The customer chose a 20% participation rate, and one observes the P/(L) of 20%. The customer has a loss of 15 bps.

[0013] Would the customer have had a lower loss if he had picked a 10% participation rate? To answer that question entails simulating the P/(L) the customer would have gotten if the customer had picked the 10% participation rate.

[0014] And for that one may:

[0015] i) take the observed prices (curve with the triangles);

[0016] ii) subtract from observed prices the impact of the execution at 20% to see what the impact-free price is (see dotted curve for Alpha Loss);

[0017] iii) calculate the average impact-free price for the execution at 10% (still on the dotted curve for Alpha Loss, but going further to the right in time because an execution at 10% takes more time than an execution at 20%);

[0018] iv) to get the P/(L) at 10% one then needs to add to the impact-free price the impact of the execution at 10%

[0019] If one did not take impact into account, the only thing that one would notice is that 10% takes more time, and if the stock moves away the customer will incur more losses. If the impact is not taken into account, one will not see how much is saved in impact by lowering the speed. Indeed, in this particular case of FIG. 36, what the customer lost in terms of impact is more than compensated for by what he gains by getting the order done earlier. But the size of the cost and benefits could have been different and the only way to know is by calculating both.

[0020] Or, in other words, transaction cost analysis is based on historical data in which what is observed is to some extent affected by the customer's strategies. To make a good assessment of alternative strategies, one may wish to first subtract out the impact of those strategies to then be able to simulate accurately alternative strategies. This applies not only to speed and participation rate analysis but also to limit price analysis.

[0021] And finally, in addition to the improvements noted above, one or more exemplary embodiments described herein may also offer the ability to “mine” and analyze historical execution data in a way that actually helps traders interpret their execution data in a manner that is useful for future trade decisions.

[0022] Therefore, in order to address these and other limitations of existing TCA products, one or more exemplary embodiments change traditional transaction cost analysis from a static, backward-looking and generic benchmark comparison to a customized, interpretive and dynamic analysis that can analyze and decompose past trades in a way that reflects the range of variables that drive execution outcomes, educate traders as to how their decisions impacted execution outcome, offer specific guidance on how past trades and/or past trade-related decisions could have been improved, and, in some embodiments, using this analysis can either suggest or assign optimal trading strategies for new orders.

[0023] Certain exemplary embodiments not only analyze and decompose various components of the execution outcomes associated with a given trade (see, for example, the section herein entitled “Implementation Shortfall Decomposition for Market Orders), but also mine and analyze trade execution data in order to identify characteristics within the orders/trades being analyzed in the form of “trade profiles” that can then be used to classify and manage new orders.

[0024] In addition, certain exemplary embodiments also offer the ability to determine what would have been the most appropriate speed(s) and limit price(s) for a particular trade or trades through evaluations of the choices that were made and simulations of alternative choices that could have been made (for examples, see the sections herein entitled “Implementation Shortfall Decomposition for Market Orders,” “Exemplary Analysis of Trade Profile,” and “Exemplary Report Regarding Trade Profile and Execution Performance,” in addition to FIGS. 44-56).

[0025] Certain embodiments also use this dynamic historical analysis and the identification of trade profiles in combination with the analysis of optimal execution speed and limit prices to aid in the identification of optimal trading strategies for new orders. Some embodiments may be used to automatically allocate trading strategies to trade profiles and carry out an execution plan (see, for example, U.S. patent application Ser. No. 13/070,852, filed Mar. 24, 2011, and incorporated herein by reference).

[0026] In certain exemplary embodiments, incoming orders are analyzed in light of current market conditions and historical patterns identified through the types of analysis described above, factors most likely to predict impact-free price movement are identified, and an “alpha profile” is assigned to the order. To generate this alpha profile in real-time, one or more exemplary embodiments analyze many (typically, hundreds of) drivers coming from both fundamentals and technical information. These drivers include but are not limited to how the market reacts to news, momentum since the open, overnight gaps, and how a stock is trading relative to the sector. Taking these statistical factors into account, the service then recommends a trading strategy predicted to maximize alpha capture and minimize adverse selection. The trader can then manually initiate trading or can enable the system to automatically initiate the execution of the trade using the strategy recommendation.

[0027] As the trade proceeds, the analytics stream provides updates to one or more users on changes in the alpha outlook, leveraging market feedback from the execution process and feeds including news and real-time order flow analysis. Furthermore, the system constantly looks at differences between the results predicted by the strategy and the actual results. For example, impact anomaly is the difference between actual return and expected return given a pre-trade model. One or more exemplary embodiments of the system may also use predictive switching between algorithms and across venues to minimize implementation shortfall and find liquidity as the trade proceeds.

[0028] In addition, one or more exemplary embodiments of the systems and methods described herein give traders an unprecedented amount of information and control regarding the process and operation of the subject system. Most execution platforms on the market today operate as “black boxes”: a trader enters an order, but he is given little to no information about how the system processes the order or how it is being executed. While this may protect the trade secrets that drive the operation of the “black box,” traders do not like this lack of information and control. Traders want to understand what is going on with the market and their order, especially if the opinion they form about an order differs from the classification produced by a quantitative system. For example, a black box might tell a trader that a given trade is a high urgency trade, but then does not tell him why. Without knowing why the black box is recommending urgency, it is impossible for the trader to understand how to incorporate the system's information into his own thinking in order to take control of the execution.

[0029] In order to reconcile these kinds of differences and to have confidence in a “black box,” a trader needs to be able to see the factors that drive the quantitative analysis to reconcile the two viewpoints. To address these limitations in the prior art, one or more exemplary embodiments of the subject system employs a graphical interface that provides users with unique insight into the factors behind the strategy assignment and selection for a given order. Then once the system begins to execute an order, an interface may also be used to allow traders to maintain unprecedented control over the system's automated trade execution. At any time, a trader may change speeds, grab a block, or change strategies, thereby allowing him to modulate the system's recommended strategy and actual order executions with his knowledge of factors driving optimal execution strategy.

[0030] One exemplary aspect comprises a method comprising: (a) receiving electronic data describing an executed trading order in a market traded security and related trade execution data; (b) calculating with a processing system an impact-free price estimate which estimates a price of the market traded security if the executed trading order had not been executed, wherein the impact-free price estimate is based in part on the data describing the executed trading order; and (c) transmitting data sufficient to describe the impact-free price estimate; wherein the processing system comprises one or more processors.

[0031] In one or more exemplary embodiments: (1) the related trade execution data comprises a list of executed fills and partial fills; (2) the list of executed fills and partial fills comprises symbol, price, and time of each fill or partial fill; (3) the calculating with a processing system an impact-free price estimate comprises comparing each fill and partial fill to one or more prevailing market quotes at the times of the fills and partial fills; (4) the method furtherr comprises classifying each fill or partial fill as a passive, aggressive, or intra-spread execution; (5) the calculating with a processing system an impact-free price estimate comprises calculating an estimated accrued price impact based on estimating price impact accrued to a tape transaction time for each fill or partial fill; (6) the calculating with a processing system an impact-free price estimate comprises calculating, for a buy trade, a difference between an observed price and an estimated accrued price impact; (7) the calculating with a processing system an impact-free price estimate comprises calculating, for a sell trade, a sum of an observed price and an estimated accrued price impact; and (8) the data sufficient to describe the impact-free price estimate comprises data sufficient to display one or more actual market prices and corresponding impact-free price estimates.

[0032] At least one other exemplary aspect comprises a method comprising: (a) receiving electronic data describing an executed trading order and related trade execution data; (b) calculating with a processing system an impact-free price estimate which estimates an execution price of the executed trading order if the executed trading order had been executed without market impact, wherein the impact-free price estimate is based in part on the data describing the executed trading order; and (c) transmitting data sufficient to describe the impact-free price estimate; wherein the processing system comprises one or more processors.

[0033] At least one other exemplary aspect comprises a method comprising: (a) receiving electronic data describing an executed trading order and related trade execution data; (b) calculating with a processing system an expected execution price of the executed trading order if the executed trading order had been executed using a specified algorithm, wherein the expected execution price is the sum of: (1) an impact-free price estimate based in part on the data describing the executed trading order, and (2) estimated market impact given the specified algorithm; and (c) transmitting data sufficient to describe the impact-free price estimate; wherein the processing system comprises one or more processors.

[0034] At least one other exemplary aspect comprises a method comprising: (a) receiving electronic data describing a potential trading order in a market traded security and related market data; (b) calculating with a processing system an impact-free price estimate which estimates a price of the market traded security if the potential trading order were not to be executed, wherein the impact-free price estimate is based in part on the data describing the potential trading order; and (c) transmitting data sufficient to describe the impact-free price estimate; wherein the processing system comprises one or more processors.

[0035] In one or more exemplary embodiments: (1) the data sufficient to describe the impact-free price estimate comprises an alpha profile; (2) calculating the impact-free price estimate comprises calculation of an average market impact; (3) calculating the impact-free price estimate comprises calculation of incremental impact from fills of portions of the potential trading order; (4) calculating the impact-free price estimate comprises calculation of incremental impact from fills of portions of the potential trading order that take place in specified time segments, and reversion from activity in prior time segments; (5) calculating the impact-free price estimate comprises calculation of net results of price effects of trading portions of the potential trading order during specified time segments; (6) calculating the impact-free price estimate comprises calculation of accrued price impacts for the potential trading order during specified time intervals; (7) calculating the impact-free price estimate when the potential trading order is a buy order comprises subtraction of an accrued price impact from a fill price for each time interval that contains that fill; and (8) calculating the impact-free price estimate when the potential trading order is a sell order comprises addition of an accrued price impact to a fill price for each time interval that contains that fill.

[0036] At least one other exemplary aspect comprises a method comprising: (a) receiving electronic data describing a potential trading order and related market data; (b) calculating with a processing system an impact-free price estimate which estimates an execution price of the potential trading order if the potential trading order were to be executed without market impact, wherein the impact-free price estimate is based in part on the data describing the potential trading order; and (c) transmitting data sufficient to describe the impact-free price estimate; wherein the processing system comprises one or more processors.

[0037] At least one other exemplary aspect comprises a method comprising: (a) receiving electronic data describing a potential trading order and related market data; (b) calculating with a processing system an expected execution price of the potential trading order if the potential trading order were to be executed using a specified algorithm, wherein the potential execution price is the sum of: (1) an impact-free price estimate based in part on the data describing the potential trading order, and (2) estimated market impact given the specified algorithm; and (c) transmitting data sufficient to describe the impact-free price estimate; wherein the processing system comprises one or more processors.

[0038] At least one other exemplary aspect comprises a method comprising: (a) receiving electronic data describing a potential trading order and related market data; (b) calculating with a processing system an expected execution price of the potential trading order if the potential trading order were to be executed using a specified algorithm, wherein the potential execution price is the sum of: (i) an impact-free price estimate based in part on the data describing the potential trading order, and (ii) estimated market impact given the specified algorithm; and (c) commencing execution of the potential trading order using the specified trading algorithm; wherein the processing system comprises one or more processors.

[0039] At least one other exemplary aspect comprises a method comprising: (a) receiving electronic data describing an executed trading order in a market traded security and related trade execution data; (b) calculating with a processing system one or more components of execution costs associated with execution of the executed trading order; and (c) transmitting data sufficient to describe the one or more components of execution costs associated with execution of the executed trading order; wherein the processing system comprises one or more processors.

[0040] In one or more exemplary embodiments: (1) the data sufficient to describe the one or more components of execution costs is received by a user terminal and displayed via a graphical user interface; (2) the graphical user interface displays the data sufficient to describe the one or more components of execution costs in a format that shows values of one or more of the components; (3) the graphical user interface displays the data sufficient to describe the one or more components of execution costs in a format that shows relative values of two or more of the components; (4) the one or more components of execution costs associated with execution of the executed trading order comprise alpha loss; (5) the one or more components of execution costs associated with execution of the executed trading order comprise market impact; (6) the one or more components of execution costs associated with execution of the executed trading order comprise alpha capture; (7) the one or more components of execution costs associated with execution of the executed trading order comprise adverse selection; (8) the one or more components of execution costs associated with execution of the executed trading order comprise opportunistic savings; (9) the one or more components of execution costs associated with execution of the executed trading order comprise speed impact; (10) the one or more components of execution costs associated with execution of the executed trading order comprise limit savings; (11) the one or more components of execution costs associated with execution of the executed trading order comprise opportunity cost; (12) the one or more components of execution costs associated with execution of the executed trading order comprise spread; and (13) the one or more components of execution costs associated with execution of the executed trading order comprise profit/loss.

[0041] At least one other exemplary aspect comprises a method comprising: (a) receiving electronic data describing an executed trading order in a market traded security and related trade execution data; (b) calculating with a processing system cost effect of one or more trade decision factors associated with execution of the executed trading order; and (c) transmitting data sufficient to describe the cost effect of one or more trade decision factors associated with execution of the executed trading order; wherein the processing system comprises one or more processors.

[0042] In one or more exemplary embodiments: (1) the one or more trade decision factors associated with execution of the executed trading order comprise limit price; (2) the one or more trade decision factors associated with execution of the executed trading order comprise choice of algorithm; (3) the one or more trade decision factors associated with execution of the executed trading order comprise level of aggression; (4) the one or more trade decision factors associated with execution of the executed trading order comprise choice of broker; (5) the one or more trade decision factors associated with execution of the executed trading order comprise participation rate; (6) the one or more trade decision factors associated with execution of the executed trading order comprise speed of execution; (7) the one or more trade decision factors associated with execution of the executed trading order comprise choice of manual versus automated execution; and (8) the one or more trade decision factors associated with execution of the executed trading order comprise choice of trading strategy.

[0043] At least one other exemplary aspect comprises a method comprising: (a) receiving electronic data describing an executed order in a market traded security and related trade execution data; (b) calculating with a processing system a decomposition of execution of the executed limit order into at least a first group of components and a second group of components, the first group of components being related to algorithm performance and the second group of components being related to trader-input parameters for the executed order; and (c) transmitting data sufficient to describe the first group of components and the second group of components; wherein the processing system comprises one or more processors.

[0044] In one or more exemplary embodiments: (1) the trader-input parameters comprise level of aggression parameters; (2) the trader-input parameters comprise limit order parameters; (3) the trader-input parameters comprise trading speed; (4) the trader-input parameters comprise participation rate; (5) the data sufficient to describe the first group of components and the second group of components is received by a user terminal and displayed via a graphical user interface; (6) the graphical user interface displays the data describing the first group of components and the second group of components in a format that shows relative values of two or more components in the first group of components and the second group of components; (7) the graphical user interface displays relative values of choices of participation rate and limit price; (8) the graphical user interface displays relative values of selected speed and benchmark speed level; (9) the graphical user interface displays relative values of market impact and alpha capture; and (10) the graphical user interface displays relative values of limit price savings and opportunity costs.

[0045] At least one other exemplary aspect comprises a method comprising: (a) receiving electronic data describing an executed trading order in a market traded security and related trade execution data; (b) calculating with a processing system one or more components of implementation shortfall associated with execution of the executed trading order; and (c) transmitting data sufficient to describe the one or more components of implementation shortfall associated with execution of the executed trading order; wherein the processing system comprises one or more processors.

[0046] At least one other exemplary aspect comprises a method comprising: (a) receiving electronic data describing an executed trading order in a market traded security and related trade execution data; (b) calculating with a processing system one or more components of profit/loss associated with execution of the executed trading order; and (c) transmitting data sufficient to describe the one or more components of profit/loss associated with execution of the executed trading order; wherein the processing system comprises one or more processors.

[0047] At least one other exemplary aspect comprises a method comprising: (a) receiving electronic data describing an executed trading order in a market traded security and related trade execution data; (b) calculating with a processing system one or more components of execution outcome associated with execution of the executed trading order; and (c) transmitting data sufficient to describe the one or more components of execution outcome associated with execution of the executed trading order; wherein the processing system comprises one or more processors.

[0048] Other aspects and embodiments comprise related computer systems and software, as will be understood by those skilled in the art after reviewing the present description.

[0089] At least one exemplary embodiment comprises a system and method for calculation, application, and display of an “impact free” price estimation on an executed trade or group of trades for the purposes of analyzing/judging the quality of the executed trade(s) and/or making and/or aiding in the selection of a trading strategy for a new trade order(s).

[0090] In an exemplary embodiment, a user provides data for an impact-free price estimation, including a list of executed partial fills giving the symbol, price, and time of every fill. This data may be automatically loaded from a database or spreadsheet by selecting the trade from a drop list. Tools known in the art for identifying the relevant trade from a set of trades (searching by date, symbol, etc.) may be provided for ease of use.

[0091] Having selected a trade of interest, the user may request from the system an impact-free price estimation using action buttons known in the art of user interface design. In a subsequent exemplary step, the system may compare each partial fill to prevailing market quotes at the time of the fill to classify each fill, distinguishing passive executions (buy on the bid or sell on the offer), aggressive executions (buy on the offer or sell on the bid) and intra-spread executions such as midpoint fills from dark pools.

[0092] Exemplary algorithms that may be used for this classification are known in the art. See, for example, “Imbalance Vector and Price Returns,” Nataliya Bershova, Christopher R. Stephens, and H. Waelbroeck, (paper submitted to J. Financial Markets; copy included in U.S. Prov. App. No. 61/322,637, which is incorporated herein by reference), and “Relating Market Impact to Aggregate Order Flow: The Role of Supply and Demand in Explaining Concavity and Order Flow Dynamics,” Christopher R. Stephens, Henri Waelbroeck, Alejandro Mendoza (dated Nov. 20, 2009, and posted to the Social Science Research Network working paper series website (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1511205) on Nov. 22, 2009; copy included in U.S. Prov. App. No. 61/322,637, which is incorporated herein by reference).

[0093] In a third exemplary step, the system may estimate price impact accrued to the time of each tape transaction from the classified fills data and the tape data, as will be described in further detail below.

[0094] In a fourth exemplary step, the system may estimate corrected market prices that would have been observed had the price impact not existed, proceeding as follows. For a buy trade, the impact-free price associated with each market transaction is the difference between the observed price and the estimated accrued price impact; for a sell trade, the impact-free price is the sum of the observed price and the estimated accrued impact.

[0095] In a fifth exemplary step, the system displays to the user the actual market prices and the hypothetical impact-free prices that would have occurred had the trade not been executed. In an exemplary embodiment these price series may be represented graphically using charts with time on the x-axis and price on the y-axis, as is known in the art. See below for a description of an exemplary system display to a user, where the term “alpha” is used to represent and refer to the calculation of impact free price.

[0096] An exemplary embodiment further enables a user to select a group of trades, such as, for example, all trades executed in May whose size was between 1% and 5% of the average daily volume in the stock, using forms known in the art of user interface design.

[0097] Having selected a set of trades, the user is enabled to request an average impact-free return estimation regarding the impact-free price for this set of trades.

[0098] In this step, the system may estimate impact free prices for each trade individually as described herein, then perform the additional step of estimating the average impact free price as follows.

[0099] In step “4a” the impact-free price returns of each stock from the midpoint at the start of the trade may be calculated in the following manner: for each print, the return (in basis points) is defined as 10,000 times the natural logarithm of the ratio of the impact-free price of this print to the starting price, multiplying by (−1) for sell trades.

[0100] In step “4b” the average and standard deviation of these returns may be calculated, as the average and standard deviation of the return at the first print following the end of minutely time periods counted from each trade start. The user may be enabled to specify whether this average return should be calculated as a flat average, in which case each trade in the selected set is given the same weight, or as a value-weighted average, in which case each return in the average is weighted by the value of the realized trade.

[0101] In this exemplary embodiment, the graphic display of impact-free price may be replaced by a similar graphic display of the impact free returns as a function of time. This description may refer to the chart of impact-free returns and standard deviation versus time from trade start below as the impact-free returns profile.

[0102] In an exemplary embodiment, the user may be further enabled to test an alternate trading strategy or schedule. In this embodiment, the user selects an alternate strategy or schedule from a drop list. Examples of a set of strategies that may be found in such a drop list include participation at 5%, 10%, 15%, 20%, etc. Another example includes a front-loaded strategy such as those known in the art. As an example, see “Optimal Execution of Portfolio Transactions,” Robert Almgren and Neil Chriss, J. Risk 3, pp. 5-39 (2000) (preprint dated Apr. 8, 1999 included in U.S. Prov. Pat. App. No. 61/322,637, which is incorporated herein by reference), and other common “benchmark” strategies such as AVWAP, TWAP, MarketOnClose, etc. Descriptions of such benchmarks can be found in reference texts on the subject—see, for example, “Optimal trading strategies,” R. Kissell and M. Glantz, AMACOM (2003).

[0103] In this exemplary embodiment, the system may estimate hypothetical prices for executing the alternate strategies in two steps. First, the trading time may be broken down into time intervals (for example, 5 minute intervals). In each interval, the number of shares that would have been filled using the alternate strategy is estimated knowing the tape volume and selected benchmark schedule. For example, if the user had chosen 5% participation, then the number of shares filled in a 5-minute interval would be 5% of the tape volume in that interval.

[0104] Second, the price impact of these hypothetical fills may be calculated using one of the methods described below for estimating impact-free price, but in reverse.

[0105] Third, the estimated prices using the alternate strategy may be estimated by adding the impact to the impact-free price. This embodiment may further enable a user to view the estimated cost of the execution using the alternate strategy, and corresponding savings or additional cost. This embodiment may also enable a user to specify a different limit price; to estimate the hypothetical prices in presence of an alternate strategy and limit price the system proceeds as above but assumes only those fills that are within the limit price would actually occur; the reduced set of fills may be used to estimate impact of the alternate strategy. Since the limit price may prevent the trade from executing in full, the number of shares filled by the alternate strategy may be displayed to the user in addition to the total cost, and the additional cost required to execute these unfilled shares at the final price may be displayed as opportunity cost associated with this choice of limit price.

[0106] In another exemplary embodiment, the system may further enable a user to initiate an automated search for groups of trades that have specific impact-free returns profiles, by selecting a profile from a drop list.

[0107] Examples of impact-free return profiles include but are not limited to “high alpha” profiles, where the impact-free returns are positive and larger than a given threshold, “negative alpha” profiles associated with negative values of the impact-free returns, or “reversion” profiles where the impact-free return has an inverted U shape exhibiting positive impact-free returns up to some point followed by reversion back to an aggregate impact-free return that becomes close to or less than zero by the end of the trading day.

[0108] In this exemplary embodiment, the system may use historical data about trades from the user, and utilize predictive data mining methods known in the art to classify the historical trades as more likely or less likely to exhibit the requested profile, given available predictive drivers. Such drivers may include the variables that are available for the users in specifying a group of trades (in the example mentioned above, size as a percentage of average daily volume) but also drivers that are calculated using market data, data about the issuer, and other sources of information (news, earnings announcements, time of day, portfolio manger's name, fund name, trader name, urgency instruction from portfolio manager, and other relevant information that can be imagined by those skilled in the art).

[0109] In this embodiment the system may display to the user each class by the drivers and values used to define the class, and show the impact-free returns profile in each class.

[0110] In an exemplary embodiment of a system that enables a classification of trades by impact-free returns profiles, the system may be further connected to real-time trade data using data feed handlers as known in the art, and enable a user to request a predicted impact-free returns profile for a hypothetical or real order to buy or sell a certain amount of stock in a given security. In this embodiment, the value(s) of all the drivers may be calculated from the data feeds in the first step; the classification model may then be used in a second step to classify the order, and a corresponding impact-free returns profile displayed to the user.

[0111] In another embodiment, the system may automatically associate an optimal trading strategy to each impact-free returns profile, selecting for this purpose a strategy that minimizes the cost of the proposed alternate strategy, as described above. For the purposes of this description the terms strategy and algorithm, while not identical in meaning, may sometimes be used interchangeably.

[0112] In another embodiment, the impact estimation may be done using mathematical models that take into consideration costs associated with information leakage, and the optimal trading strategy may be determined using an optimization program such as dynamic programming or simulated annealing to minimize a risk-adjusted cost function, as explained below under the section entitled “Optimal Execution of Portfolio Transactions: The Effect of Information Leakage” (see also “Optimal Execution of Portfolio Transactions The Effect of Information Leakage,” Adriana M. Criscuolo and Henri Waelbroeck (copy included in U.S. Prov. Pat. App. No. 61/322,637, which is incorporated herein by reference).

[0113] While the basic theory explaining impact from arbitrage arguments is known in the art of arbitrage mathematics (see “The Market Impact of Large Trading Orders: Predictable Order Flow, Asymmetric Liquidity and Efficient Prices,” J. Doyne Farmer, Austin Gerig, Fabrizio Lillo, and Henri Waelbroeck; (copy included in U.S. Prov. Pat. App. No. 61/322,637, which is incorporated herein by reference; a version is available at http://www.haas.berkeley.edu/groups/finance/hiddenImpact13.pdf)), its application to optimal execution is innovative as described in detail herein. In an exemplary embodiment, the method may also be applied to matching optimal execution profiles to classes of trades with specific impact-free returns as explained in more detail below.

[0114] Other methods for matching optimal execution strategies to impact-free returns profiles will be envisioned by those skilled in the art. An embodiment may further include an interface to a trade execution system, enabling a user who has requested an impact-free returns profile to initiate the execution of an optimal execution strategy. This may be done by means of a FIX interface to deliver an order to a trading server, as is known in the art.

[0115] In certain embodiments involving a classification of trades by their impact-free returns profile, the system may also identify hypothesis validation conditions that statistically validate or reject the class membership hypothesis in the course of execution.

[0116] For example, even though a trade may have been classified as likely to have a flat returns profile (no price movement other than the impact of the trade), if the price in fact increased by more than 40 basis points within the first 15 minutes, this may imply that further impact-free price increases are more likely than a reversal. If this were the case, such an observation would therefore invalidate the initial class membership prediction (flat returns profile) and replace it by another (rising price trend).

[0117] While exemplary embodiments of the system can be used as a stand alone offering per the above-described embodiments; it also may be used in embodiments that combine the functionality of the system with functionality covered in the patent applications listed below to enable dynamic use of impact free price estimation in a trade execution platform.

[0118] These embodiments may apply the system's ability to associate optimal trading strategies with impact free price estimations in a dynamic setting whereby the impact free price estimation is used first as a filter to help determine the best trading strategy for a given order (see, for example, U.S. patent application Ser. No. 13/070,852, filed Mar. 24, 2011, incorporated herein by reference), then may, in conjunction with hypothesis validation conditions, enable the subject system to monitor its ongoing confidence in its strategy recommendation on a real time basis and automatically switch to different execution strategies when needed (see, for example, U.S. patent application Ser. No. 11/783,250 (Pub. No. 2008/0040254), incorporated herein by reference—in particular, paragraph 0055).

[0119] In addition the system may also be used in conjunction with a block execution platform as a mechanism for providing feedback for management and placement of block orders (see, for example, U.S. patent application Ser. No. 12/463,886 (Pub. No. 2009/0281954), incorporated herein by reference, and U.S. patent application Ser. No. 12/181,028 (Pub. No. 2009/0076961), incorporated herein by reference, as well as the electronic signal notifications for order activity and price protection (see, for example, U.S. Pat. App. No. 12/181,117 (Pub. No. 2009/0089199), incorporated herein by reference, and U.S. patent application Ser. No. 12/419,867 (Pub. No. 2009/0259584), incorporated herein by reference. Finally, embodiments of the system may also be used in conjunction with the risk classification system taught by U.S. patent application Ser. No. 12/695,243 (Pub. No. 2010/0153304), incorporated herein by reference.

[0120] Exemplary Impact Free Price Estimation Embodiments

[0121] To estimate impact it is useful to capture the fact that most trades are broken down into segments, each of which involves a relatively homogeneous intended participation rate, where segments may or may not be separated by quiet periods with no trading. Research involving data from executed trades shows that impact grows non-linearly during a segment, and reverts exponentially after the completion of a segment that is followed by a quiet period. The following describes exemplary embodiments enabling an estimation of the average market impact by splitting each trade into segments and further breaking down the timeline into 5-minute intervals.

[0122] In a first step the number of shares filled in each 5-minute interval may be calculated, and this may be further refined into a number of shares filled passively, aggressively, and inside the spread.

[0123] In a second step, the incremental impact from the fills in the segment may be added and reversion from activity in prior segments subtracted, to obtain the net price effects. The incremental impact from fills in the segment may be given for example by a parametric model as follows
E(I_t)=aνπ^δ(Q_t/ADV)^α−1(MktCap[%])^−η
Where π is the participation rate, the impact factor a and exponents α, δ, η are parameters that may be estimated for different algorithmic trading styles as described below using methods known in the art of non-linear regressions for estimating exponents and taking care to control for selection bias; the shares filled up to time t are Q_t; ν is the stock's volatility and ADV is the average daily volume. The algorithmic trading style characterizes the manner in which an algorithm interacts with the market; this may be an algorithm name or aggression parameter provided by the client together with the fills data; or alternatively, it may be defined for example based on a clustering of aggregate metrics. Such metrics may include, for example, the percentage of prints by classification as aggressive, passive or midpoint. The aggregate metrics may also include short-term performance metrics. An example of such a style clustering analysis is described in Stephens and Waelbroeck, J. Trading, Summer 2009 and available at [www.alphascience.com/Portals/0/Documents/JOT_Summer_—2009_Pipeline.pdf].
After a segment is completed, the impact contribution of the segment is the sum of residual temporary impact and permanent impact, as follows. Reversion is the difference between the segment impact at the end of the segment and this residual impact.
Temporary impact at the end of the trade may be estimated, for example, as ⅓ of the total impact. This form of impact decays exponentially. The exponential decay timescale may be estimated, for example, as τ=τ₀+κ*LN(t₀+t[min]) where τ₀=0, κ=4.34, t₀=3 are parameters and the time t from the end of the segment is measured in minutes. Thus, t′ minutes after segment completion,

[0124] $E\left(I,\mathrm{end}+t^{\prime }\right)=E\left(I,\mathrm{end}\right)\left(1-\frac{1}{3}\exp \left(-t^{\prime }/\tau \right)\right)$
Permanent impact may be estimated, for example, as ⅔ of total impact at the end of the segment. The manner in which permanent impact decays may itself be estimated as a power of elapsed tape volume. The decay exponent may be taken to be the same value δ as was introduced previously regarding the scaling of total segment impact to the participation rate. Thus,

[0125] $E\left(\mathrm{PI},\mathrm{end}+t^{\prime }\right)=\frac{2}{3}E\left(I,\mathrm{end}\right){\left(\frac{\mathrm{tape}\left(\mathrm{start}->\mathrm{end}\right)}{\mathrm{tape}\left(\mathrm{start}->\mathrm{end}+t^{\prime }\right)}\right)}^{0.4}$
The accrued impact at time t may be calculated as the sum of the impact contribution of each segment.

[0126] Impact-free prices may be estimated for buys by subtracting the fill price by the accrued price impact up to the interval that contains each fill, or for sells by adding the accrued price impact.

[0127] In an alternate embodiment, the quantities inside the square root functions above may be calculated as a linear sum of weights times the quantity filled in passive, intra-spread, or aggressive executions; the three factors in this sum may be estimated using regression methods. Other embodiments including replacing the square root function with a different function or utilizing different parametric or non-parametric models in each of the steps outlined above may be easily envisioned by those skilled in the art.

[0128] Optimal Execution in Presence of Hidden Order Arbitrage

[0129] In order to enable an accurate estimation of the impact-free price for strategies with different execution speeds, it is useful to use an impact model that correctly accounts for the effect of execution speed on the cost of trading. As explained below, impact models known in the art fail to account for the possibility that arbitrage traders would be able to observe information about an algorithmic execution in the market data stream and trade on this information: if price were correctly explained by models known in the art, there would be risk-free profits available for such arbitrage strategies. Therefore it is the purpose of the present section to describe an exemplary impact model that is derived from a no-arbitrage argument, within a framework referred to herein as hidden order arbitrage theory.

[0130] The equations of hidden order arbitrage theory are a bit difficult to work with. Accordingly, to create a more readily implementable solution this description uses simplified versions in the impact model described above, using first-order expansions of some of the special functions that emerge from the theoretical framework.

[0131] It is also a purpose of this section to demonstrate how the framework of hidden order arbitrage theory enables one to calculate optimal execution solutions that minimize a risk-adjusted cost function or optimize performance relative to a VWAP benchmark.

[0132] Almgren and Chriss found that to maximize the risk-adjusted liquidation value of an asset it is optimal to trade fastest at the beginning of a trade. This result was based on simplifying assumptions including linear and stationary impact. Of these two assumptions, that of a stationary impact process is most clearly invalidated by observations: practitioners observe more price reversion after completing a long trade than a brief one. This portion of the description considers the optimal execution problem using a zero arbitrage argument for price formulation. Models and methods are described for dealing with a type of information arbitrage referred to herein as hidden order arbitrage.

[0133] Arbitrageurs detect the presence of hidden orders through the statistical properties of order flow on the market and formulate statistical models for future price and order flow imbalances. Competition between arbitrageurs keeps prices close to a level that fairly accounts for the information revealed in the order flow. When a hidden order stops, expectations of future order flow imbalances decay, and price reverts accordingly. This portion of the description explains that the shape of the impact function and subsequent reversion can be derived from the basic equations for hidden order arbitrage, the participation rate profile for executing a trade, and the statistical distribution of hidden order sizes. Numerical solutions to the optimal execution problem in the presence of hidden order arbitrage are provided.

[0134] This portion of the description considers the optimal execution of a large portfolio transaction that must be split into smaller slices and executed incrementally over time. There are many dimensions to this problem that are potentially important to the institutional trader: liquidity fluctuations, the news stream, and short-term alpha that may be associated with a fund manager's order origination process, to name a few. In response to these variables, traders make decisions including the participation rate, limit price, and other strategy attributes. As explained elsewhere in this description, one may limit the scope of the problem by adopting the definition of optimal execution from Almgren and Chriss (AC 2000): optimal execution is the participation rate profile that minimizes the risk-adjusted cost while completing the trade in a given amount of time.

[0135] To optimize the risk-adjusted cost one may first specify a model for market impact. Market impact has been analyzed by different authors as a function of time and trade size. See, for example, (Bertismas and Lo, 1998), (Almgren and Chriss, 2000), (Almgren et al., 2005), (Obizhaeva and Wang, 2006).

[0136] AC 2000 derived execution profiles that are optimal if certain simplifying assumptions hold true. These include the hypothesis that the market is driven by a random Brownian motion overlaid with a stationary market impact process. Impact is proposed to be the linear sum of permanent and temporary components, where the permanent impact depends linearly on the number of traded shares and the temporary impact is a linear function of the trading velocity. It follows that total permanent impact is independent of the trade schedule. The optimal participation rate profile requires trading fastest at the beginning and slowing down as the trade progresses according to a hyperbolic sine function.

[0137] This type of front-loaded participation rate profile is widely used by industry participants, yet it is also recognized that it is not always optimal. Some practitioners believe that the practice of front-loading executions bakes in permanent impact early in the trade, resulting in higher trading costs on average. A related concern is that liquidity exhaustion or increased signaling risk could also lead to a higher variance in trade results (Hora, 2006), defeating the main purpose of front-loading. In their paper, Almgren and Chriss acknowledge that the simplifying assumptions required to find closed-form optimal execution solutions are imperfect. The non-linearity of temporary impact in the trading velocity has been addressed previously in (Almgren, 2003), (Almgren et al., 2005); the optimization method has also been adjusted for non-linear phenomenological models of temporary impact (Loeb, 1983; Lillo et al., 2003). Most studies however share the common assumptions that short-term price formation in non-volatile markets is driven by an arithmetic Brownian motion and that the effect of trading on price is stationary, i.e., the increment to permanent impact from one interval to the next is independent of time and the temporary impact is a correction that depends only on the current trading velocity but not on the amount of time that the strategy has been in function. There are reasons to doubt the assumption of stationary impact. Practitioners find that reversion grows with the amount of time that an algorithm has been engaged; this suggests that temporary impact grows as a function of time.

[0138] Phenomenological models of market impact consistently produce concave functions for total cost as a function of trade size; this is inconsistent with linear permanent impact.

[0139] (Farmer et al., 2009) (FGLW) showed that it is possible to derive a concave shape for both temporary and permanent impact of a trade that is executed at a uniform participation rate. The basic assumption in this method is that arbitrageurs are able to detect the existence of an algorithm and temporary impact represents expectations of further activity from this algorithm. The concave shape of market impact follows from two basic equations.

[0140] The first is an arbitrage equation for a trader that observes the amount of time an execution has been in progress and uses the distribution of hidden order sizes to estimate the total size of the hidden order.

[0141] The second is the assumption that institutional trades break even on average after reversion. In other words, the price paid to acquire a large position is on average equal to the price of the security after arbitrageurs have determined that the trade is finished. The model explains how temporary impact sets the fair price of the expected future demand or supply from the algorithmic trade. When the trade ends and these expectations fade away, the model also explains how price reverts to a level that incorporates only permanent impact. The shape of the impact function can be derived from knowledge of the hidden order size distribution. If one believes the hidden order size distribution to have a tail exponent of approximately 1.5, the predicted shape of the total impact function is a square root of trade size in agreement with phenomenological models including the Barra model (Tone, 1997). See also (Chan and Lakonishok, 1993), (Chan and Lakonishok, 1995), (Almgren et al., 2005), (Bouchaud et al., 2008), (Moro et al., 2009).

[0142] This portion of the description extends hidden order arbitrage theory to estimate the impact of trades that execute with a variable participation rate, and uses the extended model to derive optimal execution solutions.

[0143] The first section below generalizes the framework for hidden order arbitrage to allow for varying-speed execution profiles. The next section describes trading solutions that minimize total trading cost with and without risk. Section 3 considers optimization with respect to the volume-weighted average price objective and shows that trade optimization with respect to the two benchmarks (implementation shortfall and VWAP) is a frustrated problem. Implications for institutional trading desks are discussed in the concluding section.

[0144] 1. Hidden Order Arbitrage

[0145] This portion of the description addresses the situation of a large institutional trade that is executed over time through a sequence of smaller transactions. For simplicity's sake, one may consider a single institutional trade executing in a market that is otherwise driven by arithmetic Brownian motion. The trade is executed according to an execution schedule with the participation rate π(n(t)) representing the probability that a market transaction n(t), at time t, belongs to the institutional trade.

[0146] 1.1 Hypotheses

[0147] 1. Hidden Order Detection. (H.1)

[0148] A hidden order executing during a period Δt with an average rate π is detected at the end of intervals of τ(π)=1/π² market transactions¹. The term “detectable interval” is used below to mean each set of

[0149] $\tau \left({\pi }_i\right)=\frac{1}{{{\pi }_i}^2}$
market transactions, for each iε, over which a hidden order is detected with a constant participation rate π_i. A detectable interval i contains

[0150] $\frac{1}{{\pi }_i}$
hidden order transactions, with 0<π_i<1, ∀i. ¹If order flow were a random walk with a bias it between buy and sell transactions, the imbalance would be detected with t-stat=1 after 1/π² transactions.

[0151] In addition, there exists a function τ_r(X, π_f) such that the end of a hidden order can be detected after a reversion time τ_r(X, π_f), where π_f is the most recently observed rate. Let be N*=N*(X)ε_>0, then N*=q+[N*], [N*]IntegerPart[N*], 0≦q≦1.

[0152] One may set τ_r(X, π_f)=qπ_f⁻². The number N* may be determined by the trade size X and [N*] represents the last detectable interval.

[0153] 2. Distribution. (H.2)

[0154] The total distribution function of the hidden order process is the product of a distribution p({right arrow over (π)}, N*(X)) of normalized execution schedules {right arrow over (π)}={π₁, π₂, . . . , π_[N*], π_f}, and the Gaussian distribution G of an arithmetic random walk.

[0155] For constant rate,

[0156] ${\int }_0^{1}p\left(\pi ,N^{*}\left(X\right)\right)d\pi \to p\left(N^{*}\right)\propto \frac{1}{N^{*\alpha +1}},$
a truncated Pareto distribution of hidden order sizes for N*≦M, where the cutoff M is very large and α=1.5 is the tail exponent (Gopikrishnan et al., 2000), (Gabaix et al., 2006).

[0157] One may call p(π₁, π₂, . . . , π_i, i≦[N*]) the probability that the hidden order was detected at least in i intervals (and i be the last detectable step [N*] or not) with a participation schedule {right arrow over (π)}={π₁, π₂, . . . , π_i}. By definition of conditional probabilities,
p(π₁, π₂, . . . , π_i, i≦[N*])=p(π_i, i≦[N*]/π₁, π₂, . . . π_i-1)p(π_i-1, i−1≦[N*]/π₁, π₂, . . . , π_i-2) . . . p(π₂, 2≦[N*]/π₁)p(π₁, 1≦[N*]).

[0158] Here p(π_i, i≦[N*]/π₁, π₂, . . . , π_i-1) is the conditional probability that the hidden order will be detected at the interval i with rate π_i given that it was detected in i−1 intervals with rate schedule {right arrow over (π)}={π₁, π₂, . . . , π_i-1}.

[0159] One may determine p(π_i, i≦[N*]/π₁, π₂, . . . , π_i-1) when all the hypotheses are given (see below). By “reversion price”, S_i, is meant the expected price after the end of the hidden order has been detected at the end of the interval i. One may denote by {tilde over (S)}_i the expected average price in the interval, where the expectation is over G, with fixed {π₁, π₂, . . . , π_i}.

[0160] 3. Price Efficiency. (H.3)

[0161] For the short term, arbitrage opportunities disappear quickly due to the efficiency of the market. This concept translates to the equation:

[0162] ${\int }_0^{1}p\left({\pi }_i,i\le \left[N^{*}\right]/{\pi }_1,{\pi }_2,\dots ,{\pi }_{i-1}\right)\left({\tilde{S}}_i-{\tilde{S}}_{i-1}\right)d{\pi }_i+p\left(i-1=\left[N^{*}\right]/{\pi }_1,{\pi }_2,\dots ,{\pi }_{i-1}\right)\left(S_{i-1}-{\tilde{S}}_{i-1}\right)==0,i\ge 2.$

[0163] Here,

[0164] $p\left(i-1=\left[N^{*}\right]/{\pi }_1,{\pi }_2,\dots ,{\pi }_{i-1}\right)=1-{\int }_0^{1}p\left({\pi }_i,i\le \left[N^{*}\right]/{\pi }_1,{\pi }_2,\dots ,{\pi }_{i-1}\right)d{\pi }_i$

[0165] is the probability that the hidden order stops at the end of the interval i−1, given that it was detected through a schedule {π₁, π₂, . . . , π_i-1}.

[0166] 4. Breakeven. (H.4)

[0167] The expected reversion price following a trade that completed after k intervals is equal to its weighted average execution price²: ²The breakeven hypothesis may seem surprising. It states that the implementation cost on average equals the information value of the trade. FLGW shows that this hypothesis is an example of the tragedy of the commons: Portfolio managers understand that their information is not exclusive and other managers will join the trade until the net profit, after impact, goes to zero.

[0168] $S_k=\frac{\sum _{i=1}^k\frac{1}{{\pi }_i}{\tilde{S}}_i}{\sum _{i=1}^k\frac{1}{{\pi }_i}},k\ge 2.$

[0169] 5. Temporary Impact. (H.5)

[0170] First Interval Impact:

[0171] The impact of first-interval, {tilde over (S)}₁−S₀, is equal to the product of a scaling factor that depends only on the volatility σ and an exponent γ of the participation rate in the first interval:
{tilde over (S)}₀=S₀={circumflex over (μ)}(σ)π₁^γ.

[0172] Impact after the First Interval:

[0173] The temporary impact out of the first interval {tilde over (S)}_k+1−S_k is a function only of the current participation rate π_k+1 and the total number of shares filled through interval k+1.

[0174] 1.2 Exemplary Impact Model

[0175] One may derive an impact model from the hypotheses above and its simplified form to first order in k⁻¹.

[0176] By hypothesis 5, temporary impact does not depend directly on the participation rate profile before the interval in consideration. Therefore, it may be modeled as the temporary impact of a constant participation trajectory that has filled the same number of shares at the current participation rate of the variable rate model. Then, one may write:
{tilde over (S)}_k+1−S_k={tilde over (S)}_jk+1−S_jk+1−1, such that
ξ_k+1=ξ_jk+1=j_k+1π_k+1⁻¹.  (1)

[0177] Here,

[0178] ${\xi }_{k+1}\stackrel{\mathrm{def}}{=}\sum _{i=1}^{k+1}\frac{1}{{\pi }_i}$
is the size executed through the end of interval k+1 in units of the average transaction size.

[0179] As shown in [FGLW], the price efficiency condition and breakeven equation in the case of a uniform participation rate and no impact-free returns can be solved recursively for the temporary impact in interval j_k+1, leading to

[0180] $\begin{array}{cc}{\tilde{S}}_{j_{k+1}}-S_{j_{k+1}-1}=\frac{1-p_1}{\left(j_{k+1}-1\right)\sum _{i=j_{k+1}}^M{p}_i}\left({\tilde{S}}_1-S_0\right).& \left(2\right)\end{array}$

[0181] Here p_ip(i=[N*]) the probability that the hidden order stops at the end of the interval i, such as defined in hypothesis 2. For a Pareto distribution of hidden order sizes (hypothesis 2), the sum in the denominator, Σ_i=1k+1p_i, is a Hurwitz zeta function ζ(α+1, j_k+1). Using equalities (1), (2) and hypothesis 5, one may write:

[0182] ${\tilde{S}}_{k+1}-S_k=\frac{1-p_1}{\left(j_{k+1}-1\right)\zeta \left(a+1,j_{k+1}\right)}\left(\hat{\mu }\left(\sigma \right){\pi }_{k+1}^{\gamma }\right),$
such that
j_k+1=ξ_k+1π_k×1.  (3)

[0183] From the asymptotic form of the Hurwitz zeta function for large j, (j−1)ζ(α+1, j)j^α+1. Therefore, one may approximately write the solution to this model as:

[0184] $\begin{array}{cc}{\tilde{S}}_{k+1}-S_k\approx \mu \left(\sigma \right){{\pi }_{k+1}^{\beta }\left(\sum _{i=1}^{k+1}\frac{1}{{\pi }_i}\right)}^{\alpha -1},k>>1.& \left(4\right)\end{array}$

[0185] Here, βγ+α−1 and μ(σ)(1−p₁){circumflex over (μ)}(σ). The temporary impact in Equation (3) only involves the most recent participation rate and shares acquired through the end of interval k+1 and is valid for all k≧1. Nevertheless, despite that equation (4) gives the temporary impact for large k, it does not invalidate Hypothesis 5 if one uses it for all the intervals k≧0. The parameters μ(σ) and β can be estimated from data on small trades, for which the shortfall {tilde over (S)}₁−S₀ can be measured with sufficient accuracy to distinguish execution strategies (see for example, Altunata et al. (2009). Trading data is consistent with β=0.3 (Gomes and Waelbroeck, 2008).

[0186] 2. Optimal Execution

[0187] Following the description above, one may write temporary impact as:

[0188] $\begin{array}{cc}{\tilde{S}}_k=S_{k-1}+{{\mathrm{\mu \pi }}_k^{\beta }\left(\sum _{i=1}^k\frac{1}{{\pi }_i}\right)}^{\alpha -1},k\ge 1.& \left(5a\right)\end{array}$

[0189] Here,

[0190] $\left[\mu \right]=\frac{\$}{\mathrm{share}},\mu >0$
for a buy and μ<0 for a sell. Combining (5a) with the breakeven hypothesis, one may derive by recursion the expression for the expected price at k, as a function of the participation rate schedule π_i, 1≦i≦k:

[0191] $\begin{array}{cc}{\tilde{S}}_k=S_0+\mu \left\{{{\pi }_k^{\beta }\left(\sum _{i=1}^k{\pi }_i^{-1}\right)}^{\alpha -1}+\sum _{i=1}^{k-1}{{\pi }_i^{\beta -1}\left(\sum _{j=1}^i{\pi }_j^{-1}\right)}^{\alpha -2}\right\},2\le k\le \left[N^{*}\right],& \left(6a\right)\\ {\tilde{S}}_1=S_0+{\mathrm{\mu \pi }}_1^{\beta -\alpha +1}.& \left(6b\right)\end{array}$

[0192] [N*]IntegerPart[N*]. By (H.1), one is considering the possibility that the total number of detectable steps N* be a non-integer value; which means the institution could finish at an “extra time” q=N*−[N], 0≦q≦1, that it is completed in less than π⁻² market transactions. In the case that q≠0, the expected price at N* will be:
{tilde over (S)}_N*=S_[N*]+μπ_N*^βξ_N*^α-1,  (5b)
Or expanding as in (6a),
{tilde over (S)}_N*=S₀+μ{π_N*^βξ_N*^α-1+Σ_i=1^[N*]π_i^β-1(Σ_j=1ⁱπ_j⁻¹)^α-2}  (6c)

[0193] where ξ_N* is the total number of transactions traded until the last detectable interval N* and it is by definition:

[0194] $\begin{array}{cc}{\xi }_{N^{*}}\stackrel{\mathrm{def}}{=}\left(\sum _{i=1}^{\left[N^{*}\right]}\frac{1}{{\pi }_i}+q{\pi }_{N^{*}}^{-1}\right).& \left(7\right)\end{array}$

[0195] Furthermore, the expected total cost of the trade (over G) is

[0196] $\begin{array}{cc}E\left(\overrightarrow{\pi },N^{*}\right)\stackrel{\mathrm{def}}{=}n{\xi }_{N^{*}}{S}_0-\sum _{i=1}^{\left[N^{*}\right]}{n}_i{\tilde{S}}_i-\mathrm{qn}{\pi }_{N^{*}}^{-1}{\tilde{S}}_{N^{*}},& \left(8\right)\end{array}$

[0197] where n_i=nπ⁻¹ is the number of shares traded in the i-segment and n is the number of traded shares in each institutional transaction with n>0 for a sell and n<0 for a buy.

[0198] In addition, one may suppose that there exists Nε, N≦N*, such that from N+1 to N* the institution participates with a constant rate π_f. Therefore, the variables ({right arrow over (π)}, N*) may be reduced to ({π_i}_i=1^N, π_f, N*).

[0199] After a routine calculation, using equations (6), the expected total cost turns out to be:

[0200] $\begin{array}{cc}E\left(\overrightarrow{\pi },N^{*}\right)=\mu n{\xi }_{N^{*}}\left\{\sum _{k=1}^{\left[N^{*}\right]}{{\pi }_k^{\beta -1}\left(\sum _{i=1}^k{\pi }_i^{-1}\right)}^{\alpha -2}+q{{\pi }_f^{\beta -1}\left(\sum _{i=1}^{\left[N^{*}\right]}{\pi }_i^{-1}+q{\pi }_f^{-1}\right)}^{\alpha -2}\right\}.& \left(9\right)\end{array}$

[0201] Note that E({right arrow over (π)}, N*)>0 always, for either a buy or a sell.

[0202] Additionally, as in (Almgren and Chriss, 2000), one may evaluate the variance of the cost V({right arrow over (π)}, N*)<(E({right arrow over (π)}, N*)−<(E({right arrow over (π)}, N*)>_G)²>_G. For that, one may sum the term representing the volatility of the asset

[0203] $\begin{array}{cc}\sigma \sum _{i=1}^k{\pi }_i^{-1}{\varsigma }_i,& \left(10\right)\end{array}$

[0204] to the equations (6). The ζ_i+1 are random variables with zero Gaussian mean and unit variance and σ is a constant with units

[0205] $\left[\sigma \right]=\frac{\$}{\mathrm{share}\times \sqrt{\mathrm{transaction}}}.$

[0206] Therefore, the variance of E({right arrow over (π)}, N*) takes the form

[0207] $\begin{array}{cc}V\left(\overrightarrow{\pi },N^{*}\right)={\sigma }^2{n}^2\sum _{k=1}^{\left[N^{*}\right]}{{\pi }_k^{-2}\left({\xi }_{N^{*}}-\sum _{j=1}^K{\pi }_j^{-1}\right)}^2.& \left(11\right)\end{array}$

[0208] One may next write the risk-adjusted cost function:
U({right arrow over (π)}, N*; λ, μ, n, σ, α, β, N)E({right arrow over (π)}, N*)+λV({right arrow over (π)}, N*),  (12)

[0209] where λ is the risk parameter with units [λ]=$⁻¹.

[0210] Applying the previous expressions, one obtains:

[0211] $\begin{array}{cc}U\left({\left\{{\pi }_i\right\}}_{i=1}^N,{\pi }_f,N^{*};\lambda ,\mu ,n,\sigma ,\alpha ,\beta ,N\right)=\mu n{\xi }_{N^{*}}\left\{\sum _{k=1}^{\left[N^{*}\right]}{{\pi }_k^{\beta -1}\left(\sum _{i=1}^k{\pi }_i^{-1}\right)}^{\alpha -2}+q{{\pi }_f^{\beta -1}\left(\sum _{i=1}^{\left[N^{*}\right]}{\pi }_i^{-1}+q{\pi }_f^{-1}\right)}^{\alpha -2}\right\}+{\mathrm{\lambda \sigma }}^2{n}^2\sum _{k=1}^{\left[N^{*}\right]}{{\pi }_k^{-2}\left({\xi }_{N^{*}}-\sum _{j=1}^k{\pi }_j^{-1}\right)}^2.& \left(13\right)\end{array}$

[0212] One may set the constraints on the total number of institutional transactions

[0213] $\begin{array}{cc}X={\xi }_{N^{*}}\stackrel{\mathrm{def}}{=}\left\{\sum _{j=1}^N{\pi }_j^{-1}+\left(N^{*}-N\right){\pi }_f^{-1}\right\},& \left(14\right)\end{array}$

[0214] and a maximum for the trading time t_N*:

[0215] $\begin{array}{cc}{T}_M\ge t_{N^{*}}\stackrel{\mathrm{def}}{=}\sum _{i=1}^N{\pi }_i^{-2}+\left(N^{*}-N\right){\pi }_f^{-2}.& \left(15\right)\end{array}$

[0216] 2.1 The Optimal Trajectory for λ=0

[0217] First, one may optimize the cost without risk, equation (9) together with the constraints (14) and (15).

[0218] Results:

[0219] Let N=10, X=100 transactions, and T_M=1000 transactions. One may define the dimensionless cost

[0220] $\frac{E}{\mu n},$
which allows an independent solution on the selection of the parameters n and μ.

[0221] The optimal solution for 100 traded lots is:

[0222] $\frac{E\left(\mathrm{minimum}\right)}{n\mu }=755.742,$

[0223] $\{{\pi }_1^{-1}=18.53,\hspace{1em}{\pi }_2^{-1}=11.62,\hspace{1em}\hspace{1em}{\pi }_3^{-1}=9.64,{\pi }_4^{-1}=8.61,{\pi }_5^{-1}==7.94,{\pi }_6^{-1}=7.45,\hspace{1em}\hspace{1em}\hspace{1em}{\pi }_7^{-1}=7.08,{\pi }_8^{-1}=6.78,{\pi }_9^{-1}==6.53,\hspace{1em}{\pi }_{10}^{-1}=6.32,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\pi }_f^{-1}=6.03,N^{*}=11.57\}.$

[0224] It satisfies the constraints ξ_N*=100, t_N*=1000, the rate in the additional step 11 and in the fractional step q=0.57 is π_f⁻¹=6.03.

[0225] This result shows that, in absence of risk aversion, it is optimal to start the trade more slowly to minimize information leakage early in the trade.

[0226] What follows shows an optimal trajectory for the cost function,

[0227] $g=\frac{E}{n\mu },$
with two variables (π₁⁻¹, π₂⁻¹), eight intervals traded with constant rate π_f⁻¹ and a fraction q of a unitary interval traded with the same constant rate π_f⁻¹. One may choose N*=10+q, X=100=Σ_i=1²π_i⁻¹+(N*−2)π_f⁻¹ and t_N*=1000=Σ_i=1²π_i⁻²+(N*−2)π_f⁻². The example was done with the purpose of showing a graph for the cost in 3D, and to understand the cost function for a number of variables >2. The constraints determine

[0228] $q=-8+\frac{{\left(100-{\pi }_1^{-1}-{\pi }_2^{-1}\right)}^2}{1000-{\pi }_1^{-2}-{\pi }_2^{-2}}\mathrm{and}{\pi }_f=\frac{100-{\pi }_1^{-1}-{\pi }_2^{-1}}{1000-{\pi }_1^{-2}-{\pi }_2^{-2}}.$

[0229] The optimization gives:

[0230] $\left\{g\left(\mathrm{minimum}\right)=758.642,\left\{{\pi }_1^{-1}=17.5318,{\pi }_2^{-1}=11.6056\right\}\right\}$

[0231] A graphic representation of the cost for the whole domain is given in FIG. 15, which depicts a three-dimensional representation of the dimensionless-cost function

[0232] $\frac{E}{n\mu },$
with two variables. The minimum is the point

[0233] $\left\{{\pi }_1^{-1}=17.5318,{\pi }_2^{-1}=11.6056,\frac{E}{n\mu }=758.642\right\}.$
The quarter of the circumference shows the divergence of the cost at 1000−π₁⁻²−π₂⁻²=0.

[0234] A closer view around the minimum is shown in FIG. 16.

[0235] FIG. 16 depicts a dimensionless-cost function around the optimal point {π₁⁻¹=17.5318, π2−1=11.6056, Enμ=758.642, following FIG. 15.

[0236] The solution satisfies ξ_N*=100, t_N*=1000, q=1. The participation rate after step 2 is π_f⁻¹=7.87. This solution is more expensive that the one with N=10, which shows that the observable N*=11.57 minimizes the cost for the selected constraints.

[0237] 2.2 Risk Adjusted Optimization

[0238] Above is provided an analysis of hidden order arbitrage methods for variable speed of trading with zero risk aversion (λ=0). The description below concentrates on finding optimal trading trajectories for a model with varying participation rate and arbitrary risk aversion. That means to minimize the total risk-adjusted cost function (13):

[0239] $\begin{array}{cc}\frac{U\left(\overrightarrow{\pi },N^{*};L\right)}{\mu n}=X\left\{\sum _{k=1}^{\left[N^{*}\right]}{{\pi }_k^{-0.7}\left(\sum _{i=1}^k{\pi }_i^{-1}\right)}^{-0.5}+q{{\pi }_f^{-0.7}\left(\sum _{i=1}^{\left[N^{*}\right]}{\pi }_i^{-1}+q{\pi }_f^{-1}\right)}^{-0.5}\right\}+L\sum _{k=1}^{\left[N^{*}\right]}{{\pi }_k^{-2}\left(X-\sum _{j=1}^k{\pi }_j^{-1}\right)}^2,& \left(16\right)\end{array}$

[0240] with the constraints (14) and (15).

[0241] The results are summarized in Table 1 for

X=100 lots, T_M=1000 transactions, α=1.5, β=0.3, for the different values of the risk constant L=λσ²|n/μ|.

[0242]

TABLE 1
L	E/|μn|	E  μ ⁢ ⁢ n  ⁢ X	λ  μ ⁢ ⁢ n  ⁢ V	U  μ ⁢ ⁢ n 	U  μ ⁢ ⁢ n  ⁢ X	N*	tN*
3 ×	925.65	9.26	527.63	1453.29	14.53	13.14	1000
10−4
1 ×	827.97	8.28	240.12	1068.09	10.68	10.99	1000
10−4
0	755.74	7.56	0	755.742	7.56	11.57	1000

[0243] Optimal Risk Adjusted Cost. Results are for X=100 lots, T_M=1000 transactions, β=1.5, β=0.3, for the different values of the risk constant L=λσ²|n/μ|. Second column is total cost, third column is cost per lot or transaction, the fourth one gives the total risk, the fifth is the total risk-adjusted cost, the sixth is per lot or transaction. N* is the total number of detectable intervals realized by the hidden order. The last column indicates that the number of market transactions reaches the maximum limit T_M. All values are dimensionless.

[0244] FIG. 17 depicts a graph of the participation rate π_k versus the detectable step k, in a continuum aproximation, for the different values of the risk constant L=λσ²|n/μ|. FIG. 17: Optimal trajectories representing the participation rate in function of the number of the detectable interval for different values of the risk constant.

[0245] Because in each step the participation rate may be constant, a detailed graph is presented of the participation rate versus the transactional time, t_k=Σ_i=1^kπ_i⁻², corresponding to each k-interval, for each L:

[0246] FIG. 18. Participation rate in function of the transactional time, t_k=Σ_i=1^kπ_i⁻², corresponding to each k-interval, considering zero risk aversion.

[0247] FIG. 19. Participation rate in function of the transactional time, t_k=Σ_i=1^kπ_i⁻², corresponding to each k-interval, considering risk aversion L=1×10⁻⁴.

[0248] FIG. 20. Participation rate in function of the transactional time, t_k=Σ_i=1^kπ_i⁻², corresponding to each k-interval, considering risk aversion L=3×10⁻⁴.

[0249] FIG. 21 depicts a comparative graph for the different values of the risk aversion in the quadratic approximation or continuum: FIG. 21 depicts a comparative graph of optimal trajectories in function of the transactional time for the different values of the risk aversion in the quadratic approximation or continuum.

[0250] 3. Optimizing Performance to the VWAP Benchmark

[0251] The section above described optimizing a sum of expected implementation shortfall and its variance. This was called a risk-adjusted cost function and optimal trading trajectories were found.

[0252] The implementation shortfall in this context is the difference between the initial book value of the shares, XS₀, and the capture of the trajectory Σ_i=1^N*n_i{tilde over (S)}_i. Nevertheless, traders may have other objectives or benchmarks rather than XS₀. These include:

• • [0253] 1) Post reversion price or closing price. This is useful to measure the effect of an entry trade on assets under management marked to market after the trade is done and post-trade reversion is complete.
  • [0254] 2) Volume-weighted average price during the transactional period or VWAP. This is the average price transacted by the market, which is useful to evaluate exit trades that are not too large relative to the ADV because it is less exposed to market effects than implementation shortfall.

[0255] The reversion price is equal to the average realized price by (H.4); hence, one may not regard this benchmark as being useful for the purpose of the schedule optimization. Therefore, one may analyze the difference α between VWAP and the capture. For a buy:

[0256] $\begin{array}{cc}\Delta \left(\mathrm{buy}\right):=-\mathrm{VWAP}+\mathrm{capture}\stackrel{\mathrm{def}}{=}\stackrel{\mathrm{def}}{=}-\frac{1}{t_{N^{*}}}\sum _{i=1}^{N^{*}}{\tau }_i{\tilde{S}}_i+\frac{1}{{\xi }_{N^{*}}}\sum _{i=1}^{N^{*}}{\pi }_i^{-1}{\tilde{S}}_i.& \left(17a\right)\end{array}$

[0257] Here, t_N* is the institutional trade duration and ξ_N* is the total number of institutional orders executed in that period. {π_i}_i=1^N* is the set of participation rates from the first to the last detectable segment and τ_i=π_i⁻² is the minimum expected transactional market period for the detection of the i^th segment. π_i⁻¹ is the expected value of the number of the institutional transactions executed in the i-detectable segment, and {tilde over (S)}_i is the price per share paid by the institution in the i-segment. For a sell,
Δ(sell):=VWAP−capture.  (17b)

[0258] Following the equations (6), and including the arithmetic Brownian noise, one has:

[0259] $\begin{array}{cc}\Delta \left(\mathrm{buy}\text{/}\mathrm{sell},{\left\{{\pi }_i\right\}}_{i=1}^{N^{*}}\right)=\left(-/+\right)\frac{\mu }{t_{N^{*}}}\left\{\sum _{k=1}^{\left[N^{*}\right]}{\pi }_k^{\beta -1}{\xi }_k^{\alpha -2}\left({\pi }_k^{-1}{\xi }_k-t_k\right)+q{\pi }_f^{\beta -1}{\xi }_{N^{*}}^{\alpha -2}\left({\pi }_f^{-1}{\xi }_{N^{*}}-t_{N^{*}}\right)\right\}+\left(-\text{/}+\right)\sigma \sum _{k=1}^{\left[N^{*}\right]}{\pi }_k^{-1}{Ϛ}_k\left(\frac{{\xi }_k}{{\xi }_{N^{*}}}-\frac{t_k}{t_{N^{*}}}\right),\mathrm{where}& \left(18a\right)\\ {\xi }_k\stackrel{\mathrm{def}}{=}\sum _{i=1}^k{\pi }_i^{-1},& \left(18b\right)\\ t_k\stackrel{\mathrm{def}}{=}\sum _{j=1}^k{\tau }_j=\sum _{j=1}^k{\pi }_j^{-2},& \left(18c\right)\end{array}$

[0260] μ, α, β, σ are constant parameters and ζ_i are random variables with zero Gaussian mean and unit variance.

[0261] Taking μ(sell)=−μ(buy) and the Gaussian mean, equation (18a) is:

[0262] $\begin{array}{cc}\Delta \left(\mathrm{buy}\mathrm{or}\mathrm{sell},{\left\{{\pi }_i\right\}}_{i=1}^{N^{*}}\right)=-\frac{\mu }{t_{N^{*}}}\left\{\sum _{k=1}^{\left[N^{*}\right]}{\pi }_k^{\beta -1}{\xi }_k^{\alpha -2}\left({\pi }_k^{-1}{\xi }_k-t_k\right)+q{\pi }_f^{\beta -1}{\xi }_{N^{*}}^{\alpha -2}\left({\pi }_f^{-1}{\xi }_{N^{*}}-t_{N^{*}}\right)\right\}.& \left(19\right)\end{array}$

[0263] Next, one may minimize (19) using Mathematica7 constrained with
X=ξ_N*=100, t_N*≦T_m=1000.

[0264] The optimal solution is:
Δ(buy or sell, minimum)/|μ|=−0.666758,

[0265] with the optimal trajectory:
π₁⁻¹=9.48, π₂⁻¹=7.23, π₃⁻¹=6.78, π₄⁻¹=6.42, π₅⁻¹=6.61, π₆⁻¹=6.90, π₇⁻¹=7.63, π₈⁻¹=9.38, π₉⁻¹=11.40, π₁₀⁻¹=14.99, π_f⁻¹=13.54, N*=10.97, ξ_N*=100, t_N*=1000.

[0266] The negative value indicates that the institution buys below the market or sells above the market, and the gain (absolute value) is maximal for the optimal trajectory.

[0267] The cost for the solution, which optimizes the VWAP, is

[0268] $\frac{E}{n\mu }=837.36.$
This is bigger than the optimal cost

[0269] $\frac{E}{n\mu }\left(\mathrm{minimum}\right)=755.74,$
for X=100, t_N*=1000, L=0, found in section 2.1. Thus, it is possible to beat the VWAP benchmark with that trajectory but at the expense of a higher shortfall³. ³For buys, the increased implementation shortfall is associated with a higher reversion price due to the breakeven condition, so the economic value of shortfall reduction may seem less evident at first glance. However, the dependence of a security price on the execution of a trade will decay over the investment horizon, so the shortfall will ultimately be a net negative contribution to the portfolio value for buys as well as for sells.

[0270] Conversely, the value for the VWAP benchmark using the optimal trajectory, which minimizes the shortfall, is:

[0271] $\frac{\Delta \left(\mathrm{buy}\mathrm{or}\mathrm{sell},\mathrm{optimal}\mathrm{for}L=0\right)}{\mu }=3.21403.$
The positive value means that the institution reduces the shortfall to the minimum at expenses of selling below or buying above the market average price.

[0272] FIG. 22 depicts a graph of participation rate versus transactional time for a VWAP-optimal solution: FIG. 22 depicts participation rate in function of the transactional time, t_k=Σ_i=1^kπ_i⁻², corresponding to each k-interval. This is the VWAP benchmark optimal trajectory.

[0273] 4. Discussion

[0274] The preceding sections of this portion of the description generalized hidden order arbitrage models and methods to non-flat execution schedules. Hidden order arbitrage methods tie the concavity of the impact function to the predictability of the order flow. As explained by Hasbrouck (Hasbrouck, J. 1991), only the unpredictable component of the order flow can cause incremental market impact. The predictable component is preemptively incorporated into the security price in the form of temporary impact. The methods described herein use this principle to estimate temporary impact: a simplifying assumption may be introduced to the effect that the predictable component of the order flow can be approximated knowing only the current trading velocity and the total number of shares accumulated so far in the trade. When the trade ends, expectations of further order flow subside and price reverts to a level that incorporates only permanent impact.

[0275] In contrast, in Kyle's classic paper (Kyle, A. 1985), the informed trader is assumed to know precisely the final price and only buys if price is lower, so there is no reversion. When the market maker sets the price to the expected informed price, since informed traders buy below the informed price and sell above it, the order flow expectations in the next step are evenly balanced. Kyle shows that in this context, where order flow is unpredictable, the participation rate optimal trajectory is constant in time and the impact function is linear in the rate of trading. There is ample evidence that, unlike the informed trades described by Kyle, algorithmic execution of hidden orders leads to a predictable order flow, and there is ample evidence for the concavity of the impact function (Bouchaud et al, 2008). Yet prior to the present invention, there had not been an optimal execution derivation that accounts for these observations.

[0276] The above description provides solutions that minimize the risk-adjusted cost for various levels of risk aversion or the performance relative to a VWAP benchmark. The optimal schedules for risk-adjusted cost are strikingly different from the classic Almgren-Chriss solutions. In particular, this description shows that it is optimal to start trades slowly. The VWAP optimal solution is more reminiscent of the A-C solution. However, it beats the VWAP in part by creating impact early in the trade, resulting in a higher price throughout the trade and a higher shortfall. The near VWAP optimality of the A-C solution may help to explain why it is so frequently used by trading desks in spite of its higher average cost. The following description considers the frustration between various optimization objectives using a concrete example.

[0277] 4.1 Example and Comparisons

[0278] Below is considered a mid-cap trade of |Xn|=25000 shares, |n|=250 shares per transaction, in a S₀=$50 security, executed at an average participation rate of 10% (π=0.1). If the security's trading volume is

[0279] $\frac{400\mathrm{transactions}}{\mathrm{hr}},$
a detectable interval will represent 15 minutes of trading. The impact for a 15-minute interval is estimated to be 10 bps for this security. From formula (6b), with β=0.3, α=1.5, one finds that a 10 bps impact for the first interval correspond to
|{tilde over (S)}₁−S₀|=10⁻³×$50=|μ|×0.1^−0.2, i.e. |μ|=0.0315 $/share.
If one take san annual volatility of 30%,

[0280] $\sigma =\frac{0.95\left(\frac{\$}{\mathrm{share}}\right)}{\sqrt{\mathrm{day}}}.$
In this case, one may work with transactions units as measure of time and τ=1/π²=100 is the average number of the market transactions in each detectable interval. If one day consists of 6 hours and 30 minutes and each detectable interval lasts 15 minutes then, 1 day=2600 market transactions. Therefore,

[0281] $\sigma =\frac{0.019\frac{\$}{\mathrm{share}}}{\sqrt{\mathrm{transaction}}.}.$

[0282] The shortfall and the VWAP benchmark of risk-adjusted cost optimal solutions are listed in Table 2 for this example and different risk aversion parameters.

[0283]

TABLE 2
L	Risk parameter λ ($−1)	Shortfall per share ( $ share )	Shortfall E ($)	Variance {square root over (V)} ($)	(Δ(buy or sell)) ( $ share )
1 × 10−4	3.49 × 10−4	0.2608	6520.5	7360.83	−0.0173
0	0	0.2381	5953.5	9192.27	0.1012
3 × 10−4	1.05 × 10−4	0.2917	7292.25	6290.65	0.2418

[0284] Shortfall and VWAP benchmark of Cost Optimal solutions. Consider a mid-cap trade of 25000 shares, in a S₀=$50 security, executed at an average participation rate of 10% (π=0.1). If the security's trading volume is

[0285] $\frac{400\mathrm{transactions}}{\mathrm{hr}},$
a detectable interval will represent 15 minutes of trading. The impact for a 15-minute interval is estimated to be 10 bps for this security, i.e. |μ|=0.0315 $/share. One may take an annual volatility of 30% or

[0286] $\sigma =\frac{0.95\left(\frac{\$}{\mathrm{share}}\right)}{\sqrt{\mathrm{day}}}.$
One day consists of 6 hours and 30 minutes or 2600 market transactions. The last column is the VWAP benchmark calculated with the risk adjusted cost optimal solution.

[0287] Exemplary result: Shortfall minimization in absence of alpha requires back-loading, which increases execution risk. Optimal solutions for risk-averse traders are less front-loaded than suggested by AC2000 and avoid an aggressive trade start. FIG. 23 depicts a graph of the optimal risk averse trajectories for the information arbitrage theory with L=10⁻⁴ in comparison to the Almgren-Chriss formulation (AC2000). For a buy,

[0288] ${\pi }_j^{A-C}=\frac{\sinh \left(\kappa T\right)}{2\times \sinh \left(\frac{\kappa \tau }{2}\right)\cosh \left(\kappa \left(j-\frac{1}{2}\right)\tau \right)},X=100,T=165\mathrm{minutes}=0.423\mathrm{day},\kappa \sim \frac{3.83}{\mathrm{day}},\tau =15\mathrm{minutes}=0.0384\mathrm{day}.$

[0289] The shortfall calculated with the optimal Almgren-Chriss trajectory for a risk averse constant

[0290] $\lambda \left(\text{Almgren-Chriss}\right)=\frac{{10}^{-5}}{\$}\mathrm{is}E\left({\left\{{\pi }_j^{A-C}\right\}}_{j=1}^{11}\right)=6879.05\$.$
This is costlier than the shortfall

[0291] $E\left(\mathrm{optimal}\right)=6520.5\$\mathrm{for}\lambda =3.49\times \frac{{10}^{-5}}{\$}.$

[0292] FIG. 23. Comparative graph of the cost optimal trajectories in function of the transactional time. The dotted line is the solution predicted by Almgren-Chriss formulation with linear impact and risk averse constant

[0293] $\lambda \left(\mathrm{Almgren}\text{-}\mathrm{Chriss}\right)=\frac{{10}^{-5}}{\$}.$
The square line is the optimal trajectory for the non-linear information arbitrage theory, as shown in FIG. 19, for risk averse constant

[0294] $L={10}^{-4}\mathrm{or}\lambda =3.49\times \frac{{10}^{-5}}{\$}.$

[0295] Additionally, one obtains a VWAP optimal value in the first column of Table 3, and the shortfall and its variance per share corresponding to the VWAP optimal trajectory in the second and third columns, respectively.

[0296]

	TABLE 3
	(Δ(buy or sell, optimal)) ( $ share )	Shortfall per share ( $ share )	Variance per share ( $ share )
	−0.0210	0.2638	0.2893
			7233.84$ for the total

[0297] Comparison of optimal VWAP and shortfall. VWAP optimal value is given in the first column, and the shortfall and its variance per share corresponding to the VWAP optimal trajectory in the second and third columns, respectively.

[0298] Comparing Tables 2 and 3, VWAP optimization increases shortfall if no risk or low risk aversion is taken into account. However, if high risk aversion applies, then VWAP optimization may be appropriate. On the other hand, to minimize implementation shortfall it is necessary to buy above or sell below the VWAP, when no risk or high risk aversion is considered. This is known in the art of multi-criteria optimization as frustration: one optimizes to one objective at the expense of the other. However, in this case only the implementation shortfall benchmark is relevant to the performance of the portfolio—so the existence of frustration simply implies that funds that create incentives for traders to perform well relative to a VWAP benchmark are in fact promoting trading behaviors that are harmful to the fund returns.

[0299] In conclusion, because the shortfalls per share are approximately the same for both optimizations, and the variance estimated with the VWAP benchmark optimal trajectory is lower, VWAP optimization is better at low risk aversion. It also provides lower shortfall and better VWAP performance than the high risk aversion solution; however, the variance of the shortfall is higher by 17% originating uncertainty.

[0300] What are the right incentives for a trading desk? The implementation shortfall is the measure of the effect of trade execution on assets under management. Other benchmarks can be used to promote trade behaviors that are more likely to result in lower shortfalls on a particular type of trade. Using the closing price for trades that have negative underlying alpha incentivizes back-loading, and will lead to lower average shortfall for exit trades. For entry trades, using the risk adjusted cost will encourage traders to reduce variance by front-loading and reduce average shortfall. For zero alpha trades VWAP benchmark provides the incentive for trades to pick favorable price points throughout the day. However, for large trades, the VWAP benchmark results in front-loaded executions schedules that increase implementation cost.

[0301] 4.2 Comments about Price Efficiency

[0302] This section shows that the density probability function, p(π_i, i≦[N*]/π₁, π₂, . . . , π_i-1), is determined by hypotheses 1 to 5 through a functional form. Reciprocally, it will be shown that given the formulae (5-6a) for Price Impact, in which (H.4) for breakeven is included, a specific probability functional form will satisfy Price Efficiency.

[0303] Proposition:

[0304] Price efficiency is satisfied

[0305] $\iff \exists f=f\left({\pi }_i,{\left\{{\pi }_j\right\}}_{j=1}^{i-1}\right)\mathrm{such}\mathrm{that}f\left({\pi }_i,{\left\{{\pi }_j\right\}}_{j=1}^{i-1}\right){|}_{{\pi }_i=0}^1=1\bigwedge \frac{\left({\pi }_{i-1}^{\beta }{\xi }_{i-1}^{\alpha -1}-{\pi }_{i-1}^{\beta -1}{\xi }_{i-1}^{\alpha -2}\right)}{\left({\pi }_i^{\beta }{\xi }_{i-1}^{\alpha -1}\right)}\frac{\partial f\left({\pi }_i\right)}{\partial {\pi }_i}=p\left({\pi }_i,i\le \left[N^{*}\right]/{\pi }_1,{\pi }_2,\dots ,{\pi }_{i-1}\right),\forall i>2;{\pi }_j\ne 0,1\le j\le i,\mathrm{and}p\left({\pi }_i=0,i\le \left[N^{*}\right]/{\pi }_{1,}{\pi }_2,\dots ,{\pi }_{i-1}\right)=0,\forall i.$

[0306] Proof:

[0307] Hypothesis of Price Efficiency (H.3) says:

[0308] $\begin{array}{cc}{\int }_0^{1}p\left({\pi }_i,i\le \left[N^{*}\right]/{\pi }_1,{\pi }_2,\dots ,{\pi }_{i-1}\right)\left({\tilde{S}}_i-{\tilde{S}}_{i-1}\right)d{\pi }_i+p\left(i-1=\left[N^{*}\right]/{\pi }_1,{\pi }_2,\dots ,{\pi }_{i-1}\right)\left(S_{i-1}-{\tilde{S}}_{i-1}\right)=0,i\ge 2.& \left(H\mathrm{.3}\right)\end{array}$
Using

[0309] $\begin{array}{cc}p\left(i-1=\left[N^{*}\right]/{\pi }_1,{\pi }_2,\dots ,{\pi }_{i-1}\right)=1-{\int }_0^{1}p\left({\pi }_i,i\le \left[N^{*}\right]/{\pi }_1,{\pi }_2,\dots ,{\pi }_{i-1}\right)d{\pi }_i\mathrm{in},& \left(H\mathrm{.3}\right)\end{array}$
one may write:

[0310] $\begin{array}{cc}\iff \left(S_{i-1}-{\tilde{S}}_{i-1}\right)+{\int }_0^{1}p\left({\pi }_i,i\le \left[N^{*}\right]/{\pi }_1,{\pi }_2,\dots ,{\pi }_{i-1}\right)\left({\tilde{S}}_i-S_{i-1}\right)d{\pi }_i=0,i\ge 2.& \left(A\right)\end{array}$

[0311] Using equations (5a) and (6a), one finds:

[0312] $\begin{array}{cc}{S}_{i-1}-{\tilde{S}}_{i-1}=-\mu \left({\pi }_{i-1}^{\beta }{\xi }_{i-1}^{\alpha -1}-{\pi }_{i-1}^{\beta -1}{\xi }_{i-1}^{\alpha -2}\right),i\ge 2,& \left(B\right)\end{array}$

[0313] Introducing (B) and (5a) in (A),

[0314] $\begin{array}{cc}\iff \left({\pi }_{i-1}^{\beta }{\xi }_{i-1}^{\alpha -1}-{\pi }_{i-1}^{\beta -1}{\xi }_{i-1}^{\alpha -2}\right)=={\int }_0^{1}p\left({\pi }_i,i\le \left[N^{*}\right]/{\pi }_1,{\pi }_2,\dots ,{\pi }_{i-1}\right){\pi }_i^{\beta }{\xi }_i^{\alpha -1}d{\pi }_i,i\ge 2.& \left(C\right)\\ \iff \exists f=f\left({\pi }_i,{\left\{{\pi }_j\right\}}_{j=1}^{i-1}\right)❘\left({\pi }_{i-1}^{\beta }{\xi }_{i-1}^{\alpha -1}-{\pi }_{i-1}^{\beta -1}{\xi }_{i-1}^{\alpha -2}\right)[f\left({\pi }_i,{\left\{{\pi }_j\right\}}_{j=1}^{i-1}\right)-f\left(0,{\left\{{\pi }_j\right\}}_{j=1}^{i-1}\right)]=={\int }_0^{{\pi }_i}p\left({\pi }_i,i\le \left[N^{*}\right]/{\pi }_1,{\pi }_2,\dots ,{\pi }_{i-1}\right){\pi }_i^{\beta }{\xi }_i^{\alpha -1}d{\pi }_i,\bigwedge f\left({\pi }_i,{\left\{{\pi }_j\right\}}_{j=1}^{i-1}\right){|}_{{\pi }_i=0}^1=1.& \left(D\right)\\ \iff \exists f=f\left({\pi }_i,{\left\{{\pi }_j\right\}}_{j=1}^{i-1}\right)❘f({\pi }_i,{\left\{{\pi }_j\right\}}_{j=1}^{i-1}){|}_{{\pi }_i=0}^1=1\bigwedge \left({\pi }_{i-1}^{\beta }{\xi }_{i-1}^{\alpha -1}-{\pi }_{i-1}^{\beta -1}{\xi }_{i-1}^{\alpha -2}\right)\frac{\partial f\left({\pi }_i\right)}{\partial {\pi }_i}=p\left({\pi }_i,i\le \left[N^{*}\right]/{\pi }_1,{\pi }_2,\dots ,{\pi }_{i-1}\right)\left({\pi }_i^{\beta }{\xi }_i^{\alpha -1}\right),i>2.& \left(E\right)\end{array}$

[0315] Because each detectable step i means by definition that p(π_i=0, i≦[N*]/π₁, π₂, . . . , π_i-1)=0, ∀i; then, one may take:

[0316] $\begin{array}{cc}\frac{\left({\pi }_{i-1}^{\beta }{\xi }_{i-1}^{\alpha -1}-{\pi }_{i-1}^{\beta -1}{\xi }_{i-1}^{\alpha -2}\right)}{\left({\pi }_i^{\beta }{\xi }_i^{\alpha -1}\right)}\frac{\partial f\left({\pi }_i\right)}{\partial {\pi }_i}==p\left({\pi }_i,i\le \left[N^{*}\right]/{\pi }_1,{\pi }_2,\dots ,{\pi }_{i-1}\right),\forall i>2;{\pi }_j\ne 0,1\le j\le i.& \left(F\right)\end{array}$

[0317] Examples

[0318] Let

[0319] $\begin{array}{cc}f\left({\pi }_i,{\left\{{\pi }_j\right\}}_{j=1}^{i-1}\right)=\hspace{1em}\frac{\sum _{m=1}^n{a}_m\left({\left\{{\pi }_j,{\xi }_j\right\}}_{j=1}^{i-1}\right){\pi }_i^{{\gamma }_m}}{\sum _{m|{\gamma }_m\ne 0}^n{a}_m^{\prime }}n\epsilon {\gamma }_m\epsilon \ge 0⩓\sum _{m|{\gamma }_m\ne 0}^n{a}_m\ne 0.& \left(G\right)\end{array}$

[0320] The functions α_m=α_m({π_j, ξ_j}_j=1^i-1) and coefficients γ_m could be partially determined by (H.2),

[0321] ${\int }_0^{1}p\left(\pi ,i=N^{*}\left(X\right)\right)d\pi ->p\left(i=N^{*}\right)\propto \frac{1}{i^{\alpha +1}},$
when π_j=constant, ∀j. It can be shown that if α_m are constants for all m, then α_m and γ_m are arbitrary.

[0322] If one takes f(π_i)=π_i then,

[0323] $p\left({\pi }_i,i\le \left[N^{*}\right]/{\pi }_1,{\pi }_2,\dots ,{\pi }_{i-1}\right)=\frac{\left({\pi }_{i-1}^{\beta }{\xi }_{i-1}^{\alpha -2}{\xi }_{i-2}\right)}{\left({\pi }_i^{\beta }{\xi }_i^{\alpha -1}\right)},i>2.$

[0324] Also, because

[0325] ${\int }_0^{1}p\left({\pi }_1,1\le \left[N^{*}\right]\right)d{\pi }_1\stackrel{\mathrm{def}}{=}p\left(1\le \left[N^{*}\right]\right)=C,$
where C is a normalization constant; then, one may choose p(π₁, 1≦[N*])=C. Let be p(π₂, 2≦[N*]/π₁)=π₂^βξ₂^−α+1. Using (H.2),

[0326] $\begin{array}{cc}p\left({\pi }_1,{\pi }_2,\dots ,{\pi }_i,i\le \left[N^{*}\right]\right)=C\frac{{\pi }_1^{-1}}{{\pi }_i^{\beta }{\xi }_i^{\alpha -1}{\xi }_{i-1}}.& \left(H\right)\end{array}$

[0327] For constant participation rate, (H) becomes:

[0328] $\begin{array}{cc}p\left(\pi ,i\le \left[N^{*}\right]\right)=C\frac{{\pi }^{\alpha -\beta -1}}{i^{\alpha -1}\left(i-1\right)},i>1.& \left(I\right)\end{array}$

[0329] Additionally,

[0330] $\begin{array}{cc}p\left(i\le \left[N^{*}\right]\right)={\int }_0^{1}p\left(\pi ,i\le \left[N^{*}\right]\right)d\pi =\frac{c}{i^{\alpha -1}\left(i-1\right)},i>1.\frac{C}{\left(k-1\right)k^{\alpha -1}}=p\left(\left[N^{*}\right]\ge k\right)\stackrel{\mathrm{def}}{=}\sum _{i=0}^{\infty }{p}_{k+i},k>1.& \left(J\right)\end{array}$

[0331] Subtracting p([N*]≧k+1) from p([N*]≧k), one obtains:

[0332] $\begin{array}{cc}{p}_k=C\left[\frac{1}{\left(k-1\right)k^{\alpha -1}}-\frac{1}{{k\left(k+1\right)}^{\alpha -1}}\right],k>1,.& \left(K\right)\end{array}$

[0333] Equation (K) turns out to be:

[0334] $\frac{C}{k^{\alpha }\left(1-k^{-1}\right)}-\frac{C}{{k^{\alpha }\left(1+k^{-1}\right)}^{\alpha -1}}\stackrel{k⪢1}{->}\hspace{1em}{\mathrm{Ck}}^{-\alpha }\left(1+k^{-1}\right)-{\mathrm{Ck}}^{-\alpha }\left(1-\left(\alpha -1\right)k^{-1}\right)=\hspace{1em}\hspace{1em}\alpha {\mathrm{Ck}}^{-\left(\alpha +1\right)},$
which is the Pareto distribution. It can be shown that results (J), (K) are also valid for any f such as (G), with a_mε for all m.

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