CA2264758A1 - Method and apparatus for automated opening of options exchange - Google Patents

Abstract

A computer-based system for determining a set of opening prices for a number of series of options traded on an options exchange and for allocating public order imbalances at the opening of trading. Market makers input a current position, a desired target position and market maker orders (104) for options series from market maker terminals. An order entry system receives public orders (102) for options series. A controller (2) determines a set of implied volatilities (prices) for each options series that will maximize a weighted volume of trades across all options series at the opening. Contra orders than can be matched at the opening price are then executed. If there is a residual imbalance of non-executed public orders, the residual imbalance of non-executed public orders is assigned to individual ones of the plurality of market makers so as to minimize a cumulative measure of deviation between the desired target position and the current position of each market maker. The system is applicable to an options exchange, this term including any facility operating an over-the-counter market in options.

Description

CA 02264758 1999-03-03W0 98/12659 PCTIUS97/15665101520253035Method and Apparatus For Automated Opening ofOptions Exchange. 1: E !.The present invention is directed to a method andapparatus for the automated opening or reopening of anoptions exchange, and more particularly, to a method andapparatus for opening all option series substantiallysimultaneously when opening or reopening trading on anoptions exchange and for optimization of the allocationof public order imbalances to market makers.: . 1! H I.A portion of the disclosure of this patent documentcontains material which is subject to copyrightprotection. The copyright owner has no objection to thefacsimile reproduction by anyone of the patent documentor patent disclosure as it appears in the Patent andTrademark Office patent file or records, but otherwisereserves all copyright rights whatsoever.SUBSTITUTE SHEET (RULE 26)101520253035WO 98/12659CA 02264758 1999-03-03mQmmm There are presently five equity options exchanges in theUnited States and approximately fifty exchanges that tradein options throughout the world. Options are traded on anumber of financial such as,instruments, for example,stocks, currencies, Treasury instruments, interest rates,market indices, commodities and the like.When an options exchange opens trading each morning, orreopens trading after a trading halt in the underlyinginstrument during the trading day, the exchange conducts anopening “rotation” procedure to determine the opening pricefor each option. The opening rotation may take upwards of45 minutes, during which time the price of the underlyinginstrument may change dramatically. Presently, the openingrotation consumes a significant portion of the trading day.Additionally, present methods used by options exchanges toallocate the residual imbalance in public orders to marketmakers at the opening often results in undesirable andinefficient allocations.To better explain the problems associated with opening andreopening of trading in an options market, an explanationof options is in order. For the purpose of clarity, thispatent will discuss United States exchange-traded equityoptions. However, it will be appreciated that thediscussion herein also applies to (a) options on otherfinancial instruments traded on U.S. and non-U.S.exchanges, and (b) options of all types that are traded onnon—U.S. exchanges.‘ An equity option is a securities contract which conveys toits owner the right, but not the obligation, to buy or sella particular stock (called the underlying)price (called the strike price)at a specificon or before a given date.SUBSTITUTE SHEET (RULE 25)PCT/US97/15665101520253035W0 98/ 12659CA 02264758 1999-03-033Typically,Put options and Call options.there are two basic types of options, namely,An American-style equityCall option gives its owner the right to buy 100 shares ofthe underlying stock at the strike price on or before agiven date. An American-style equity Put option gives itsowner the right to sell 100 shares of the underlying stockat the strike price on or before a given date.United States,(In theone option contract normally equals 100shares.) For American—style options, the owner of theoption can exercise the contract at any time prior toexpiration. For European-style options, the option cannotbe exercised until thelast trading day prior toexpiration.Equity options are generally traded on United Statesoptions exchanges at any time a pricing mechanism existsfor the underlying instrument, for example, approximatelyduring the normal hours of operation of U.S. securitiesexchanges.The expiration for an option contract is typically theSaturday following the third Friday of the expiration monthfor the particular contract. Thus, the third Friday of themonth is the last trading day for all equity optionsexpiring that month. If the owner of the option does notexercise the option prior to expiration, then the optionexpires thereafter giving no rights to the owner andplacing no obligation on the writer. (The writer is theperson who assumes, for a Call option, the obligation tosell stock, or for a Put option, the obligation to buystock.)Assume that PQR Corp. is a publicly traded stock which alsohas publicly traded options. A typical option for thisstock might be a PQR October 70 Call. A PQR October 70SUBSTITUTE SHEET (RULE 25)PCT/US97/ 15665101520253035WO 98/12659CA 02264758 1999-03-034Call option is a contract giving the owner of the contractthe right to buy 100 shares of PQR Corp. stock at $70 pershare until the third Saturday in October, if the buyerchooses to exercise that right.Generally, there are four expiration months available foreach equity option. Moreover, often there are three ormore strike prices available for each expiration month ofThus, for a single stock, there willlikely be at least 24 and very possibly many more optionstraded for a particular underlying.each equity option.(It is not unusual tohave 60 different options available for a single stock.)for PQR Corp.may be traded on an options exchange:PQR January 70 PutPQR April 70 PutPQR July 70 PutPQR January 75 PutPQR April 75 PutPQR July 75 PutPQR January 80 PutPQR April 80 PutPQR July 80 PutFor example, the following Put option seriesThere also would be a similar number of call option series.Thus, it will be apparent that for each underlying stock,there could be dozens of option series, each of which wouldbe differently priced.Therefore, for each underlyingstock, there are many option series that must be pricedwhen trading on an option exchange opens or reopens.The following terms are often used by options traders. Anoption “type” is either a Put or Call. An option “class”consists of option contracts having the same underlyingSUBSTITUTE SHEET (RULE 26)PCTIUS97ll5665101520253035W0 98/ 12659CA 02264758 1999-03-03PCTfUS97/ 156655security. An option “series” consists of option contractsof the same class having the same strike price andexpiration month. For example, PQR October 60 Callsconstitute a series.A premium is the price an option buyer pays for the rightto buy or sell the underlying security. The premium for anoption contract is usually quoted on a per share basis,e.g., PQR October 60 Call $5% --premiunt is $5%,would be $525.in this example, theand so the cost of the option contractAn option may be “in-the-money,” “at—the-money” or“out—of—the-money.” A Call option is in-the-money if theunderlying stock price is above the strike price, i e., theowner of the Call option has the right to buy stock at aprice which is less than the price the owner would have topay to buy the shares in the open market. A Put option isin—the-money if the underlying stock price is below thestrike price. An option is at-the—money when it has astrike price that is approximately equal to the currentmarket price of the underlying stock. A Call option isout—of—the—money if the underlying stock price is below thestrike price. A Put option is out—of—the—money if theunderlying stock price is above the strike price.The intrinsic value of anoption contract is thein-the-money portion of the option's premium. The timevalue of an option contract is the part of an option'stotal premium that exceeds its intrinsic value -— it is theamount the buyer is willing to pay for an option, above itsintrinsic value, in the hope that prior to expiration itsvalue will increase because of a favorable change in theThus,out—of-the-money option consists entirely of time value.price of the underlying. the premium for anSUBSTITUTE SHEET (RULE 25)101520253035WO 98/12659CA 02264758 1999-03-036Accordingly, the premium for an option contract (the totalprice of the option) is its intrinsic value plus its timevalue.In unusual market conditions, it may happen that the marketpremium for a deep-in—the—money option is actually lowerthan its intrinsic value. This condition may result frominadequate liquidity in the underlying security, whichinduces options market makers to purchase these contractsonly at discount prices with respect to their theoreticalvalues. (This phenomenon has no deleterious impact on theutility of the present invention.)There are five quantifiable factors that influence anoption's price. These factors are:* the underlying security price* the strike price of the option* the time to expiration* the volatility of the underlying security* the current “risk—free” interest rate.Using these five factors as input to a theoretical optionpricing model, the Black-Scholesone can determinesuch as, for example,model or the Cox-Ross-Rubenstein model,the theoretical fair option value. Option traders use theAdditionally,given the current market value of an option, one can use atheoretical option value as a pricing guide.theoretical option pricing model to derive the impliedvolatility of the underlying.There are other non—quantifiable factors that influence anoption's price, such as:* market participants’ estimates of futurevolatility* estimates of future performance of the underlyingSUBSTITUTE SHEET (RULE 25)PCTIU S97/ 15665CA 02264758 1999-03-03101520253035W0 98/12659 PCT/US97/156657stock* supply and demand of the option and of theunderlying* depth of market for the optionTheoretical option pricing models produce values thatreflect an option’s sensitivity to changes in one of thefive quantifiable. factors. These sensitivities areassigned Greek names, such as delta, gamma, theta, rho andvega. Delta is a measure of the rate of change in anoption's theoretical value for a one-unit change in theprice of the underlying security. Thus, delta is thetheoretical amount by which the option price can beexpected to change for a small change in the price of theunderlying. As such, it provides a local measure of theequivalent position risk of an option position with respectto a position in the underlying security. Delta may beexpressed as a percentage, e.g. 63% (or simply “63” withthe percentage symbol implied.) Every option contract hasits own unique theoretical delta determined by the fivequantifiable factors described above.Gamma is a measure of the rate of change in an option'sdelta for a one-unit change in the price of the underlyingsecurity. Gamma expresses how umch the option's deltashould theoretically change for a $1 change in the price ofthe underlying. Gamma is largest when the option isat—the—money. As the underlying stock's price moves awayfrom the option’s strike price (in either direction), thegama of that option will decrease. Gamma provides a localmeasure of the rate of change of delta with respect to theprice of the underlying.Theta is a measure of the rate of change in an option'stheoretical value for a one-unit change in time to theSUBSTITUTE SHEET (RULE 26)101520253035W0 98/12659CA 02264758 1999-03-03option's expiration date. Vega (also known as kappa) is ameasure of the rate of change in an option's theoreticalvalue for a one—unit change in the volatility of theRho ischange in an option's theoretical value for a one-unitunderlying security. a 1neasure of the rate ofchange in the risk-free interest rateDelta and gamma are the primary measures used by those whotrade in options. For example, if a trader has a portfoliothat has a large absolute value of delta and. a largeabsolute value of gamma, the trader would have a bigposition that is very sensitive to movements in theunderlying’s price, and thus, the trader would have a highrisk exposure.Volatility is a measure of stock price fluctuation.Mathematically, volatility is the annualized standarddeviation of a stock's daily price changes.volatility is theImpliedthattheoretical premium value of an option to match a marketvolatility would cause thepremium value, given fixed values of the remaining fourquantifiable factors.It will be appreciated that trading in options is a complexmatter, particularly in light of the number of differentoption contracts that are listed on an options exchange,their interrelationships, and their relationship with theunderlying stock. Each series will have differentpremiums, deltas, gammas and public order supply/demandcharacteristics. Increasing the complexity of thesituation. is that options can also be traded. on stockindices, such as the S&P5OO index.United States options markets are typically conducted usingan “open outcry" trading method, by which competing floorSUBSTITUTE SHEET (RULE 26)PCT/U S97/ 15665101520253035WO 98/12659CA 02264758 1999-03-03PCT/U S97] 156659brokers, representing public orders, and market makerstrading for their own accounts make bids and offers on thetrading floor. Typically, trading takes place in a tradingpit -- a specific location on the trading floor of anexchange designated for the trading of a specific optionclass. A market maker is an exchange member on the tradingfloor who buys and sells options for his or her own accountand who has the responsibility of making bids and offersand maintaining a fair and orderly market. A floor brokeris a trader on. an exchange floor" who executes tradingorders for the public.At the opening (or reopening) of an options exchange, theexchange conducts an opening rotation procedure todetermine the opening price for each option. It will beappreciated that at an opening or reopening, there are adisproportionate number of public buyers and sellersrepresented in each series at any particular opening price.There are many series to open; presently, the opening priceof each is determined in rotation. For each series, takinginto account public order supply and demand, marker makerssignal their bid and offering prices, which converge to anagreed opening price for that series. All public ordertrades that can be matched at that price are executed, andtheregenerally will be, at thisimbalance of non-matched. orders.stage, a residualThe opening for allseries in an options class may take upwards of 45 minutes,during which time the price of the underlying instrumentmay change dramatically, resulting in price discrepanciesacross series.when opening multiple series of options in a commonunderlying security, variations in public order bids andoffers across the series also will produce discrepancies inopening prices with respect to the theoretical prices thatSUBSTITUTE SHEET (RULE 25)101520253035W0 98/12659CA 02264758 1999-03-0310would correspond to a single implied volatility across allseries. For example, a large imbalance of buy orders inone particular series might cause it to open at asignificantly higher implied volatility than that of otherseries. Imbalances between public bids and offers at theopening must be rectified by the market makers.stated above,Also, asvariations in the price of the underlyingsecurity during the time required to cycle through theopening can result in further significant discrepancies inopening prices across the series. For example, if theunderlying stock or index: price is falling rapidly, alater-opening call that would have been in-the-money at thebeginning of the opening rotation might actually be openedat a lower price than an earlier-opening call that wasopened at—the-money.On some exchanges, once a series is opened, it cannot betraded until the end of the opening rotation. Since ittakes time to perform the opening rotation, regular tradingin a particular series may be delayed for a significantperiod of time.Thus, in summary, the present opening method used by optionexchanges takes an undue amount of time and results ininconsistencies in pricing between related series.There are additional problems related to the presentopening methods used by options exchanges,example,such as, forthe mechanism by which the residual imbalance inpublic orders is allocated to market makers.makers asThe marketa group have an obligation to satisfy theresidual imbalance in public orders at the opening price.The present method ofaccomplishing this goal is around-robin assignment of the residual contracts to eachmarket maker. This method of assignment often results inSUBSTITUTE SHEET (RULE 26)PCT/US97I15665101520253035W0 98/12659CA 02264758 1999-03-03PCTIU S97/ 156651 lundesirable and inefficient allocations. For example, onemarket maker who is short delta might like to be a buyer ofoptions contracts and another market maker who is long maywant to increase his long position. Each market makercomes to the opening with his or her own current positionand his or her desired target position after the opening.No attempt is made to meet these desires in the allocationof public order residual balances. As each market makerhas unique desires, the present round-robin allocation ofthe residual balance in public orders may not improve, andcould possibly worsen, the actual position of each marketmaker with respect to their desired position.Accordingly, an opening method for trading on an optionsexchange is needed that enables the simultaneous opening ofan options market and that takes into account public ordersupply and demand and the consistency in pricing betweenvarious series. Most desirably, an opening method isneeded that simultaneously determines an opening price forall series, where the opening price arrived at is areasonable compromise between (a) having the opening priceconsistent across all series with respect to impliedvolatility and (b) having an opening price that caters tothe variations in public order supply and demand. Further,there is a need for a method of allocation of residualpublicorder imbalancesamongst market makers at theopening that optimizes each market marker's position withrespect to his or her desired position.flm The present invention is a computer-implemented method andsystem that provides an automated simultaneous opening oftrading on an options exchange. The present inventionpermits flexibility in the degree to which public orderSUBSTITUTE SHEET (RULE 26)101520253035W0 98l12659CA 02264758 1999-03-0312supply and demand imbalancesacross a seriesimposevariations in opening implied volatilities, while at thesame time providing for optimal assignment of market makersto offset the residual imbalances.As used herein, the term “opening” also includes“reopeningJ’ Accordingly, the present invention can beused to reopen trading on an options exchange after ithalts trading during thetrading day. The presentinvention can also be used to facilitate trading when acustomer requests a call market in one or more optionseries,i.e., a demand call.The present invention determines the opening prices foreach option series so that all option series can be openedsimultaneously on the option exchange.The present invention allows all option series to be openedsimultaneously at a price that:I. corresponds reasonably well to a single value ofimplied. volatility (or in the more general case, to alinked set of implied volatilities that correspond to apredetermined implied volatility skew vs. strike pricerelationship);II. optimizes volume (or weighted measures of volume)across all series; andIII. enables options market makers to balance thevariations in supply and demand of public orders in eachseries at prices that provide an incentive to the marketmakers.The first feature (I) avoids gross inconsistencies inimplied volatility at the opening, while still allowingsome latitude for variations in market demand. The secondfeature (II) meets the general purpose of an optionSUBSTITUTE SHEET (RULE 26)PCT/US97/15665101520253035W0 98/ 12659CA 02264758 1999-03-0313exchange, i.e., to maximize the mutual satisfaction amongall participants to the greatest degree possible.The third feature (III)current methods for the assignment of market makers to theprovides an improvement overrequired positions in the opening. Each market maker comesto the opening with his or her own current position (asspecified by delta and game) and his or her desired targetposition. after the opening (as specified. by delta andgamma). The desired target position may be dependent uponthe absolute and relative values of implied volatilitiesdetermined at the opening. These current and targetpositions impact the market maker'spreferences onparticipation in the rectification of imbalances in publicorders among the different series. Thus, according to theprinciples of the present invention, the assignment ofpublic orders can be optimized across all market makers.The present invention proceeds in two stages. At the firststage, the present invention determines a reasonablyconsistent set of implied volatilities that will maximizea weighted volume of trades across all series at theopening. At the completion of the first stage,theregenerally will be a residual imbalance in the public ordersin each series that do not match off between buyers andsellers. These residual imbalances among public orders arerequired to be offset by assigning contra positions to themarket makers. Accordingly, at the second stage, thepresent invention assigns residual public orders to marketmakers so as to minimize a cumulative measure of deviationbetween the post—opening desired target positions and theactual positions of each market maker at the conclusion ofthe first stage.In the representative embodiment of the present invention,SUBSTITUTE SHEET (RULE 25)PCT/U S97/ 15665101520253035W0 98/ 12659CA 02264758 1999-03-03PCTIU S97/ 1566514the first stage can be formulated as an optimizationproblem. At one extreme, there is the optimum impliedvolatility that can be determined so that there is absoluteconsistency in implied volatility across all series. Atthe other extreme, there is the optimum volatility of eachindividual series that can be determined to satisfy marketsupply and demand. The present invention computes a set ofopening implied volatilities that set acompromise between these extremes.reasonableFrom these impliedvolatility value(s), the corresponding price is determinedfor each option series. The present invention also enablesan exchange (or other entity) to determine the compromisepoint between these two positions. Alternatively, thiscompromise point can be market driven at the opening by anumber of predetermined variables and/or be required tofall within specified bounds.It will be appreciated that, with the appropriate computerhardware and software,for allsimultaneously.the opening volatilities and priceseries can be determined substantiallyThe opening prices and corresponding volatilities, oncedetermined by the present invention, can be output tomarket makers(and, if seen as desirable, to otherinterested parties) so as to assist market makers todetermine their post-opening desired target position.In the representative embodiment of the present invention,the second stage can be regarded as a subsequentoptimization problem that minimizes a cumulative measure ofdeviation between post-opening desired target and actualpositions of market makers, subject to the constraint thatall public order imbalances across the series must beoffset. This second optimization problem can be solvedSUBSTITUTE SHEET (RULE 25)101520253035W0 98/12659CA 02264758 1999-03-03PCT/US97ll56651 5according to the present invention as a quadratic integerprograming problem with linear equality constraints.At the second stage, each market maker supplies as inputhis or her current delta and gama positions prior to theopening and his or her desired delta and gamma positionsafter the opening.theta,variables.)(If required, other measures, such asalso could be included as targetPublic orders are allocated to market makersrho and Vega,according to the solution to this second optimizationproblem.The first and the second stage are independent and eithercould be implemented without the other. For example, anoptions exchange may implement only the first stage todetermine the opening price for each option but decide toallocate residual public order imbalances using the presentrobin—robin assignment of the residual contracts to eachmarket maker. Alternatively, an options exchange mayimplement only the second stage, using the present openingrotation procedure to determine the opening price for eachoption but utilizing the present invention to allocateresidual public order imbalances to market makers. Byextension, the present invention can also be used to effecttrading in periodic or event driven call market structures(the latter not to be confused with call option types).It also will be appreciated that the principles of thepresent invention can be used to open trading on any marketwhere there are multiple instruments that each derive theirvalue from an underlying instrument. For example, thepresent invention could be adapted for use in the openingof trading on a futures market. The present inventioncould also be adapted for use in the bond market (withinterest rates taking the place of implied volatilities.)SUBSTITUTE SHEET (RULE 26)1015202530W0 98/12659CA 02264758 1999-03-03PCT/US97ll56651 6Thus, as used herein, the term “option” includes, whereappropriate, futures, bonds and instruments that derivetheir value from an underlying instrument or parameter.lE.E: I” EH1.Fig. 1 is a block diagram of the overall system of thepresent invention.Fig. 2 summarizes the inputs and outputs of the presentinvention.Figs. 3A and 3B show values relating to example one.Figs. 4A and 4B show a plot of the optimum and round-robin assignment errors for example one.Figs. 5A and 5B show values relating to example two.Figs. 6A and 6B show a plot of the optimum and round-robin assignment errors for example two.I .1 1 I . I.Referring now to the drawings, and initially Fig. 1,there is illustrated a block diagram of an overallarchitecture according to a representative embodiment ofthe present invention. A central controller 2 controlsthe opening and allocation process of the presentinvention. Coupled to the controller 2 are a number ofmarket maker terminals 4-10 that can be used to provideinformation to and receive information from theSUBSTITUTE SHEET (RULE 26)101520253035WO 98/12659CA 02264758 1999-03-03PCT/US97/1566517controller 2. Also coupled to the controller 2 are oneor more input/output devices 12 that can be used by anexchange to provide information to, control the operationof and receive information from the controller 2.Information about public orders can be input to thecontroller 2 from an order entry system 14. The orderentry system 14 can comprise regular options order entryterminals or, for example, the order entry terminals asdescribed in PCT Application No. PCT/US96/07265, filedApril 26, 1996 titled “Crossing Network UtilizingSatisfaction Density Profile With Price DiscoveryFeatures” which is expressly incorporated in full hereinby reference.At the market maker terminals 4-10, market makers caninput their orders, their current position, i.e , theirposition at the conclusion of the first stage, and theirdesired target position, i.e., the position they desireto be in after the opening. The current position anddesired target position can be input in the form ofdeltas and gammas.At the first stage, the controller 2 determines a set ofimplied volatilities that will maximize a weighted volumeof offsetting public orders at the opening across alloption series.After determining the set of implied volatilities, thecontroller 2, at the second stage, utilizing the currentposition and the desired target position of each of themarket makers, assigns any residual public orders toindividual market makers so as to minimize a cumulativemeasure of deviation between the desired target positionand the actual position of each market maker after thisSUBSTITUTE SHEET (RULE 26)101520253035W0 98/ 12659CA 02264758 1999-03-03PCT/US97l1566518assignment.For exchanges with large volumes, the controller 2 ispreferably a computer that can perform calculations atrates of multiple gigaflops, such as, for example withpresent technology, an IBM SP2 computer.device 3 is coupled to the controller 2.A storageThe storagedevice 3 can comprise a database for storing informationreceived from the market makers and for storing theresults of processing by the controller 2. Informationreceived from the market makers can be received in theform of files, and stored as such in the storage device3.Typically, each market maker on the exchange will haveaccess to a market maker terminal, such as one of 4—10.The market maker terminals 4-10 can be high poweredpersonal computers or workstations or hand held wirelessentry terminals. The market maker terminals 4-10 cancomunicate with the controller 2. For example, marketmaker terminals may be coupled to the controller 2 over alocal area network (LAN), a wide area network (WAN), viawireless communication protocols or over the Internet.Each market maker terminal includes one or moreinput/output devices that allow for the entry of ordersand current and desired positions and can display output,such as opening volatilities and public orders assignedto the market makers.According to the present invention, the controller 2solves two optimization problems. For purposes ofexplanation, the operation of the present invention canbe considered in two stages, each involving a differentoptimization problem.SUBSTITUTE SHEET (RULE 25)101520253035W0 98/ 12659CA 02264758 1999-03-03PCTlUS97/ 1566519Fig. 2 summarizes the inputs and outputs of each stage ofthe present invention. In stage one, public orders (102)are input to the controller 2 via the order entry system14 and market maker orders (104) are input to the(4-10). Theoutput of the first stage is a set of prices (impliedvolatilities) for each series (106).occur at that price are executed,will be,matched orders.controller 2 via market maker terminalsAll trades that canand there generallyat this stage, a residual imbalance of non-In stage two, each market maker inputsvia the market maker terminals (4-10) his or her actualposition and desired position (108). This input to thepresent invention can be in the form of two sets ofdeltas and gammas from each market maker.(Alternatively, each market maker can input thisinformation as the desired change in delta and gamma.)If stage one was not performed by the present invention,(110)e.g., via the order entry system 14. Theoutput of the second stage is the assignment of theresidual public orders to the market makers (112).residual public orders also must be input to thecontroller 2,First Stage OptimizationThe first stage is to determine a reasonably consistent(which will,incorporate a skew in implied volatility versus strikeprice)set of implied volatilities in general,that will maximize the mutual satisfactionweighted volume of offsetting public orders across allseries at the opening. By “reasonably consistent", it ismeant that some variation in implied volatility amongindividual series is acceptable to accommodate theimbalance in public orders (i.e., a higher impliedvolatility for buy imbalances and lower impliedvolatility for sell imbalances).SUBSTITUTE SHEET (RULE 26)101520253035WO 98/12659CA 02264758 1999-03-03PCT/US97/156652 0At one extreme, the present invention could allow eachseries to open independently, using a single price callthat is based solely on supply and demand in each series,without regard to any resulting implied volatilityinconsistencies. One method that enables each series toopen independently using a single price call is describedin PCT Application No. PCT/US96/07265, filed April 26,1996 titled “Crossing Network Utilizing SatisfactionDensity Profile With Price Discovery Features”. Usingsuch method, the controller 2 would determine an openingprice to maximize mutual satisfaction weighted volumebased upon public supply and demand, i.e., determinewhich implied volatilities will maximize the weightedvolume in each series individually. Alternatively, ifsuch a method was not used, an opening price could bedetermined which maximizes trading volume.Allowing each series to open individually, using a singleprice call that is based solely on supply and demand ineach series would permit the market makers to offset thepublic order imbalances at prices most favorable to them(i.e., buying low, selling high), but might result inlarge implied volatility discrepancies across seriesand/or lower traded volume.At the other extreme, the present invention could insiston the absolute consistency of a single impliedvolatility (or a set of implied volatilities satisfying aprescribed skew in volatility versus strike price),without regard to any corresponding imbalances in publicorders. This would eliminate pricing discrepancies inpubic orders among the individual series, but mightrequire the market makers to offset large buy and sellimbalances at prices less favorable to them.SUBSTITUTE SHEET (RULE 26)1015202530W0 98/12659CA 02264758 1999-03-03PCT /US97/ 156652 lIdeally, the present invention enables control of theoperating point between these extremes in order toprovide a compromise between these conflicting desires.To this end, suppose that a prescribed implied volatilityskew relationship is known, such that for any givenimplied volatility corresponding to a particular strikeprice, the “appropriate” implied volatilities of allother strike prices can computed. This allows consistentprospective opening implied volatilities across theseries, against which to measure public order induceddeviations. Let Q be a vector representing one suchconsistent set of implied volatilities. At the otherextreme, let Q‘ be the vector of implied volatilitiesthat maximizes the respective weighted volumes traded ineach series individually.that Og}xi_<_l .Let Ad be a parameter suchFor the i“ option series, the presentinvention defines the opening implied volatilities as afunction of Q to be6'.=(1-}»I.)OI. + ligi (1)which has the effect of allowing the opening impliedvolatility to vary from one extreme of maximizing theweighted volume traded in each series individually, tothe other extreme of forcing consistency of impliedvolatility across series, as the parameter A varies from0 to 1.This parameter can be set uniformly across all series(e.g., by the options exchange), or can be market-drivenby a number of possible variables at the opening. Forexample, suppose that u and m are the mean and standarddeviation, respectively, of the elements of the vectorQ‘. Then one could specify A1 by the formulaSUBSTITUTE SHEET (RULE 25)1015202530CA 02264758 1999-03-03W0 98l12659 PCT/US97/1566522/2,; ll(0k1.=1-tanh a (2)where tanh(x) is the hyperbolic tangent function. SinceAiin this equation will tend to zero as Qfl deviatessignificantly from u, this has the effect in equation (1)of “pushing” the opening implied volatility in the inoption series toward Q1‘ when the latter variable is anoutlier among the set of individual series’ impliedvolatilities.(The parameter ‘a’ in equation (2)controls the rate at which Ai'varies between 0 and 1, as afunction of the remaining portion of the argument.)The final step is to determine the (coordinated) value(s)of Q5 in equation (1) for which the corresponding valuesof 6i maximize the total weighted volume traded over alloption series. For these particular volatility values(s),the corresponding price is determined for each series,and the two nearest trading increments above and belowthis price are identified. If there exists at both priceincrements a corresponding residual imbalance of buyersamong the public orders, the controller 2 sets as theopening price the higher of the two, and vice versa forthe case of a seller imbalance at both prices. If theimbalance switches from buyers to sellers between the twoprice increments, the controller 2 selects as the openingprice the one yielding the higher mutual satisfactionweighted volume in that series. (In the pathologicalcase where there existed no public buy (sell) orders inany series that could be used to establish an openingprice, the present invention could select the respectiveprices just below (above) the minimum (maximum) impliedvolatility across all series corresponding to a unitysatisfaction value among all limit sell (buy) orders.)SUBSTITUTE SHEET (RULE 25)W0 98/126591015202530CA 02264758 1999-03-03PCT/US97/156652 3Another method of choosing the A1 has the objective ofmaximizing the profit potential of the market makers inthe process of offsetting public order imbalances,perhaps within some limits on the allowable deviationfrom a consistent set of implied volatilities. Thecontroller 2 determines each set of consistent impliedvolatilities Q over the range of possible opening impliedFor each gg in this set, the controller 2determines the side (buy or sell) of the correspondingvolatilities.public order imbalance in the series. For a buyimbalance, the controller 2 steps upward in priceincrements from the price pj corresponding to gj, and ateach price pfim computes the product of the remaining buyimbalance times the price deviation (pyy — pj), whichrepresents the potential profit available to the marketmakers by selling at price pfix and unwinding their shortposition at price pj. The analogous procedure is followedfor sell imbalances, stepping down in price. In eachseries, the controller 2 finds the price for which thisprofit potential is maximized, perhaps with an exchange-specified bound on the maximum deviation from pj to beallowed. The controller 2 then picks the opening pricesas the set from the above procedure that maximizes theweighted volume traded across all series. The fii valuescorresponding to these prices are then computed.Once the values of 51 are determined by one of the aboveprocedures, they are substituted into the 0 arguments inequations (3) through (10) below. At the exchanges'soption, these opening volatilities could be output to themarket makers (or other interested persons) prior toproceeding into the second stage optimization of marketmaker assignments to public order imbalances. This wouldgive the market makers more information upon which to1015202530W0 98/ 12659CA 02264758 1999-03-03PCT/US97/ 156652 4determine their delta and gamma targets as detailedbelow.Second Stage OptimizationAfter the above step, there generally will be a residualimbalance in the public orders in each series that doesnot match off between buyers and sellers.exchange rules,According tothese residual imbalances among publicorders must be offset by assigning the contra positionsto the market makers.The market maker assignment problem at the opening can beformulated as an optimization problem that minimizes acumulative measure of deviation between the post-openingdesired target and actual positions of the market makers,subject to the constraints that all public orderimbalances across the series must be offset. To thisend, define the following variables:N number of option series in the underlying security(calls and puts combined)K number of market makers in the option classDihfl size imbalance (in the number of contracts) inpublic orders for series i at implied volatility6 (positive integer for more buyers, negativeinteger for more sellers)Jg(G) delta position of j“ market maker prior toopening at implied volatility oxj(U) desired delta position of j“ market maker afteropening at implied volatility o}g(o) gamma position of j” market maker prior toopening at implied volatility o§j(o) desired gamma position of j” market maker afteropening at implied volatility oAi(o ) delta value of i“ series at implied volatility GSUBSTITUTE SHEET (RULE 25)1015202530WO 98112659CA 02264758 1999-03-03PCT/U S97I 156652 5Pi(o ) gamma value of i“ series at implied volatility oFor the i” series, assume the j“ market maker changes hisposition by m” contracts at the opening assignment(positive if buying, negative if selling). The resultingchange in his delta position is given by6U=100n%:A16), (3)and the corresponding change in his gamma position isgiven byyU=100n%rF(6) (4)The net change in the j“ market maker's delta positionover all series is thenN N 5,] = loom”.-A,(o,), <5)and the corresponding change in the j“ market maker'sgamma position over all series isN N27,] = V;1oom,.J_-1‘,(a,.) (5)i=1 1:1LetNeA].(6,) =)§ — x]. + 1O0mU°Aj(6,.) (7)be the "error" between the j“ market maker's desireddelta position after the opening and his actual positionwere he to change his positions by mg in each of therespective series. Similarly, letNeU(a,) : yj. - y} + 100mU.-1"J.(6,.) (8)be the "error" between the j“ market maker's desiredgama position after the opening and his actual positionwere he to change his positions by mg in each of therespective series.Assuming there are K market makers, the market makerassignment problem is to find the values of mgSUBSTlTUTE SHEET (RULE 25)CA 02264758 1999-03-03W0 98l12659 PCT/US97/156652 6(i=1,..,N;j=1,..,K) that minimize the sum-squared deviationerrors of all the market makersN;[eAJ.(a,)2 + e,J.(a,.)2], (9)subject to the constraintsMEm,.J.+D,(a,.):o i=1,..,N, (10)1:1which reflect the requirement that public orderimbalances in each series be offset exactly by the marketmakers. The present invention can be augmentedstraightforwardly to extend the target variables in (9)10 to include other Greek parameters associated with options(e.g., theta, rho, Vega) and also to extend (10) toinclude market maker specified inequality constraints oncapital commitment.15 According to the present invention, this secondoptimization problem is a quadratic integer programmingproblem with linear equality constraints. In order torewrite equations (7)—(10) in matrix notation, we define11 MIXM: ]: .. <11)'1NI MN};I]M</> : Mk : I 20 NJA=[A,(6l) AN(6N)]"' (13)I‘=[I‘,(61) 1"N(6N)]T <14)C=[£1-xl £K—xk]T (15)E=[}71-yl _}7K—yK]T (16)25 D= [D1(a,) DN(6N)]T (17)Equations (7) and (8) can be written in matrix notationSUBSTITUTE SHEET (RULE 26)10152025CA 02264758 1999-03-03W0 98/121559 PCTfUS97Il566527aseA=C—1OO(ATM)T (18)e,=:-1oo(I‘TM)’ (19)The optimization problem then becomes, from (9) and (10),to findmin(cT — 1ooATM)(c — 100M TA) + ((7 — 1ooI‘TM)(§ — 100M 71‘) (2 0,Msubject to the constraintMIX = —Da (2 1)where 1K denotes a K-dimensional column vector, all ofwhose elements are unity.(21)Adjoining the constraintequation to the objective function (20) using avector Lagrange multiplier A, and then setting thepartial derivatives of this expression with respect toMk(the rows of M)and A to zero, results in thefollowing equations for the optimum M:—200Ak(C - 1 ooA31TA) —20o1“k(g — IOOMTI‘) +1 1k K=0, k=1..N, <22)A5I1K+D=0 <23)pre-multiplying (22) with the vector 1; and employing(23), results inA = 2_I‘:")(1,7;c+1ooATD)A +(1,Z<:+1oo1"’D)1*J. (24)Substituting this expression into (22) yields thefollowing matrix equation for M1XUW=Y§ (25)whereXEAAT+PF{ (26)1 1 1=]—O—6-{[C—E(1fC+100ATD)1k]AT+[E--IZ(1ZE+100A7T')1k]I‘7} (27)SUBSTITUTE SHEET (RULE 25)10152025W0 98/12659CA 02264758 1999-03-03PCT/US97/156652 8Note that both X? and YT are rank 2 matrices whosecolumns are linear combinations of the vectors A and F.Thus 2? and the augmented matrix [AF:YT] have the samerank (=2) and therefore a solution of (25) exists.This solution may be expressed in the formM=M+V, (28)where.M is a rank 2 matrix whose columns are linearcombinations of the vectors A and P, and V is a matrixwhose columns are orthogonal to A and F. Thus M can bewritten asM= [AEI‘]A, (29)where A is a 2xK matrix representing the coefficients ofthe linear combinations of A and F embodied in thecolumns of M. From (26) and (29), we haveXTA71=(AAT+I‘1“7)[A 21‘]A=[|A!’A+(A7I‘)1‘s(A7I‘)A+|I‘[’I‘]A <30)= [(A,.J.) A I 2 +A2}A7I")A +(A UAT]."+A2}.]I"|2)I‘]where the term within the brackets above represents thej“ column of an NxK matrix. Using (25), and equatingcoefficients of A and P between (27) and (30), we obtain:7-l(1,f §+100ATD)l T2 T k‘A! A F 1 k (31)3A=1—:—ATF |1‘|2 10°57-%(1,f g+1oo1‘TD)1,fOrSUBSTITUTE SHEET (RULE 25)1015202530CA 02264758 1999-03-03 W0 98/1255929A = B "ZwherecT~l(1,f c+1ooA'*'D)1,f(AP ATI‘ d 1 k: an :___A71‘ IP12’ 100 gT—.]1;(1,f g+10oI‘TD)1,fThus, from (29) and (32), we can write1v7=[A21‘]B“ZFrom (23) and (28), we have(A/~I+V)1k=-Dthus showing, after some algebra, thatA711k=CD,where C is an NxN rank 2 matrix given byC: ——|%|-(A[|I‘l2 AT-(ATI')I‘TJ+n)A|21‘T-(ATI‘)ATJ}.PCT/US97/ 15665(32)(33)(34)Therefore, from (35) and (36), we have the followingconstraint equation for V:V1k=—(IN + C)D,(38)where IN is the N-dimensional identity matrix, along withthe orthogonality constraints implicit in the definitionOf V:ATV: 0,PTV = 0.Determining the minimum—norm solution V,(39)is equivalentto finding the solution to the optimization problemSUBSTITUTE SHEET (RULE 26)CA 02264758 1999-03-03W0 98/12659 PCT/US97I1566530rnintrace(VTV), (40)Vsubject to the constraints in (38) and (39). Again usingthe technique of Lagrange multipliers, results in thefollowing set of equations satisfied by the optimum:2VW61Z+ApT+PnT=Ql71k+(IN+C)D=0ATI?'=o,5 FTI7=0.(41)Since the latter two equations require the columns of 9to be orthogonal to both A and P, the term in the firstequation involving 8 must exactly cancel the two termsinvolving p and n, and thus we have10 2I?+e’1{,=o (42>Where 9' is orthogonal to A and F.Postmultiplying by 1k and using the second equationyields26’=—(IN +C)D. (43)K"15 Therefore1?: - _‘.(1N +C)D1,f. (44)KHence, combining (28), (34) and (44), results in aclosed-form optimal solution for the assignments of20 offsetting contracts to each market maker in order torectify the imbalances in public orders:A3I=[AsI‘]B"Z—%(I,+C)D1,:. (45)From (18), (19) and (43) that the optimal errors aregiven byéA = i(1,:C+100A7D)1K,K"1 T (46)~ _ 7er _ E(1Kg+10oI‘ D)1K25SUBSTITUTE SHEET (RULE 26)1015202530W0 98/ 12659CA 02264758 1999-03-03PCTIU S97/ 156653 1Note the following implications of these expressions: A)the optimal errors are identical for each market maker,since the error vector is a constant times the unitvector; and E) the optimal error magnitude summed overall market makers represents the sum of all delta (gama)changes desired by the market makers and the sum of alldelta (gamma) changes dictated by the public orderimbalances across the option series. Thus, it ispossible (although unlikely) that the error could bereduced to zero if the aggregate market maker desire wereto offset exactly the aggregate delta (gamma) imbalancein public orders.In general, this optimum solution will not result ininteger values of the elements of the.M matrix, and thusthe optimization is completed by finding the nearestinteger point in NK-dimensional space lying in thesubspace defined by the intersection of the N linearmanifolds specified by equation (21), that results inminimum squared error as given by (20). Rather thanusing exhaustive or exotic search approaches to thisrelatively immaterial problem, the controller 2 couldfind an approximately optimal integer point solution bytaking into account the nature of the constraintequations (21), which require thatk2 A4,, =—D‘.(6,), :=l..N, (47)/"=1i.e , the row sum of the i“ row of the matrix M mustequal the integer value -Di(6) ..1A simple procedure thatwill produce a near—optimal solution can be obtained byrounding up or down the elements of the j“ row kg of thematrix M to their nearest integer values, and thenmaking adjustments of plus or minus one to the roundingover the row elements as needed to remain compliant with(44). For example, if the row sum of the i“ row afterSUBSTITUTE SHEET (RULE 26)1015200.114 77-0.164 0.03 -970.205 0.085 10_ 0.11 0.103 -41A_ 0.037 F: 0.15 D: 340.309 0.034 73—O.276 0.098 -260.411 0.14 -1980.738 0.03 40_O.462_ 0.028 _30 _CA 02264758 1999-03-03W0 98/ 12659 PCT/US97/1566532"D.(5.)+f,J .1the rounding is then for r2=1, controller 2identifies the r row elements with the smallestfractional parts that were rounded up, and rounds themdown. For rs-1, the controller 2 performs the reverseprocedure, rounding up the r row elements with thelargest fractional parts that were rounded down. Thiswill assure that at most a change of $1 with respect tothe optimum number contracts in a given series will beassigned to any market maker.ExamplesTo illustrate the relative performance of the optimumassignment with respect to the round-robin assignment ofpublic order imbalances to market makers, we consider thefollowing examples. We assume there are ten optionseries, with 30 market makers.Suppose that the openingvalues of A, P and D in the ten series are given by thevectors:'—0.212WFurther assume that the 30 market maker's Q and E valuesare shown in the plots in Figs. 3A and 3B.The optimum and round-robin assignment errors for C and Eare shown in Figs. 4A and 4B respectively.SUBSTITUTE SHEET (RULE 26)10152025W0 98l12659CA 02264758 1999-03-03PCT/US97l156653 3The root mean square (rms) error per market maker in Q is721 for the optimum assignment, and, 4,529 for the round-The corresponding rms error per marketmaker in E is 135 and 544, respectively.robin assignment.As a second example, let A, P and D in the 10 optionseries be given by the vectors:'0.437i0.196” 20.458 0.059 -173-0.372 0.175 192-0.258 0.02 46-0.269 0.029 -53A: 0.327 P"O.O62 D: 144—0.473 0.163 -1040.692 0.163 -180-0.314 0.18 -132_0.413_ _0.054_ _—112_Further assume that the 30 market maker's Q and E valuesare as shown in the plots in Figs. 5A and SBrespectively. The optimum and round—robin assignmenterrors for Q and Q are shown in Figs.6A and 6Brespectively. The rms error per market maker in Q is 74for the optimum assignment, and 5,514 for the round-robinassignment. The corresponding rms error per market makerin Q is 158 and 527, respectively.These examples demonstrate the substantial improvement inthe rms market maker target errors using the optimalassignment method of the present invention versus theprior art round-robin assignment method.The controller 2 of the representative embodiment of thepresent invention can be implemented utilizing a logiccircuit or a computer memory comprising encoded computer-readable instructions, such as a computer program. TheSUBSTITUTE SHEET (RULE 25)1015W0 98/ 12659CA 02264758 1999-03-03PCT/US97/156653 4functionality of the logic circuit or computer memory isdescribed in detail above.Generally, the present invention has practicalapplication as it enables options market to openessentially instantaneously and it efficiently allocatesresidual public orders to market makers. It should beappreciated that the solution posed by the presentinvention could not be manually accomplished by a human.According to the present invention, input can be receivedfrom many market maker terminals (that may be located ina multitude of various locations), and this input iselectronically transmitted and stored in a centrallocation for processing. No human could possibly performthe communications, calculations and optimizationsrequired to concurrently determine the impliedvolatilities for each of the many series of options andthe efficient assignment of residual public orders.SUBSTITUTE SHEET (RULE 25)

Claims (36)

WHAT IS CLAIMED IS:

1. A computer-based system for opening trading in options, comprising:
a plurality of input devices to receive from each one of a plurality of market makers a current position, a desired target position and market maker orders for options series;
an order entry system for receiving public orders for options series; and a controller, coupled to the plurality of input devices and the order entry system, the controller comprising (a) means for determining a set of prices for each options series that will maximize a weighted volume of trades across all option series at the opening, (b) means for executing all orders that can occur at the set of prices, (c) means for determining a residual imbalance of non-executed public orders, and (d) means, utilizing the current position and the desired target position of each one of the plurality of market makers, for assigning the residual imbalance of non-executed public orders to individual ones of the plurality of market makers so as to minimize a cumulative measure of deviation between the desired target position and the current position of each market maker.

2. The system of claim 1 wherein the set of prices comprises a set of implied volatilities.

3. The system of claim 1 wherein the set of prices for each option series is simultaneously determined.

4. The system of claim 1 further comprising a plurality of output devices, coupled to the controller, to output the set of prices to the plurality of marker makers.

5. The system of claim 1 wherein means for determining the set of prices further comprises means for determining a set of opening volatilities for each series that is a reasonable compromise, according to a predetermined parameter, between consistency in pricing across each series and public order demand across each series.

6. The system of claim 5 wherein the parameter is set by an options exchange.

7. The system of claim 5 wherein the parameter is determined on-the-fly at the opening.

8. The system of claim 1 wherein the current position of each one of the plurality of market makers is input as a delta measure and a gamma measure;

9. The system of claim 1 wherein the desired target position of each one of the plurality of market makers is input as a delta measure and a gamma measure.

10. A computer-based system for determining a set of opening prices for a plurality of series of options, comprising:
a plurality of input devices to receive from each one of a plurality of market makers orders for options series;
an order entry system for receiving public orders for options series; and a controller, coupled to the plurality of input devices and the order entry system, utilizing the market maker orders and public orders to determine a set of prices for each options series that will maximize a weighted volume of trades across all option series at the opening.

11. The system of claim 10 wherein the controller further comprises means for executing all public orders and market maker orders that can occur at the determined set of prices.

12. The system of claim 10 wherein the set of prices for each option series is simultaneously determined.

13. The system of claim 10 further comprising a plurality of output devices, coupled to the controller, to output the set of prices to the plurality of marker makers.

14. The system of claim 10 wherein means for determining the set of prices further comprises means for determining a set of opening volatilities for each series that is a reasonable compromise, according to a predetermined parameter, between consistency in pricing across each series and public order demand across each series.

15. The system of claim 14 wherein the parameter is set by an options exchange.

16. The system of claim 14 wherein the parameter is determined on-the-fly at the opening.

17. A computer-based system for allocating public order imbalances at the opening of trading of in options, comprising:
a memory storing a plurality of orders for options representing a residual imbalance of public orders in each options series that do not match an opening price corresponding to said options series;

a plurality of input devices to receive from each one of a plurality of market makers a current position and a desired target position; and a controller, coupled to the memory and the plurality of input devices, utilizing the current position and the desired target position of each one of the plurality of market makers to assign each public order in the residual imbalance of public orders to individual ones of the plurality of market makers so as to minimize a cumulative measure of deviation between the desired target position and the current position of each market maker.

18. The system of claim 17 wherein the current position of each one of the plurality of market makers is input as a delta measure and a gamma measure.

19. The system of claim 17 wherein the desired target position of each one of the plurality of market makers is input as a delta measure and a gamma measure.

20. A computer-implemented method for opening trading in options by a plurality of market makers, comprising:
receiving from each one of a plurality of market makers a current position, a desired target position and market maker orders for options series;
receiving public orders for options series; and determining a set of prices for each options series to maximize a weighted volume of trades across all option series at the opening;
executing all orders that can occur at the set of prices determined in the previous step;
if there is a residual imbalance of non-executed public orders, assigning the residual imbalance of non-executed public orders to individual ones of the plurality of market makers so as to minimize a cumulative measure of deviation between the desired target position and the current position of each market maker.

21. The method of claim 20 wherein the set of prices comprises a set of implied volatilities.

22. The method of claim 20 wherein the set of prices for each option series is simultaneously determined.

23. The method of claim 20 further comprising the step of outputting the determined set of prices.

24. The method of claim 20 wherein the step of determining the set of prices further comprises the step of determining a set of opening volatilities for each series that is a reasonable compromise, according to a compromise parameter, between consistency in pricing across each series and public order demand across each series.

25. The method of claim 24 further comprising the step of setting the compromise parameter in advance by an options exchange.

26. The method of claim 24 further comprising the step of determining the compromise parameter on-the-fly at the opening.

27. The method of claim 20 wherein the step of receiving from each one of a plurality of market makers a current position further comprises receiving from each one of the plurality of market makers a delta measure and a gamma measure corresponding to the current position.

28. The method of claim 20 wherein the step of receiving from each one of a plurality of market makers a desired target position further comprises receiving from each one of the plurality of market makers a delta measure and a gamma measure corresponding to the desired target position.

29. A computer-implemented method for determining a set of opening prices for a plurality of series of traded options, comprising:
receiving orders for options series from a plurality of market makers;
receiving public orders for options series; and determining a set of prices for each options series, wherein the set of prices maximizes a weighted volume of trades across all option series at the opening.

30. The method of claim 29 further comprising the step of executing all possible public orders and market maker orders at the determined set of prices.

31. The method of claim 29 wherein the step of determining a set of prices for each options series further comprises the step of simultaneously determining a set of prices for each options series.

32. The method of claim 29 wherein the step of determining a set of prices for each options series further comprises the steps of:
providing a parameter; and determining a set of opening volatilities for each series that is a reasonable compromise, according to the parameter, between consistency in pricing across each series and public order demand across each series.

33. A computer-implemented method for allocating public order imbalances at the opening of trading in options, comprising:

providing a plurality of orders for options representing a residual imbalance of public orders in each options series that do not match an opening price corresponding to said options series;
receiving from each one of a plurality of market makers a current position and a desired target position;
and automatically assigning each public order in the residual imbalance of public orders to individual ones of the plurality of market makers so as to minimize a cumulative measure of deviation between the desired target position and the current position of each market maker.

34. The method of claim 33 wherein the step of receiving from each one of the plurality of market makers a current position further comprises receiving from each one of the plurality of market makers a delta measure and a gamma measure corresponding to the current position.

35. The method of claim 33 wherein the step of receiving from each one of a plurality of market makers a desired target position further comprises receiving from each one of the plurality of market makers a delta measure and a gamma measure corresponding to the desired target position.

36. A computer-implemented method for opening trading in options, comprising:
receiving from each one of a plurality of market makers a current position, a desired target position and market maker orders for options series;
receiving public orders for options series; and concurrently determining a set of opening prices for each options series;
executing all possible public orders and market maker orders at the set of opening prices determined in the previous step;
if there is a residual imbalance of non-executed public orders, optimally assigning the residual imbalance of non-executed public orders to individual ones of the plurality of market makers taking into account the desired target position and the current position of each market maker.