WO2006031807A2

WO2006031807A2 - Securities trading system and method

Info

Publication number: WO2006031807A2
Application number: PCT/US2005/032560
Authority: WO
Inventors: Kai Huang
Original assignee: Susquehanna International Group, Llp
Priority date: 2004-09-13
Filing date: 2005-09-12
Publication date: 2006-03-23
Also published as: WO2006031807A3

Abstract

The invention relates to the trading of securities [101]. In one respect of the invention, a method of optimizing a desired trade of a security or plurality of securities is provided. The method includes forming a cost equation for the desired trade, wherein the cost equation includes at least one risk factor and a market impact factor associated with each security. The method further includes solving the cost equation to derive a trading solution, wherein the trading solution includes a trading strategy for each security [203]. Preferably the trading solution is provided in the form of a percentage of market volume [105]. In another aspect of the invention, a set of multiple cost equations are provided, with each cost equation having a weight risk factor or set of risk factors. The cost equations are solved to provide a plurality of trading strategies for the desired trade.

Description

SECURITIES TRADING SYSTEM AND METHOD CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of prior U.S. provisional application 60/608,904, filed September 13, 2004. TECHNICAL FIELD

This invention relates to a system and method for executing trades of securities such as stocks, and more particularly to a system for estimating, optimizing and/or executing one or more preferred strategies for executing a transfer of a security or portfolio of securities. BACKGROUND

In many transfers of securities, buyers or sellers must concern themselves with the market impact of their own intended trading, as well as the risk inherent in delaying trades to minimize market impact. For example, when executing a stock trade, a prospective buyer can frequently cause an increase in the market price simply by placing a buy order. Likewise a seller can cause a stock price to decrease by placing a sell order. This is particularly true for institutional investors, or other investors who may trade large volumes of a particular stock or stocks. On the other hand, metering large trades over time in order to minimize market impact increases the risk that the price of the security will move during the trading period, either with the market or relative to the market.

Another issue faced by many investors is how to execute trades of a "basket" or "portfolio" of securities. That is, in what sequence and/or volume should the investor seek to execute trades of multiple securities. This problem may arise in many situations, for example where two securities are within the same market sector (such as technology stocks), or other situations in which the market price of one security can positively or negatively affect the other. Even where the individual securities in a portfolio are not highly correlated, investors face the issue of how to conduct trading of each security to minimize expected market impact and risk. In many cases, investors are also provided a single trade execution strategy based on current impact or risk measurements. While in some situations a single execution strategy might be sufficient for an investor, in other cases such limited options may not fully accommodate the investor's needs. For example, if an investor requires rapid trading of a security or portfolio of securities based on a high estimated return (for example given a temporarily high valuation relative to existing market prices), a single trading strategy for execution over a period of days may not be adequate. Conversely, if an investor wishes to minimize market impact and believes long-term risk is low, the investor may be dissatisfied with the same strategy for opposite reasons. In either case, this dissatisfaction may exist even if the trading strategy is optimized under a given model.

SUMMARY hi one aspect of the present invention, a method of optimizing a desired trade of a plurality of securities is provided. The method includes forming a cost equation for the desired trade, wherein the cost equation includes at least one risk factor and a market impact factor associated with each security. One or more of the factors may be weighted, as described below. The method further includes solving the cost equation to derive a trading solution, wherein the trading solution includes a trading strategy for at least one of the securities, and preferably for each of the plurality of securities. hi another aspect of the invention, a method of trading is provided in which multiple trading options are provided. The method includes forming a plurality of cost equations for a desired trade (where the "desired trade" may include a single security or multiple securities), wherein each cost equation includes a set of input factors, the input factors including at least one risk factor and at least one market impact factor, wherein one or more of the factors may be weighted. The method further includes solving each of the cost equations to derive a plurality of trading solutions for the desired trade, wherein each trading solution includes a trading strategy for at least one of the securities in the desired trade, and preferable each individual security in the desired trade. In additional forms, the invention further includes selecting one of the trading solutions to execute the trade.

The details of one or more embodiments of the invention are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims.

DESCRIPTION OF DRAWINGS

FIG. 1 is a flow diagram showing an exemplary method of trading securities according to the invention. FIG. 2 is a flow diagram showing a second exemplary method of trading securities according to the invention.

FIG. 3 is a schematic diagram of a computer system according to the present invention.

Like reference symbols in the various drawings indicate like elements. DETAILED DESCRIPTION

In one aspect of the invention, a method of optimizing a desired trade is provided. As used herein, a "desired trade" may include, in various forms, a trade of a single security or a trade of a plurality of securities. In an exemplary form described herein, the invention is sometimes described with respect to a stock or plurality of stocks. It is understood, however, that the term security is construed broadly, and may include various properties such as stocks, bonds, commodities, and the like, as well as options, calls, futures, etc. of such properties. It is further understood that, unless specified, the terms "security," "stock," etc. should be construed to include both the single and the plural. A plurality of securities may also be referred to as a "portfolio" or "basket" of securities.

With respect to Figure 1, in one exemplary aspect, the invention includes forming a cost equation for the desired trade (101), wherein the cost equation reflects the potential market impact of a desired trade as well as risks of a desired trade. In a basic form, a cost equation may be depicted as equation [I]: [1] C = R + MI

where C is the total cost of the trade, R is the risk, and MI the market impact, as further discussed below. Once formed, the cost equation may be solved to derive a trading solution, which may define a trading strategy for at least one of the securities. In a preferred form, the trading solution defines a trading strategy for each security or plurality of securities in the desired trade, or substantially all of such securities, as described below. It is understood that such a cost equation provides an estimate or prediction of the true costs of a trade, based on historical pricing information and other data, which underlie both the estimated risk and estimated market impact. For clarity and convenience, these factors are not repeatedly referred to as estimates herein, but are simply referred to as "cost," "risk," etc.

As used in equation [I]₅ R represents one or more risk factors that may be selected from a number of risk factors associated with securities trades. The invention is not limited to any particular selection of risk factors, but instead may include any useful group of risk factors. Such factors may include, but are not limited to, volatility, variance, dollar exposure, expected drift (a measure of an investor's perception of the difference between the market price of a security and the true value of the security), interest rates, interest rate volatility, governmental or political events, market sector events, corporate events and other news events.

Some of these risk factors may be expressed as functions of an intended trading strategy, for example as a function of a volume percentage (as described below), as a function of an expected time to complete the trade, and the like. In this arrangement, the trading strategy variables represent the independent variables in the cost equation. When the function is solved as described below, the trading solution may then be expressed directly as a collection of trading strategies for the securities of the desired trade. If suitable, however, risk factors may be expressed as constants, functions of other variables or as other types of functions, including functions defining correlations between the various securities in a desired trade, as understood in the art.

Preferred risk factors for use with the invention include volatility, variance, exposure and expected drift. When applied to a portfolio of securities, each of these factors may be derived or calculated on a portfolio basis, as understood in the art, to obtain a single function or value for the portfolio as a whole. In other words, for an exemplary trade involving a portfolio of multiple securities, one may derive a function representing, for example, the volatility of the entire portfolio of securities. The derivation of such functions is within the skill of one of ordinary skill in the art, and as understood in the art, these functions may be derived based on historical information. In the particular case of trading a portfolio of stocks, for example, the historical information may include historical stock prices for each of the stocks. These factors may also be calculated for a desired trade of a single security. For example, for an exemplary trade involving one security, one may derive a function representing the volatility for the single security.

A risk factor for use in the present invention may preferably include various mathematical components related to the securities within a desired trade. For example, a function representing volatility may be constructed for a portfolio of three stocks. This function may include various mathematical components related to each stock, and/or to the relationship between the stocks.

The function may include, for example, three mathematical components representing each stock's contribution to the dollar volatility of the portfolio as a whole. Each of these mathematical components can be formed as a function of the trading strategy for the particular stock. A trading strategy resulting in a relatively fast purchase of stock, for example, will increase the dollar value of the portfolio quickly, leading to a higher dollar volatility than a trading strategy resulting in a slower purchase of stock (other things being equal) because a higher volume of stock is in the investor's possession at an earlier time. On the other hand, a trading strategy resulting in a relatively slow purchase of stock will increase the dollar value of the portfolio slowly, leading to a lower dollar volatility than a fast trading strategy. Conversely, a trading strategy resulting in a relatively fast sale of stock will decrease the dollar value of the portfolio quickly, leading to a lower dollar volatility than a trading strategy resulting in a slow sale of stock. A trading strategy resulting in a relatively slow sale of stock will decrease the dollar value of the portfolio more slowly, leading to a higher dollar volatility.

It is understood that these components are only one type of mathematical component that may contribute to volatility. The volatility function may also preferably include, for example, mathematical components representing the historical movement of each security. In the case of the example of the three stock portfolio, the volatility function may include three separate mathematical components representing the historical movement of each of the three stocks. This movement may be viewed in any useful manner, for example in simple numerical terms, relative to price, relative to the market as a whole, relative to a market segment, etc.

Preferably, where the invention is applied to a desired trade involving a plurality of securities, risk factors used with the invention include mathematical components representing correlations between the movement of the various securities. In the three-stock example described above, a volatility function may include, for example, three such mathematical components representing the correlations between each pair of two stocks. In a desired trade involving five securities, the volatility function may include ten such mathematical components representing the correlations between each different pair of stocks. Regardless of its particular form, a volatility function, once derived, may be utilized as a risk factor in the cost equation.

It is understood that the term "mathematical component" is used merely for clarity, may include any type of function or constant, and should not be construed as a limitation on the invention. Moreover, different mathematical components may be utilized for different securities in the desired trade. Various mathematical components also may be grouped to form a variety of functions or factors related to risk or market impact. The invention is not limited to the exemplary form described herein, and a cost equation according to the invention may include such components as parts of other types of factors or functions. It is understood that a cost equation having components that correspond to risk factors and/or market impact factors described herein is within the scope of the invention.

Likewise, one may calculate a variance for the exemplary security or portfolio, as understood in the art, to obtain a single function or value for use in the cost equation. In a preferred form, the variance may be constructed by squaring each individual mathematical component in a volatility function, but any form of variance may be utilized. It is also understood that a risk function representing dollar exposure may be constructed, as understood in the art, for use in a cost equation according to the invention. As understood in the art, dollar exposure relates to the volume and price of the securities being traded. Expected drift refers to a measure of the expected movement of a stock relative to the market or market sector, as perceived by an investor or other individual. In practice, the value for expected drift may be selected by the investor seeking to execute the desired trade, or may be selected by others, for example a broker for an investor. It is understood that any useful individual risk factor or combination of risk factors may be used with the present invention. In an exemplary form, at least two of these four risk factors (volatility, variance, exposure and expected drift) are utilized in a cost equation according to the invention. In one preferred form, all four of volatility, variance, exposure and expected drift are utilized. In a preferred form, each risk factor may be assigned a weight, which may be based on historical data, the preference of the entity desiring to execute the trade, or other information. Such weight or weights may be assigned to one or more risk factors, regardless of the number of risk factors utilized or the specific risk factors utilized, as further described below. In equation [1], MI represents a market impact factor for the security or securities involved in the desired trade. Such market impact factors may be expressed as functions of time or a related variable, in which a shorter trade time leads to a higher market impact (and hence higher cost due to market impact), whereas a longer trade time leads to a lower market impact (and hence lower cost due to market impact). Market impact factors may also be expressed as a function of the desired trading strategy, such as time to trade or volume percentage, as described below. In an exemplary form, each security in the desired trade is associated with a corresponding market impact factor. In other words, if there are six securities involved in the desired trade, then the exemplary cost equation will include six market impact factors, one corresponding to each of the six securities. A cost equation may omit one or more market impact factors for a given security or securities, however, if useful. It may also be useful to include additional factors or various modifications or combinations of such individual factors, and any such useful combination is within the scope of the invention. As understood in the art, there are multiple methods of deriving or calculating a market impact function for any given security. In a preferred form, a market impact factor for use with the invention is a market impact function derived from historical information including price, average daily volume, and other fundamental information. Such market impact functions may be purchased commercially from a variety of sources, or may be derived separately based on any useful information, as understood in the art. It is understood that the invention is not limited to any particular market impact factor or function for a given security, but instead includes any such useful factor or function. As with risk factors, market impact factors may preferably be provided as functions of the intended trading strategy, so that a trading solution is likewise directly provided as one or more trading strategies. As further discussed below, however, one or more factors of the cost equation (including market impact factors) may be formed as functions of some other variable, and then converted to a preferred fonn of market impact factor after the cost equation is solved.

An exemplary form of the invention can be described with respect to a cost equation for trading a portfolio of n securities, where n is the number of individual securities in the portfolio, hi this embodiment, the cost equation may be formed as equation 2:

[2] C = R + MIi + MI₂+ . . . + MI_n where C is the cost, R is the risk factor or group of risk factors, MI₁ is the market impact function associated with the first security of the portfolio, MI₂ is the market impact function of the second security of the portfolio, and so on.

If, for example, four risk factors are utilized, the equation may be formed as [3]:

[3] C = Rl + R2 +R3 + R4 + MI₁ + MI₂ + . . . + MI_n

where Rl through R4 are the exemplary risk factors, and the market impact factors are as described with respect to equation [2]. It should be understood that in this exemplary form, risk factors Rl through R4 may include mathematical components related to each security in the desired trade, but are, taken as a whole, associated with the entire portfolio, as described above.

If desired, the risk factors may be assigned weights in any useful manner, so that one or more risk factors are emphasized or de-emphasized relative to the others. An exemplary cost equation with such weights may be provided in the form of equation [4] :

[4] C = wl^»Rl + w2^»R2 + w3*R3 + w4*R4 + MI₁ + MI₂ + . . . + MI_n

where wl'Rl represents a weighted first risk factor, w2^#R2 represents a second weighted risk factor, and so on. If desired, some risk factors may be weighted, while others remain unweighted. Any useful weights may be applied, and it is understood that the invention is not limited to any particular weight or weights. In a preferred form, weights are chosen so that the relative impact of two different risk factors within a given cost equation is within five orders of magnitude, hi another preferred form, weights are selected to substantially correspond (i.e., within the same order of magnitude) to a historical relative impact of various risk factors in similar trades or in trades by similar investors. In this exemplary form, market impact factors are not weighted, although such weights can be applied if useful.

Referring again to Figure 1, once the cost equation is constructed, it may then be solved to produce a trading solution for the desired trade (103), where the trading solution includes a trading strategy for at least one security in the desired trade. In a preferred form, the trading solution includes a trading strategy for each security in the desired trade, or for substantially all such securities. Preferably, solving the cost equation to produce the trading solution means solving for the lowest expected cost, so that the cost of the desired trade is minimized. Other "solutions" to the cost equation may be derived if desired, for example solutions that minimize risk or market impact, or solutions that minimize cost given certain constraints. Such constraints may include, for example, a maximum time to execute trading of a given security in the desired trade.

Minimizing or otherwise solving a cost equation may be performed in any useful manner, as understood in the art. In certain cases it may be useful to solve the equation using calculus or other mathematical procedures. In a preferred embodiment, the cost equation is solved electronically, using a computer having a processor and memory, and running software having the capability to minimize such functions. A computer package such as SAS®, S-Plus®, or MATLAB® may be employed, for example, to obtain a solution to a cost function according to the invention. Such programs typically seek an estimation or prediction of minimum cost, for example, by inserting multiple possible values for each independent variable, and selecting the set of values that provide the lowest cost. It is understood that these methods of solving the cost equation are within the capability of one skilled in the art, and the invention is not limited by the choice of method of solving the cost equation.

Once the cost equation has been solved, the desired trade may be executed according to the trading strategies (105). In one form, if the trading solution is for some reason not acceptable, then another cost equation may be formed with different factors and/or weights, and this second cost equation may be solved to obtain another trading solution. This process may continue, for example based on information derived from each prior cost equation and solution, until an acceptable trading solution is obtained.

In this exemplary form, the trading solution to the cost equation will provide trading strategies for at least one security in the trade, and preferably for each of the securities involved in the trade. It is understood that the invention is not limited to any particular form of trading strategy. One such strategy may be a Volume Weighted Average Price (known as "VWAP") strategy for each security or securities. As understood in the art, such a strategy will seek to match historical trading patterns of the individual security using a weighted average price, thereby minimizing impact. Another type of trading strategy usable for executing the desired trade is a trading time, which provides a measure of the time expected to complete trading of a security of the desired trade.

In a preferred form of the invention, the trading strategies are provided as a volume percentage. In this form, the trading strategy for a given security is in the form of a percentage of actual market trading volume of the security. In other words, a given security within the desired trade is traded so that the number of shares offered for purchase or sale at a given time (referred to herein as the "volume of orders") is maintained as a selected percentage of the total market volume of the security at that time. As the total market volume increases, the volume of orders for that security is increased proportionally, and as the total market volume decreases, the volume of orders for that security is decreased, so that the volume of orders compared to the total market volume remains substantially at the selected percentage.

It is understood that a given volume percentage will result in a given expected trading time for a security. For example, if the desired trade involves a trade of N shares of a stock, and the stock has a historical market volume of ION shares per day, then a 10% trading strategy will result in the execution of the trade of that stock in approximately one day. A 2% trading strategy would result in execution of the desired trade of that stock in approximately five days. These simplified examples do not address variations in market volumes, such as variations at different times of the day, but such modifications are within the skill of one of ordinary skill in the art. These simplified examples are merely provided for clarity, and should not be construed as a limitation on the invention.

In particular, a trading strategy (a volume percentage or other trading strategy) may be a dynamic function itself, that changes with given market conditions or constraints. For example, a cost equation for a desired trade may be constructed so that risk factors and/or market impact factors are expressed as functions of volume percentage, where the volume percentage is allowed to vary over time or in response to certain market conditions. A corresponding trading strategy for a security within the desired trade may then be provided, with the trading strategy itself being expressed as volume percentage function which varies the volume percentage based market conditions or other constraints.

In this exemplary embodiment, once a selected percentage has been determined, trades of the security are then executed at that percentage. Specifically, the market trading volume of the security may be tracked, and orders for the security are placed so that the volume of orders compared to the market trading volume is substantially equivalent to the corresponding percentage (i.e., is substantially equal to the trading strategy). It is understood that the actual trading volume can be maintained more or less precisely as the desired percentage, to the extent useful or practical, and that perfect precision is not required. It is understood that the actual time period to complete trading of a particular security may vary, given variations between actual and historic trading volumes for the security, and the precision with which the trading strategy is executed.

It is understood that the trading solution may be in one form, and then converted to any other useful form such as a preferred trading strategy. Any trading solution that provides a trading strategy directly, or which is presented in a form that may be converted to a useful trading strategy, is understood to "include" the trading strategy.

In particular, it is understood that the invention may be implemented so that the trading solution(s) are provided in any useful manner. In some cases, it may be useful to simply solve the cost equation for the preferred trading strategy directly. In other cases, it may be useful to solve for one type of trading strategy, or some other variable, and then further calculate other trading strategies from the initial form. Such transformations are within the skill of one or ordinary skill in the art. For example, a cost equation may be formed and solved to provide a solution in the form of a trading time. The specific trading times may then be converted to another form of trading strategy, such as a volume percentage or "VWAP" described above. As another example, a cost equation may be formed to provide a solution directly in the form of a volume percentage, which may be converted, for example, to a trading time. These examples are for clarity only, and it is understood that they are not construed as limitations on the invention.

In another aspect of the invention, a method of trading is provided in which multiple cost equations for a desired trade are derived. These multiple cost equations may then be solved to provide a selection of trading strategies.

In summary, with respect to Figure 2, the exemplary method includes forming a plurality of cost equations for a desired trade (201). It is understood that the invention may preferably apply to a desired trade of multiple securities, but may include desired trades of a single security, hi the illustrated form of Figure 2, each of the cost equations includes a weighted set of input factors, understanding that a "weighted set" of input factors may include weights for some factors in the set, and no weights for other factors in the set. The input factors include at least one risk factor and at least one market impact factor. The method further includes solving each of the cost equations to derive a plurality of trading solutions for the desired trade (203). Each trading solution includes a trading strategy for each individual security in the desired security trade. As described above, the trading strategy may be in the form of a trading time, a VWAP strategy, a percentage as described above, or other useful form, including combinations of such forms. The invention may further include selecting one of the trading solutions (205), and executing the trade according to the selected trading solution (207).

For clarity, this exemplary aspect of the invention is discussed with respect to a desired trade involving a portfolio of n securities, where n is the number of individual securities in the portfolio. For purposes of example only, a set of six cost equations will be described, labeled with subscripts "A" through "F." It is understood that any particular number of cost equations may be provided in a set of cost equations according to the invention, but preferred sets include between 3 and 10 cost equations. The labeling A through F is provided by way of example only, for clarity and convenience of description. hi this exemplary form, a set of cost equations may be formed as set [5]:

C_A = R_A + MIi + MI₂+ . . . + MI_n C_B = R_B + MI₁ + MI₂+ . . . + MI_n

[5] Cc = Rc + MI₁ + MI₂+ . . . + MI_n

C_D = R_D + MI₁ + MI₂ + . . . + MI_n

C_E - R_E + MI₁ + MI₂+ . . . + MI_n

C_F = R_F + MI₁ + MI₂ + . . . + MI_n

where: (1) CA is the cost for equation A, R_A is the risk factor or group of risk factors for equation A, MI₁ is the market impact function associated with the first security of the portfolio, MI₂ is the market impact function of the second security of the portfolio, and so on for equation A; (2) C_B is the cost for equation B, R_B is the risk factor or group of risk factors for equation B, MI₁ is the market impact function associated with the first security of the portfolio, MI₂ is the market impact function of the second security of the portfolio, and so on for equation B; (3) and so on for each cost equation C through F. hi this exemplary form, the same market impact factors are utilized for each cost equation A through F, and no weight is applied to any market impact factor, although the invention is not limited to this form.

The set of equations [5] is preferably constructed with weighted risk factors. Using the same four exemplary risk factors of equations [3] and [4] above, the set of cost equations A through F may be provided as shown in set of equations [6]: C_A = wl _A'RI + w2_A*R2 + w3_A'R3 + w4_A*R4 + MI₁ + MI₂ + . . . + MI_n C_B = W1_B"R1 + W2B'R2 + w3_B'R3 + w4_B ^«R4 + MI₁ + MI₂ + . . . + MI_n [6] C_c = wlc'Rl + w2c'R2 + w3_C"R3 + w4_c-R4 + MI₁ + MI₂ + . . . + MI_n C_D = wl_D ^«Rl + w2_D ^«R2 + w3_D ^«R3 + w4_D ^»R4 + MI₁ + MI₂ + . . . + MI_n

C_E = WI_E'RI + w2_E*R2 + w3_E ^»R3 + w4_E ^»R4 + MI₁ + MI₂ + . . . + MI_n Cp = wlp'Rl + w2_F ^»R2 + w3_F*R3 + w4_F ^«R4 + MI₁ + MI₂ + . . . + MI_n

where (1) C_A is the cost for equation A; WIA'RI is the first weighted risk factor of equation A, W2_A ^ΦR2 is the second weighted risk factor of equation A, etc. ; MI₁ is the market impact function associated with the first security of the portfolio, MI₂ is the market impact function of the second security of the portfolio, and so on for equation A; (2) C_B is the cost for equation B, wl_B'Rl is the first weighted risk factor of equation B, W2_B*R2 is the second weighted risk factor of equation B, etc.; MI₁ is the market impact function associated with the first security of the portfolio, MI₂ is the market impact function of the second security of the portfolio, and so on for equation B; (3) and so on for cost equations C through F.

A set of cost equations according to this aspect of the invention may preferably be formed so that each cost equation in the set includes a unique set of weighted risk factors. As used herein, a "unique" set of weighted risk factors does not require that each weight is itself unique, but only that the group of weights for the various cost equations are not identical. For example, assuming three separate risk factors are used in a set of cost equations, the group of weights {0.4, 0.5, 0.6} is considered different from the group of weights {0.3, 0.5, 0.6}. A set of two cost equations having these groups of weights is understood to have a "unique" set of weights associated with each cost equation. It is further understood that a weight can be zero, which corresponds to the absence of a given risk factor in a given cost equation. Once a set of cost equations is formed, each of the cost equations maybe solved as described above to provide a set of trading solutions. In the example described above, solving the cost equations will provide six separate trading solutions, associated with cost equations A through F, respectively. Each of these trading solutions will include trading strategies for each of the securities in the desired trade. Each trading strategy may take any useful form, as described above, for example a trading time, a VWAP strategy, a percentage as described above, or other useful form, including combinations of such forms.

Once the trading solutions have been calculated, an investor or other person or persons may select one of the trading solutions to execute the trade, as described above. This decision may be based on any useful information. Frequently, investors may wish to keep the bases for such decisions confidential from other parties such as brokers or competitors. A plurality of trading solutions provides an opportunity to obtain a preferred, optimized trading strategy without revealing the reasons for the preference. It should be understood that the system and method can be executed on an iterative basis, so that if an initial set of cost equations fails to generate an acceptable trading solution, a new set of cost equations may be formed with different factors or weights, thereby providing a new set of trading solutions for a desired trade.

It is understood that the labeling of the above equations and solutions is for clarity of description only, and does not limit the invention. In an alternative nomenclature, any two cost equations in a set of multiple cost equations may be designated as a first and second cost equation, which may be solved to produce first and second solutions. An investor or other user may then select one of the first and second solutions to execute the trade. It is understood that even in the case of more than two cost equations, two equations and solutions may be designated so that the selected trading solution is among the first and second trading solutions.

Figure 3 illustrates an exemplary system according to the invention, such as a computer system having, for example, a processor 301 (or multiple processors) running useful software, and associated memory 303 (or multiple memory devices). Processor 301 and memory 303 may, for example, be connected through a local bus 305. In an alternative embodiment, a system according to the invention may be a distributed system or network, in which processor 301, memory 303, and/or other parts of the system are geographically or topographically distributed, for example over a Local Area Network, Wide Area Network, the Internet, or other network or communication link. Alternatively, memory 303 may be present as a part of processor 301. As understood in the art, software may be provided to perform one or more of the functions described above, using relevant data. Data may be input directly from an external source (such as a keyboard, touch screen, modem, etc.) or may be provided from memory 303. Processor 301 may cooperate with memory 303, as understood in the art, to run software for performing one or more of the steps described above. For example, processor 301, in cooperation with memory 303, may calculate one or more trading solutions as described above, and an investor may then select one of the trading solutions to execute the desired trade. A number of embodiments of the invention have been described.

Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. For example, a set of multiple cost equations may include equations with weighted market impact factors, if desired. Accordingly, other embodiments are within the scope of the following claims.

Claims

What is claimed is:

1. A method of optimizing a desired trade of a plurality of securities, comprising: forming a cost equation for the desired trade, the cost equation including at least one risk factor and including a plurality of market impact factors, wherein each of the plurality of market impact factors is associated with a corresponding security; solving the cost equation to derive a trading solution, the trading solution including a trading strategy for at least one of the plurality of securities.

2. The method according to claim 1, wherein the trading solution includes a trading strategy for each of the plurality of securities.

3. The method according to claim 1, further comprising: executing the desired trade by implementing the trading solution.

4. The method according to claim 1, wherein each trading strategy is defined as a percentage of market trading volume of the corresponding security.

5. The method of claim 1, wherein the plurality of securities includes a plurality of stocks.

6. The method of claim 1 , wherein each market impact factor is a function of a volume percentage.

7. The method of claim 1, wherein the cost equation includes a plurality of risk factors, the plurality of risk factors including at least two of the group consisting of a portfolio volatility, a portfolio variance, a portfolio exposure, and a portfolio expected drift.

8. The method of claim 7, wherein each market impact factor is a function of a volume percentage.

9 The method of claim 1, wherein the cost equation includes a plurality of risk factors including of a portfolio volatility, a portfolio variance, a portfolio exposure, and a portfolio expected drift.

10. A method of trading, comprising: forming a plurality of cost equations for a desired trade, wherein the desired trade includes at least one individual security, and wherein each cost equation includes a set of input factors, the input factors of each cost equation including at least one weighted risk factor and at least one market impact factor; solving each of the cost equations to derive a plurality of trading solutions for the desired trade, wherein each trading solution includes a trading strategy for at least one individual security in the desired trade.

11. The method according to claim 10, wherein each trading solution includes a trading strategy for each individual security in the desired trade.

12. The method according to claim 10, further comprising: selecting one of the plurality of trading solutions, and executing the desired trade according to the selected trading solution.

13. The method according to claim 10, wherein the desired trade includes a plurality of individual securities.

14. The method according to claim 10, wherein each trading strategy is defined as a percentage of market trading volume of the corresponding security.

15. The method according to claim 10, wherein the desired trade includes trading at least one stock.

16. The method according to claim 10, wherein the risk factors include at least two of the group consisting of a portfolio volatility, a portfolio variance, a portfolio exposure, and a portfolio expected drift.

17. The method according to claim 10, wherein the risk factors include a portfolio volatility, a portfolio variance, a portfolio exposure, and a portfolio expected drift.

18. The method according to claim 10, wherein at least one of the risk factors is weighted, and wherein each cost equation includes a unique set of weighted risk factors.

19. A method of trading at least one security, comprising: forming a first cost equation for a desired trade, the first cost equation including a set of input factors including a risk factor, the risk factor having a first weight; forming a second cost equation for the desired trade, the second cost equation including the set of input factors including the risk factor, the risk factor having a second weight; solving the first cost equation to derive a first trading solution for the desired trade; solving the second cost equation to derive a second trading solution for the desired trade.

20. The method according to claim 19, further comprising: selecting one of the first trading solution and the second trading solution, and executing the desired trade according to the selected trading solution.

The method according to claim 19, wherein the desired trade includes a plurality of individual securities.

21. The method according to claim 19, wherein each trading solution includes a trading strategy for each individual security in the desired security trade, and wherein each trading strategy is defined as a percentage of market trading volume of the corresponding security.

22. The method according to claim 19, wherein the set of input factors includes a plurality of risk factors, the plurality of risk factors including at least two of the group consisting of a portfolio volatility, a portfolio variance, a portfolio exposure, and a portfolio expected drift.