System for providing a correlation index

A system includes a memory and a processor communicatively coupled to the memory. The memory stores a correlation index that is based at least in part on a portfolio of index options, a plurality of stock options, and a plurality of deltas computed using a plurality of constant flat implied volatilities. The processor determines a value of the correlation index by selecting a plurality of strike prices for each of the index and stock options, computing a constant flat implied volatility for an index and a plurality of stocks on a first date, and computing the plurality of deltas using the plurality of constant flat implied volatilities.

TECHNICAL FIELD OF THE INVENTION

This disclosure generally relates to trading and more specifically to a system for providing a correlation index.

BACKGROUND OF THE INVENTION

Traditional trading systems allow traders to pursue various investment schemes, including correlation trading. In correlation trading, an investor may buy or sell options on an index and on the individual constituents of the index. In addition, an investor may buy or sell variance swaps on the index and on the individual constituents. However, correlation trading is often complicated and is thus prohibitive to many investors.

SUMMARY OF THE INVENTION

In accordance with the present invention, the disadvantages and problems associated with prior trading systems with respect to correlation trading have been substantially reduced or eliminated.

In some embodiments, a system includes a memory and a processor communicatively coupled to the memory. The memory stores a correlation index that is based at least in part on a portfolio of index options, a plurality of stock options, and a plurality of deltas computed using a plurality of constant flat implied volatilities. The processor determines a value of the correlation index by selecting, using a predetermined standard deviation range, a plurality of strike prices for each of the index and stock options, computing, using the selected strike prices, a flat implied volatility for an index and a plurality of stocks, and holding the computed flat implied volatilities constant over a predetermined time period, thereby creating the plurality of constant flat implied volatilities. The processor further determines the correlation index value by computing the plurality of deltas using the plurality of constant flat implied volatilities, computing a daily profit and loss (P&L) based on a change in a value of the portfolio of index and stock options, the plurality of deltas, an interest on cost to borrow stock, and an interest on funding, and determining the correlation index value based at least in part on a previous correlation index value and the computed daily P&L.

The invention has several important technical advantages. Various embodiments of the invention may have none, some, or all of these advantages. One advantage is that the trading system provides a correlation index that packages correlation trading into a single line item. The correlation index replicates variance swaps using, among other things, listed options and deltas computed using constant flat implied volatilities. As such, the correlation index reduces the complexity of correlation trading and thus provides access to correlation trading for investors who might otherwise be prohibited.

Other advantages of the present invention will be readily apparent to one skilled in the art from the description and the appended claims.

DETAILED DESCRIPTION OF THE INVENTION

Correlation is the bridge that links index and single stock volatilities and can be approximated by the following formula:

correlation≈Index⁢⁢VolatilityAverage⁢⁢Single⁢⁢Stock⁢⁢Volatility
This formula shows that the relative level of implied index and implied single stock volatilities give the approximate implied correlation.

Implied correlation in the equity market tends to trade at a significant premium to subsequent realized correlation due to the relative richness of index implied volatility. This makes selling correlation and/or buying dispersion a well performing trade with attractive return/risk ratios. Dispersion trades are usually structured as variance-to-variance strategies by selling index variance swaps and buying single stock variance swaps thus capturing the disconnect between index and single stock volatilities.

Traditional trading systems allow traders to pursue various investment schemes, including correlation trading. In correlation trading, an investor may buy or sell options on an index and buy or sell options on the individual constituents of the index. In addition, an investor may buy or sell variance swaps on the index and buy or sell variance swaps on the individual constituents. However, correlation trading is often complicated and is thus prohibitive to many investors.

The teachings of the disclosure recognize that it would be desirable to provide access to correlation trading that is less complicated and easier to execute and maintain. The following describes a system and method of providing a correlation index that packages correlation trading into a single line item. The correlation index replicates variance swaps using, among other things, listed options and deltas computed using constant flat volatilities. As such, the correlation index reduces the complexity of correlation trading and thus provides access to correlation trading for investors who might otherwise be prohibited.

FIG. 1illustrates a trading system10, according to certain embodiments. Trading system10may comprise one or more clients20, an index server30, one or more market data servers40, and one or more market centers50communicatively coupled by one or more networks60.

Trading system10provides a correlation index18. In general, correlation index18is based at least in part on stock options26, index options24, and deltas42. As described in more detail below, deltas42are computed using constant flat volatilities that are computed on a start date (e.g., the tranche start date). The flat volatilities are held constant until an end date (e.g., the tranche end date).

Trading system10comprises one or more clients20. Client20represents any suitable local or remote end-user device that may be used by traders14to access one or more elements of trading system10, such as index server30. Trader14may use client20to submit deposits, make withdrawals, request information, and/or communicate with various components of trading system10. In some embodiments, trader14may use client20to trade correlation index18that is managed by index server30. A particular client20may comprise a computer, workstation, telephone, Internet browser, electronic notebook, Personal Digital Assistant (PDA), pager, or any other suitable device (wireless, wireline, or otherwise), component, or element capable of receiving, processing, storing, and/or communicating information with other components of trading system10. Client20may also comprise any suitable user interface such as a display, microphone, keyboard, or any other appropriate terminal equipment according to particular configurations and arrangements. It will be understood that trading system10may comprise any number and combination of clients20. In some embodiments, client20may comprise a graphical user interface (GUI)36.

GUI36is generally operable to tailor and filter data presented to trader14. GUI36may provide trader14with an efficient and user-friendly presentation of trading orders12, market data38, and/or other suitable information. GUI36may comprise a plurality of displays having interactive fields, pull-down lists, and buttons operated by trader14. In one example, GUI36presents relevant market data38to trader14and conceals the remaining information to reduce visual clutter. Then, upon receiving a request from trader14, GUI36expands the visual representation of market data38to display account information, market information, and/or other suitable information. GUI36may include multiple levels of abstraction including groupings and boundaries. It should be understood that the term graphical user interface may be used in the singular or in the plural to describe one or more graphical user interfaces36and each of the displays of a particular graphical user interface36.

Although clients20are described herein as being used by “traders,” it should be understood that the term “trader” is meant to broadly apply to any user of trading system10, whether that user is an agent acting on behalf of a principal, a principal, an individual, a legal entity (such as a corporation), or any machine or mechanism that is capable of participating in transactions in trading system10.

Trader14may use client20to communicate with index server30. Index server30is generally operable to provide correlation index18. Based at least in part on the movement of market data38, index server30is operable to update correlation index18.

Index server30is operable to update correlation index18periodically (e.g., daily and/or any other suitable time period). In some embodiments, index server30re-calculates correlation index18on a daily basis. Index server30is operable to transmit a closing level of correlation index18to market data server40.

Index server30may comprise any suitable combination of hardware and/or software implemented in one or more modules to provide the described functions and operations. In some embodiments, index server30may comprise a general-purpose personal computer (PC), a Macintosh, a workstation, a Unix-based computer, a server computer, or any suitable processing device. In some embodiments, the functions and operations described above may be performed by a pool of multiple index servers30. A particular index server30may comprise an index memory44and an index processor46.

Index memory44comprises any suitable arrangement of random access memory (RAM), read only memory (ROM), magnetic computer disk, CD-ROM, or other magnetic or optical storage media, or any other volatile or non-volatile memory devices that store one or more files, lists, tables, or other arrangements of information such as market data38. AlthoughFIG. 1illustrates index memory44as internal to index server30, it should be understood that index memory44may be internal or external to index server30, depending on particular implementations. Also, index memory44may be separate from or integral to other memory devices to achieve any suitable arrangement of memory devices for use in trading system10.

Index memory44is generally operable to store index logic48. Index logic48generally comprises rules, algorithms, code, tables, and/or other suitable instructions for updating correlation index18. Index memory44is further operable to store market data38, correlation index18, selected strikes32, deltas42, flat implied volatilities34, daily profit and loss (P&L)62, and interest on funding64.

Index memory44is communicatively coupled to index processor46. Index processor46is generally operable to execute index logic48stored in index memory44to calculate correlation index18. Index processor46may comprise any suitable combination of hardware and software implemented in one or more modules to provide the described function or operation.

Index server30may transmit trading orders12to one or more market centers50. Market center50is generally operable to receive and execute trading orders12. Once a particular trading order12is executed, market center50is operable to generate and transmit a trade confirmation message to index server30. Market center50is further operable to transmit trading data52to market data server40. Trading data52may comprise information regarding trading activities in market center50. In particular, trading data52may comprise information regarding best bid prices, best offer prices, trading volumes, volatility, and/or any other suitable information regarding trading activity in market center50. In some embodiments, trading data52represents raw data regarding conditions in market center50.

Market centers50may comprise all manner of order execution venues including exchanges, Electronic Communication Networks (ECNs), Alternative Trading Systems (ATSs), market makers, or any other suitable market participants. Each market center50may maintain a bid and offer price for at least one investment instrument by standing ready, willing, and able to buy or sell that investment instrument at publicly quoted prices, also referred to as market center prices. Different market centers50may provide different market center prices for particular investment instruments. For example, a particular market center50may offer a particular bid price and/or offer price for a particular investment instrument, while another market center50may offer a different bid price and/or offer price for the same investment instrument.

Market center50may be communicatively coupled via network60to market data server40. Market data server40is generally operable to receive and process trading data52from market center50. Market data server40may process trading data52to generate market data38. Market data38may comprise current and/or historical information regarding any suitable index, financial instrument, mutual fund, hedge fund, exchange traded fund (“ETF”), interest rate, investment instrument, trader14, and/or any suitable number and combination of indicators regarding trading system10. In particular, market data38may comprise current and/or historical values of stocks22, index options24, stock options26, and strikes28. In some embodiments, market data server40may be operated by a financial news service organization. In other embodiments, market data server40may be operated by a market maker, brokerage firm, bank, market center50, and/or any suitable financial services entity.

Market data server40may comprise any suitable combination of hardware and/or software implemented in one or more modules to provide the described functions and operations. In some embodiments, market data server40may comprise a general-purpose personal computer (PC), a Macintosh, a workstation, a Unix-based computer, a server computer, or any suitable processing device. In some embodiments, the functions and operations described above may be performed by a pool of multiple market data servers40. A particular market data server40may comprise a market data memory54and a market data processor56.

Market data memory54comprises any suitable arrangement of random access memory (RAM), read only memory (ROM), magnetic computer disk, CD-ROM, or other magnetic or optical storage media, or any other volatile or non-volatile memory devices that store one or more files, lists, tables, or other arrangements of information. AlthoughFIG. 1illustrates market data memory54as internal to market data server40, it should be understood that market data memory54may be internal or external to market data server40, depending on particular implementations. Also, market data memory54may be separate from or integral to other memory devices to achieve any suitable arrangement of memory devices for use in trading system10.

Market data memory54is generally operable to store trading data52from market centers50. Market data memory54is further operable to store market data logic58. Market data logic58generally comprises rules, algorithms, code, tables, and/or other suitable instructions for generating market data38based at least in part on trading data52.

Market data memory54may be communicatively coupled to market data processor56. Market data processor56is generally operable to execute market data logic58to generate market data38. Market data processor56comprises any suitable combination of hardware and software implemented in one or more modules to provide the described function or operation.

As explained above, clients20, index server30, market data servers40, and market centers50may be communicatively coupled via one or more networks60. Network60may represent any number and combination of wireline and/or wireless networks suitable for data transmission. Network60may, for example, communicate internet protocol packets, frame relay frames, asynchronous transfer mode cells, and/or other suitable information between network addresses. Network60may include one or more intranets, local area networks, metropolitan area networks, wide area networks, cellular networks, all or a portion of the Internet, and/or any other communication system or systems at one or more locations.

It should be understood that the internal structure of trading system10and the servers, processors, and memory devices associated therewith is malleable and can be readily changed, modified, rearranged, or reconfigured to achieve the intended operations of trading system10.

In operation, some embodiments of correlation index18are provided by index server30as described below. In general, correlation index18(hereinafter, the “Index”) is designed to replicate the payoff of an investment in a variance-to-variance dispersion trade. Embodiments of the Index result in implied correlation through a position in a portfolio of single stock options26and a portfolio of S&P 500 (SPX) index options24. In addition, the Index includes the result of “delta hedging” such a portfolio on a daily basis thus experiencing a cost of realized correlation. Each year, a new tranche (i.e., series) of the Index is created using a portfolio of SPX index options24, single stock options26and deltas42(i.e., a new series of the Index may be launched each year, or any other appropriate time period). The choice of stocks22and strikes28as well as trading mechanics is outlined in what follows.

The following notation is used herein:

Calculationscheduled trading day t on the Chicago Board Options ExchangeDay t(CBOE), New York Stock Exchange, American Stock Exchangeand NASDAQnifloat-adjusted shares outstanding of stock imnumber of assets minus one (SPX + number of stocks − 1) in theIndexiindexing for an asset i where i = 0 corresponds to SPX and i ≧ 1corresponds to stockjindexing for strikes jSitclosing spot price of stock i on Calculation Day tSi=0tclosing spot price of SPX on Calculation Day tATMAt-the-moneyOTMOut-of-the-moneyITMIn-the-moneyσi*implied volatility for each asset i to the Option Expiration Date,as calculated using the CBOE VOLATILITY INDEX (VIX)methodology and all available strikes with non-zero bids onTranche Start Dateσiimplied volatility for each asset i to the Option Expiration Date,as calculated using the CBOE VIX methodology and boundedstrikesσi=0implied volatility for SPX to the SPX Option Expiration Date,as calculated using the CBOE VIX methodology and boundedstrikesTBtOn Calculation Day t, the number of Business Days to OptionExpiration divided by 252TCtOn Calculation Day t, the actual number of calendar days toOption Expiration divided by 365Kstrike (i.e., strike price)Kit,0ATM strike for asset i on Calculation Day tRtrisk-free interest rate, the yield of Treasury bills or notes to therelevant Option Expiration date. If no treasury maturity matchesthe Option Expiration date, then the interpolated yield of the twotreasury contracts immediately preceding and following theOption Expiration dateTBilltrisk-free interest rate, the yield of the On-The-Run 6-monthTreasury billSpreadtFunding spread, where the annualized default spread is 0.55%.Spreadtwill be updated at each Tranche Start Date to reflect themost recent market value. However, if the Index Sponsordetermines that funding costs have increased significantly priorto Tranche End Date, then Spreadt will equal the new rate as ofCalculation Day t.Q(Ki,j)midmarket price of option with strike Ki,jΔKi,jinterval between strike prices for asset i at strike jΔKi,j=Ki,j+1-Ki,j-12(ΔK for lowest strike is the difference between the lowest strikeand the next higher strike. AK for highest strike is the differencebetween the highest strike and the next lower strike. ΔK for theatm strike is the difference between the next higher strike andthe next lower strike, multiplied by 0.5. )Cit,0midmarket price of call option with ATM strikePit,0midmarket price of put option with ATM strikeditdividend amount on ex-date t for asset iN(x)the standard normal cumulative distribution functionRborrowit−1Borrow Cost on Calculation Day t-1, where the annualizeddefault borrow cost is 0.10%. However, if the Index Sponsordetermines that the cost to borrow stock Si′ has increasedsignificantly, then RBCitwill equal the new rate as ofCalculation Day t. Note that there is no Borrow Cost for shortpositions in SPX.daystNumber of calendar days from and including Calculation Day t-1 to but excluding Calculation Day tI0Index closing level on Tranche Start Date

In some embodiments, a stock tracking basket is created. In certain embodiments, on the close of each tranche start date, the components of a particular market index (e.g., the SPX) are ranked by market capitalization, with the top N stocks22selected as the tracking basket. In some embodiments, the top 50 stocks22from the SPX are selected as the SPX tracking basket. In other embodiments, another amount of top stocks22and/or another market index may be used. While the following examples may use the top 50 stocks22from the SPX, it should be understood that any appropriate number of top stocks22and/or any appropriate market index may be used.

For each selected stock22, a weight may be computed by dividing the individual market capitalization by the total market capitalization of the 50-stock basket according to the following equation:

If a certain predetermined financial stock is in the top 50 (e.g., BAC), it may be excluded and its weight proportionally allocated among other financial stocks, for example, JPM, WFC, C and GS, on a market capitalization weighted basis. Thus, if the predetermined financial stock is in the top 50, there will be 49 stocks in the SPX tracking basket.

In certain embodiments, options24and26are selected to be used in correlation index18. In some embodiments, quantities are determined for the selected options24and26and VIX implied volatilities34are computed. For example, on each Tranche Start Date, a new option portfolio is selected using the following criteria for the expiration, strikes, option type and quantity of options:Tranche Start Date=November Listed Expiry of year YTranche End Date=November Listed Expiry of year Y+1Single Stock Option Expiration=January Listed Expiry of year Y+2SPX Option Expiration=December Listed Expiry of year Y+1
For example, on the Tranche Start Date in November 2010, the expiration date used for the single stock components is January 2012 and for the Index components December 2011. The Tranche End Date and the following Tranche Start Date would be in November 2011. While the following examples may use the months listed above for Tranche Start Date, Tranche End Date, Single Stock Option Expiration and SPX Option Expiration, it should be understood that any appropriate date may be used.

At each Tranche End Date, the positions held are liquidated and cash settled with the single stock option positions sold on the bid and the SPX option positions bought back on the offer. At each Tranche Start Date, new positions start at mid-market value, which is the average of bid and ask.

For SPX and each stock i=0, . . . , m, some embodiments use all strikes between an upper standard deviation and a lower standard deviation for the option portfolio. The standard deviation may be between any appropriate upper and lower limit. For example, the upper limit of the standard deviation may be up to +3, inclusive, and the lower limit of the standard deviation may be as low as −4, inclusive. As a specific example, strikes between −2.5 standard deviations and +1.5 standard deviations are used:

Fit=0×ⅇ(-2.5×σi*×TC)≤K≤Fit=0×ⅇ(+1.5×σi*×TC)
Where the implied volatility, σ*i, is calculated in some embodiments using the CBOE VIX methodology for all available listed strikes with non-zero bids on the Tranche Start Date. While the CBOE VIX methodology is discussed herein as an example, any other appropriate method of determining volatility may be used.

In some embodiments, the initial implied volatility34, σi, is then calculated, also using the CBOE VIX methodology (or any other appropriate method of determining volatility), but based on the bounded option portfolio on Tranche Start Date. If two consecutive single stock put (call) options have bid premiums strictly less than a certain amount, such as $0.02, some embodiments may remove both contracts and all puts with a lower strike (calls with a higher strike) from the portfolio. If the lowest (highest) strike single stock put (call) in the portfolio is less than $0.02, some embodiments may remove the contract. If two consecutive SPX put (call) options have bid premiums strictly less than another amount such as $0.05, some embodiments may remove both contracts and all puts with a lower strike (calls with a higher strike) from the portfolio. If the lowest strike SPX put (highest strike SPX call) in the portfolio is less than $0.05, some embodiments may remove the contract. The implied volatility34is defined as:

σi2=2TBt⁢∑j⁢Δ⁢⁢Ki,jKi,j2⁢eRt⁢TCt⁢Q⁡(Ki,j)-1TBt⁡[FitKit,0-1]2
The VIX formula used in some embodiments is described in the CBOE white paper, available on the CBOE website at www.cboe.com/micro/vix/vixwhite.pdf, which is incorporated herein by reference. Note that the above formula for implied volatility uses both business and calendar days whereas VIX only uses calendar days.

On each Calculation Day t, the forward computation for SPX and stocks, is defined as:
Fit=Strike Priceit+eiRtTt(Cit,0−Pit,0)

Some embodiments select as the StrikePriceit, the strike of the Call and Put with the lowest absolute price difference based on all available listed strikes on Calculation Day t.

Some embodiments next determine Kit,0, which is the listed strike immediately below the forward, Fit.

In some embodiments, a quantity of options24and26are selected. For SPX, stocks and strikes above, OTM calls or OTM puts, centered around Kit=0,0, are selected on the Tranche Start Date. OTM calls have strikes>Kit=0,0, OTM puts have strikes<Kit=0,0and calls and puts are selected for strike Kit=0,0. Thus, both calls and puts are selected for strike Kit=0,0whereas either a call or a put is selected at every other strike.

In some embodiments, the implied volatilities of each put and call pair are weighted through a linear interpolation to arrive at a single at-the-money implied volatility, σi. Once selected on the Tranche Start Date, the quantity of options and implied volatility will be fixed until the Tranche End Date, unless removed due to a small premium.

In some embodiments, the Quantity of options for each strike is defined as:

QuantityStock=i,Strike=jt=0=100×weighti×Vega⁢⁢Notionalstocksσi×Δ⁢⁢Ki,jTB×Ki,j2
where σiis the implied volatility for stock i=1, . . . , m on the Tranche Start Date using the bounded option portfolio.

Quantityi=0,Strike=jt=0=-100×Vega⁢⁢NotionalSPXσi=0×Δ⁢⁢Ki,jTB×Ki,j2
In certain embodiments, the quantity of options is rounded to the nearest integer.

The Vega Notional for SPX from the computed volatilities is given by:

Vega⁢⁢Notionali=1=∑i=1m⁢weighti⁢σiσi=0×VegaNotionalStocks
where Vega NotionalStocksis determined at the launch of each Tranche.

In some embodiments, daily option portfolio maintenance is performed and option deltas42are calculated. For example, on each calculation day, at the scheduled closing time of the relevant Exchange, starting from but excluding the Tranche Start Date up to but excluding the Tranche End Date, options may be removed from the portfolio based on the following criteria.

Some embodiments include a small premium exclusion for OTM options. For example, if two consecutive single stock put (call) options have bid premiums strictly less than an amount such as $0.02 over two consecutive Calculation Days, some embodiments may remove both contracts and all puts with a lower strike (calls with a higher strike) at a price of zero on Calculation Day t and each subsequent Calculation Day up to and including Tranche End Date:If AND(Bid_Pricei,j-1t, Bid_Pricei,jt, Bid_Pricei,j-1t-1, Bid_Pricei,jt-1)<0.05, then Quantityi,pt=0 for i>0 and all p such that Ki,p≦Ki,jfor put strikes or Kt,p≧Ki,j-1for call strikes.

In addition, if the lowest strike put (highest strike call) in the remaining portfolio is less than an amount such as $0.02 over two consecutive Calculation Days, some embodiments may remove the contract.

If two consecutive SPX put (call) options have a bid premiums strictly less than an amount such as $0.05 over two consecutive Calculation Days, some embodiments may remove both contracts and all puts with a lower strike (calls with a higher strike) at a price of zero on Calculation Day t and each subsequent Calculation Day up to and including the Tranche End Date:If AND(Bid_Pricei,j-1t, Bid_Pricei,jt, Bid_Pricei,j-1t-1, Bid_Pricei,jt-1)<0.05, then Quantityi,pt=0 for i=0 and all p such that Ki,p≦Ki,jfor put strikes or Ki,p≧Ki,j-1for call strikes.

In addition, if the lowest strike put (highest strike call) in the remaining portfolio is less than an amount such as $0.05 over two consecutive Calculation Days, some embodiments may remove the contract.

Some embodiments include a small premium exclusion for deep ITM options. For example, if two consecutive calls (puts) are deep ITM and the corresponding consecutive puts (calls) with the same strikes fall under the above criteria for SPX or single stock contract omission over two consecutive Calculation Days, some embodiments may then remove the calls (puts) and any calls (puts) with a lower (higher) strike on Calculation Day t and each subsequent Calculation Day up to and including the Tranche End Date. In addition, if the highest strike put (lowest strike call) in the remaining portfolio falls under the above criteria over two consecutive Calculation Days, some embodiments may remove the contract. Deep ITM options are unwound at closing market bid or ask prices on Calculation Day t and the cash settlement amount, OptionUnwindi,jt, added to Daily P&Ltand Fundingt.

In the case of an ITM put for i≧1:If AND[(Ki,j-1−Sit)>0, Calli,j-1t<0.02, (Ki,j−Sit)>0, Calli,jt<0.02, (Ki,j-1−Sit-1)>0, Calli,j-1t-1<0.02, (Ki,j−Sit-1)>0, Calli,jt-1<0.02]tthen Quantityi,p-1t=0 for all put strikes Ki,p-1≧Ki,j-1.

In the case of an ITM call for i≧1:If AND[(Sit−Ki,j-1)>0, Puti,j-1t<0.02, (Sit−Ki,j)>0, Puti,jt-1<0.02, (Sit-1−Ki,j-1)>0, Puti,j-1t-1<0.02, (Sit-1−Ki,j)>0, Puti,jt-1<0.02]tthen Quantityi,p-1t=0 for all call strikes Ki,p-1≦Ki,j.

For all other strikes that were not omitted from the option portfolio, the quantity is fixed from Tranche Start Date.

In some embodiments, the deltas42held against the options portfolio are computed. The implied volatility used in the delta computation is the volatility determined above on Tranche Start Date. As used herein, “deltas” may include index deltas and stock deltas. Index deltas may refer to changes in prices of index options24with respect to the underlying index. Stock deltas may refer to changes in prices of stock options26with respect to the underlying stocks. For i≧0 and strikes j, deltas42in some embodiments are then calculated at the close using the Black-Scholes formula:

In some embodiments, daily Profit and Loss (the “Daily P&L”)62is computed based on the change in the value of the option portfolio, the cumulative delta and interest on funding64. Funding includes change of value of the option portfolio, cash generated from unwind of deep ITM options, cost to borrow stock and SPX or stock delta trading.

In some embodiments, the closing level of correlation index18(excess return) is computed. The Index closing level on Index Calculation Day t, It, is defined as:

In some embodiments, the calculations described above may be adjusted for corporate actions, including, but not limited to, stock splits, special dividends, mergers, acquisitions, spin-offs, bankruptcies and delistings. Corporate actions may be applied after the close of trading on the Calculation Day prior to ex-date. Removal of a component due to bankruptcy may be done the day after the official announcement, at the close. For example, if the bankruptcy filing is on Tuesday, removal may occur on Wednesday at the market close.

The following table lists example adjustment guidelines for various corporate actions.

Removals may be made at the close on the Calculation Day immediately following the announcement and executed at market bid and ask prices with cash settlement added to Fundingtas part of OptionUnwindt. Any Corporate Actions not listed above may be adjusted by an Index Sponsor in a commercially reasonable manner consistent with the methodologies above.

If the total weight of stocks removed from the basket exceeds 3% (or any other appropriate percentage), then the corresponding amount of SPX options may be removed. For example, if 5% of stocks have been removed, then 5% of the quantity of SPX options at each strike will also be removed, rounded to the nearest integer.

If a Market Disruption Event occurs, as determined by the Index Sponsor in its sole discretion, on a Calculation Day, the Index closing level may be adjusted in certain embodiments using the most recently published market prices.

Furthermore, due to widening market bid and ask prices from such an event, the Index Sponsor may opt to include options that would have otherwise been removed during Option Portfolio Maintenance. The Index Sponsor, in its sole discretion, may adjust the methodology of the Index such that it can be calculated in a practical manner. If a market disruption occurs for five or more consecutive Calculation Days, the Index Sponsor will determine the level of the Index in a commercially reasonable manner.

Should the level of the Index on any Calculation Day be at or below 10% of the initial Tranche level, the Index Sponsor may, but is not obligated, to unwind the Index tranche at the prevailing market bid and ask prices. The final level at which the Index is unwound will be determined by the Index Sponsor in its sole discretion.

In some embodiments, the source of option prices used in the Index methodology is the CBOE. Source of stock prices is the primary exchange on which the stock is listed.

FIG. 2is block diagram illustrating a portion of a processing/computing system220that may be used to provide correlation index18according to one embodiment. Computing system220performs one or more steps of one or more methods described or illustrated herein. In particular embodiments, one or more computing systems220provide functionality described or illustrated herein. In particular embodiments, software running on one or more computing systems220(i.e., a computer program250) performs one or more steps of one or more methods described or illustrated herein or provides functionality described or illustrated herein. Particular embodiments include one or more portions of one or more computing systems220.

Computing system220may be in any suitable physical form. As example and not by way of limitation, computing system220may be a virtual machine (VM), an embedded computer system, a system-on-chip (SOC), a single-board computer system (SBC) (e.g., a computer-on-module (COM) or system-on-module (SOM)), a desktop computer system, a laptop or notebook computer system, an interactive kiosk, a mainframe, a mesh of computer systems, a server, an application server, or a combination of two or more of these. Where appropriate, computing system220may include one or more computing systems220; be unitary or distributed; span multiple locations; span multiple machines; or reside in a cloud, which may include one or more cloud components in one or more networks. Where appropriate, one or more computing systems220may perform without substantial spatial or temporal limitation one or more steps of one or more methods described or illustrated herein. As an example and not by way of limitation, one or more computing systems220may perform in real time or in batch mode one or more steps of one or more methods described or illustrated herein. One or more computing systems220may perform at different times or at different locations one or more steps of one or more methods described or illustrated herein, where appropriate.

In some embodiments, computing system220may execute any suitable operating system such as IBM's zSeries/Operating System (z/OS), MS-DOS, PC-DOS, MAC-OS, WINDOWS, UNIX, OpenVMS, an operating system based on LINUX, or any other appropriate operating system, including future operating systems. In some embodiments, computing system220may be a web server running web server applications such as Apache, Microsoft's Internet Information Server™, and the like.

In particular embodiments, computing system220includes a processor222, memory224, storage device226, an input/output (I/O) interface228, a communication interface232, and a bus234. Although this disclosure describes and illustrates a particular computing system220having a particular number of particular components in a particular arrangement, this disclosure contemplates any suitable computing system220having any suitable number of any suitable components in any suitable arrangement.

In particular embodiments, processor222includes hardware for executing instructions, such as those making up computer program250. As an example and not by way of limitation, to execute instructions, processor222may retrieve (or fetch) the instructions from an internal register, an internal cache, memory224, or storage device226; decode and execute the instructions; and then write one or more results to an internal register, an internal cache, memory224, or storage device226. In particular embodiments, processor222may include one or more internal caches for data, instructions, or addresses. This disclosure contemplates processor222including any suitable number of any suitable internal caches, where appropriate. As an example and not by way of limitation, processor222may include one or more instruction caches, one or more data caches, and one or more translation lookaside buffers (TLBs). Instructions in the instruction caches may be copies of instructions in memory224or storage device226, and the instruction caches may speed up retrieval of those instructions by processor222. Data in the data caches may be copies of data in memory224or storage device226for instructions executing at processor222to operate on; the results of previous instructions executed at processor222for access by subsequent instructions executing at processor222or for writing to memory224or storage device226; or other suitable data. The data caches may speed up read or write operations by processor222. The TLBs may speed up virtual-address translation for processor222. In particular embodiments, processor222may include one or more internal registers for data, instructions, or addresses. This disclosure contemplates processor222including any suitable number of any suitable internal registers, where appropriate. Where appropriate, processor222may include one or more arithmetic logic units (ALUs); be a multi-core processor; or include one or more processors222. Although this disclosure describes and illustrates a particular processor, this disclosure contemplates any suitable processor.

In particular embodiments, memory224includes main memory for storing instructions such as computer program250for processor222to execute, or data such as text105for processor222to operate on. As an example and not by way of limitation, computer system220may load instructions from storage device226or another source (e.g., another computing system220) to memory224. Processor222may then load the instructions from memory224to an internal register or internal cache. To execute the instructions, processor222may retrieve the instructions from the internal register or internal cache and decode them. During or after execution of the instructions, processor222may write one or more results (which may be intermediate or final results) to the internal register or internal cache. Processor222may then write one or more of those results to memory224. In particular embodiments, processor222executes only instructions in one or more internal registers or internal caches or in memory224(as opposed to storage device226or elsewhere) and operates only on data in one or more internal registers or internal caches or in memory224(as opposed to storage device226or elsewhere). One or more memory buses234(which may each include an address bus and a data bus) may couple processor222to memory224. Bus234may include one or more memory buses, as described below. In particular embodiments, one or more memory management units (MMUs) reside between processor222and memory224and facilitate accesses to memory224requested by processor222.

In certain embodiments, instructions executed by processor222may reside in one or more computer programs250. A computer program generally refers to instructions, logic, rules, algorithms, code, tables, or other suitable instructions for performing the described functions and operations. In some embodiments, computer program250may be stored in memory224, storage device226, or any other location accessible to computing system220. Where appropriate, a computer program may include one or more computer programs; be unitary or distributed; span multiple locations; span multiple machines; or reside in a cloud.

In particular embodiments, storage device226includes mass storage for data or instructions. As an example and not by way of limitation, storage device226may include an HDD, a floppy disk drive, flash memory, an optical disc, a magneto-optical disc, magnetic tape, a Universal Serial Bus (USB) drive, a solid-state drive (SSD), or a combination of two or more of these. Storage device226may include removable or non-removable (or fixed) media, where appropriate. Storage device226may be internal or external to computing system220, where appropriate. In particular embodiments, storage device226is non-volatile, solid-state memory. In particular embodiments, storage device226includes read-only memory (ROM). Where appropriate, this ROM may be mask-programmed ROM, programmable ROM (PROM), erasable PROM (EPROM), electrically erasable PROM (EEPROM), electrically alterable ROM (EAROM), or flash memory or a combination of two or more of these. This disclosure contemplates storage device226taking any suitable physical form. Storage device226may include one or more storage control units facilitating communication between processor222and storage device226, where appropriate. Where appropriate, storage device226may include one or more storage device226. Although this disclosure describes and illustrates particular storage, this disclosure contemplates any suitable storage.

In particular embodiments, I/O interface228includes hardware, software, or both providing one or more interfaces for communication between computing system220and one or more I/O devices. System220may include one or more of these I/O devices, where appropriate. One or more of these I/O devices may enable communication between a user and computing system220. As an example and not by way of limitation, an I/O device may include a keyboard, keypad, microphone, monitor, mouse, printer, scanner, speaker, still camera, stylus, tablet, touchscreen, trackball, video camera, sensor, another suitable I/O device or a combination of two or more of these. This disclosure contemplates any suitable I/O devices and any suitable I/O interfaces228for them. Where appropriate, I/O interface228may include one or more devices or software drivers enabling processor222to drive one or more of these I/O devices. I/O interface228may include one or more I/O interfaces228, where appropriate. Although this disclosure describes and illustrates a particular I/O interface228, this disclosure contemplates any suitable I/O interface228.

In particular embodiments, communication interface232includes hardware, software, or both providing one or more interfaces for communication (e.g., packet-based communication and facsimile communication) between computing system220and one or more other computing systems220. As an example and not by way of limitation, communication interface232may include a network interface controller (NIC) or network adapter for communicating with an Ethernet or other wire-based network or a wireless NIC (WNIC) or wireless adapter for communicating with a wireless network, such as a WI-FI network. This disclosure contemplates any suitable network and any suitable communication interface232for it. As an example and not by way of limitation, computing system220may communicate with an ad hoc network, a personal area network (PAN), a local area network (LAN), a wide area network (WAN), a metropolitan area network (MAN), one or more portions of the Internet, a PSTN, or a combination of two or more of these. One or more portions of one or more of these networks may be wired or wireless. As an example, computing system220may communicate with a wireless PAN (WPAN) (e.g., a BLUETOOTH WPAN), a WI-FI network, a WI-MAX network, a cellular telephone network (e.g., a Global System for Mobile Communications (GSM) network), or other suitable wireless network or a combination of two or more of these. Computing system220may include any suitable communication interface232for any of these networks, where appropriate. Communication interface232may include one or more communication interfaces31, where appropriate. Although this disclosure describes and illustrates a particular communication interface, this disclosure contemplates any suitable communication interface.

In particular embodiments, bus234includes hardware, software, or both coupling components of computing system220to each other. As an example and not by way of limitation, bus234may include an Accelerated Graphics Port (AGP) or other graphics bus, an Enhanced Industry Standard Architecture (EISA) bus, a front-side bus (FSB), a HYPERTRANSPORT (HT) interconnect, an Industry Standard Architecture (ISA) bus, an INFINIBAND interconnect, a low-pin-count (LPC) bus, a memory bus, a Micro Channel Architecture (MCA) bus, a Peripheral Component Interconnect (PCI) bus, a PCI-Express (PCI-X) bus, a serial advanced technology attachment (SATA) bus, a Video Electronics Standards Association local (VLB) bus, or another suitable bus or a combination of two or more of these. Bus234may include one or more buses234, where appropriate. Although this disclosure describes and illustrates a particular bus, this disclosure contemplates any suitable bus or interconnect.

The components of system220may be integrated or separated. In some embodiments, components of system220may each be housed within a single chassis. The operations of system220may be performed by more, fewer, or other components. Additionally, operations of system220may be performed using any suitable logic that may comprise software, hardware, other logic, or any suitable combination of the preceding.

FIG. 3illustrates a method300for providing a correlation index according to certain embodiments. The provided correlation index may refer to correlation index18described above. Method300begins in step310where a plurality of strikes for each of a plurality of index and stock options is selected. In some embodiments, the strikes are selected using a predetermined standard deviation range as described above. In certain embodiments, the plurality of strikes may refer to strikes32described above.

In step320, a flat implied volatility for each index and stock option is computed using the selected strikes of step310. In some embodiments, the flat implied volatilities are computed on a first tranche day. In some embodiments, the flat implied volatilities may refer to flat implied volatilities34described above.

In step330, a plurality of constant flat implied volatilities are created by holding the computed flat implied volatilities constant over a predetermined time period. In step340, a plurality of deltas is computed on a daily basis using the plurality of constant flat implied volatilities of step330. In some embodiments, the deltas may refer to deltas42described above.

In step350, a daily profit and loss (P&L) is computed based on a change in a value of a portfolio of index options, stock options, the plurality of deltas of step340, interest on cost to borrow stock, and an interest on funding. In some embodiments, the interest on funding may refer to interest on funding64described above.

In step360, a correlation index value is determined based at least in part on a previous correlation index value and the computed daily P&L of step350. After step360, method300ends.

Herein, reference to a computer-readable storage medium encompasses one or more non-transitory, tangible computer-readable storage media possessing structure. As an example and not by way of limitation, a computer-readable storage medium may include a semiconductor-based or other integrated circuit (IC) (such, as for example, a field-programmable gate array (FPGA) or an application-specific IC (ASIC)), a hard disk, an HDD, a hybrid hard drive (HHD), an optical disc, an optical disc drive (ODD), a magneto-optical disc, a magneto-optical drive, a floppy disk, a floppy disk drive (FDD), magnetic tape, a holographic storage medium, a SSD, a RAM-drive, a SECURE DIGITAL card, a SECURE DIGITAL drive, or another suitable computer-readable storage medium or a combination of two or more of these, where appropriate. A computer-readable non-transitory storage medium may be volatile, non-volatile, or a combination of volatile and non-volatile, where appropriate.