# Patent

## Title
Method and system for creating and trading derivative investment products based on a statistical property reflecting the variance of an underlying asset

## Abstract

Systems and methods for creating a limited risk derivative based on a realized variance of an underlying equity is disclosed. In one implementation, a limited risk derivative product includes a capped value for a statistical property reflecting a variance of the underlying equity is calculated based on a pari-mutuel action. The capped value includes a dynamic value and a cap. The dynamic value reflects an average volatility of prices returns of the underlying equity over a predefined period of time and the cap reflects a maximum value of the dynamic value. The limited risk derivative product additionally includes an average of a summation of each squared daily return of the underlying equity included in the value for the statistical property reflecting the variance of the underlying equity.

## Background
RELATED PATENT APPLICATIONS
This application is a Continuation-In-Part of pending U.S. application Ser. No. 11/122,748, filed May 4, 2005, entitled “Method of Creating and Trading Derivative Investment Products Based on a Statistical Property Reflecting the Variance of an Underlying Asset,” the entirety of which is hereby incorporated herein by reference.


FIELD OF THE DISCLOSURE
The present disclosure relates to derivative investment markets. More specifically, the present disclosure relates to aspects of actively disseminating, trading and auctioning derivatives.
BACKGROUND
A derivative is a financial security whose value is derived in part from a value or characteristic of another security, known as an underlying asset. Two exemplary, well known derivatives are options and futures.
An option is a contract giving a holder of the option a right, but not an obligation, to buy or sell an underlying asset at a specific price on or before a certain date. Generally, a party who purchases an option is referred to as the holder of the option and a party who sells an option is referred to as the writer of the option.
There are generally two types of options: call options and put options. A holder of a call option receives a right to purchase an underlying asset at a specific price, known as the “strike price,” such that if the holder exercises the call option, the writer is obligated to deliver the underlying asset to the holder at the strike price. Alternatively, the holder of a put option receives a right to sell an underlying asset at a specific price, referred to as the strike price, such that if the holder exercises the put option, the writer is obligated to purchase the underlying asset at the agreed upon strike price. Thus, the settlement process for an option involves the transfer of funds from the purchaser of the underlying asset to the seller, and the transfer of the underlying asset from the seller of the underlying asset to the purchaser. This type of settlement may be referred to as “in kind” settlement. However, an underlying asset of an option does not need to be tangible, transferable property.
Options may also be based on more abstract market indicators, such as stock indices, interest rates, futures contracts and other derivatives. In these cases, in kind settlement may not be desired, or in kind settlement may not be possible because delivering the underlying asset is not possible. Therefore, cash settlement is employed. Using cash settlement, a holder of an index call option receives the right to “purchase” not the index itself, but rather a cash amount equal to the value of the index multiplied by a multiplier such as $100. Thus, if a holder of an index call option elects to exercise the option, the writer of the option is obligated to pay the holder the difference between the current value of the index and the strike price multiplied by the multiplier. However, the holder of the index will only realize a profit if the current value of the index is greater than the strike price. If the current value of the index is less than or equal to the strike price, the option is worthless due to the fact the holder would realize a loss.
Similar to options contracts, futures contracts may also be based on abstract market indicators. A future is a contract giving a buyer of the future a right to receive delivery of an underlying commodity or asset on a fixed date in the future. Accordingly, a seller of the future contract agrees to deliver the commodity or asset on the specified date for a given price. Typically, the seller will demand a premium over the prevailing market price at the time the contract is made in order to cover the cost of carrying the commodity or asset until the delivery date.
Although futures contracts generally confer an obligation to deliver an underlying asset on a specified delivery date, the actual underlying asset need not ever change hands. Instead, futures contracts may be settled in cash such that to settle a future, the difference between a market price and a contract price is paid by one investor to the other. Again, like options, cash settlement allows futures contracts to be created based on more abstract “assets” such as market indices. Rather than requiring the delivery of a market index (a concept that has no real meaning), or delivery of the individual components that make up the index, at a set price on a given date, index futures can be settled in cash. In this case, the difference between the contract price and the price of the underlying asset (i.e., current value of market index) is exchanged between the investors to settle the contract.
Derivatives such as options and futures may be traded over-the-counter, and/or on other trading facilities such as organized exchanges. In over-the-counter transactions the individual parties to a transaction are free to customize each transaction as they see fit. With trading facility-traded derivatives, a clearing corporation stands between the holders and writers of derivatives. The clearing corporation matches buyers and sellers, and settles the trades. Thus, cash or the underlying assets are delivered, when necessary, to the clearing corporation and the clearing corporation disperses the assets as necessary as a consequence of the trades. Typically, such standard derivatives will be listed as different series expiring each month and representing a number of different incremental strike prices. The size of the increment in the strike price will be determined by the rules of the trading facility, and will typically be related to the value of the underlying asset.
Additionally, there are two widely utilized methods by which derivatives are currently traded: (1) order-matching and (2) principal market making. Order matching is a model followed by exchanges such as the CBOT or the Chicago Mercantile Exchange and some newer online exchanges. In order matching, the exchange coordinates the activities of buyers and sellers so that “bids” to buy can be paired off with “offers” to sell. Orders may be matched both electronically and through the primary market making activities of the exchange members. Typically, the exchange itself takes no market risk and covers its own cost of operation by selling memberships to brokers. Member brokers may take principal positions, which are often hedged across their portfolios.
In principal market making, a bank or brokerage firm, for example, establishes a derivatives trading operation, capitalizes it, and makes a market by maintaining a portfolio of derivatives and underlying positions. The market maker usually hedges the portfolio on a dynamic basis by continually changing the composition of the portfolio as market conditions change. In general, the market maker strives to cover its cost of operation by collecting a bid-offer spread and through the scale economies obtained by simultaneously hedging a portfolio of positions. As the market maker takes significant market risk, its counterparties are exposed to the risk that it may go bankrupt. Additionally, while in theory the principal market making activity could be done over a wide area network, in practice derivatives trading is today usually accomplished via the telephone. Often, trades are processed laboriously, with many manual steps required from the front office transaction to the back office processing and clearing.
The return to a trader of a traditional derivative product is, in most cases, largely determined by the value of the underlying security, asset, liability or claim on which the derivative is based. For example, the value of a call option on a stock, which gives the holder the right to buy the stock at some future date at a fixed strike price, varies directly with the price of the underlying stock. In the case of non-financial derivatives such as reinsurance contracts, the value of the reinsurance contract is affected by the loss experience on the underlying portfolio of insured claims. The prices of traditional derivative products are usually determined by supply and demand for the derivative based on the value of the underlying security (which is itself usually determined by supply and demand, or, as in the case of insurance, by events insured by the insurance or reinsurance contract).
While standard derivative contracts may be based on many different types of market indexes or statistical properties of underlying assets, current standard derivative contracts do not provide investors with sufficient tools to hedge against greater than expected or less than expected volatility in an underlying asset. Nor do such standard derivative contracts cap the amount of risk undertaken. Thus, there is a need for a standard derivative contract based on a statistical property that reflects the volatility of an underlying asset, but that is capped to limit the amount of risk undertaken.
BRIEF SUMMARY
Accordingly, the present disclosure relates to a method and system for creating and auctioning derivative contracts based on a statistical property that reflects variance of an underlying asset. A first aspect of the invention relates to a limited risk derivative product based on a realized variance of an underlying equity and an average of a summation of each squared daily return of the underlying equity included in a value for a statistical property reflecting the variance of the underlying equity. The realized variance of the underlying equity includes a capped value for a statistical property reflecting the variance of the underlying equity. The capped value includes a dynamic value and a cap. The dynamic value reflects an average volatility of price returns of the underlying equity over a predefined time period and the cap reflects a maximum value of the dynamic value.
The value of the statistical property reflecting the variance of the underlying equity may be calculated according to the formula:



  
    
      Realized
      ⁢
      
        
      
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      Variance
    
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    ,
  



Pi is an initial value of the underlying equity used to calculate a daily return, Pi+1 is a final value of the underlying equity used to calculate the daily return, Ne is a number of expected underlying equity values needed to calculate daily returns during a variance calculation period, Na is an actual number of underlying equity values used to calculate daily returns during the variance calculation period; and AF is an annualization factor.

A second aspect of the invention relates to a method and system for creating limited risk derivatives based on a realized variance of an underlying equity. The method includes calculating from a parimutuel auction a capped value for a statistical property reflecting the variance of the underlying equity. The capped value includes a dynamic value and a cap. The dynamic value reflects an average volatility of price returns of the underlying equity over a predefined time period and the cap reflects a maximum value of the dynamic value.
The method may additionally include calculating an average of a summation of each squared daily return of the underlying equity included in the value for the statistical property reflecting the variance of the underlying equity. The value of the statistical property may be calculated according to the formula:



  
    
      Realized
      ⁢
      
        
      
      ⁢
      Variance
    
    =
    
      A
      ⁢
      
        
      
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Pi is an initial value of the underlying equity used to calculate a daily return, Pi+1 is a final value of the underlying equity used to calculate the daily return, Ne is a number of expected underlying equity values needed to calculate daily returns during a variance calculation period, Na is an actual number of underlying equity values used to calculate daily returns during the variance calculation period; and AF is an annualization factor.

The method may further include displaying at least one limited risk variance derivative based on the statistical property reflecting variance on a trading facility display device coupled to a trading platform and transmitting at least one limited risk variance derivative quote of a liquidity provider from the trading facility to at least one market participant.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a flow chart of one embodiment of a method of creating and trading a variance futures contract;
FIG. 2 is a diagram showing a listing of a variance futures contract on a trading facility;
FIG. 3 is a block diagram of one embodiment of a system for creating and trading variance futures contracts;
FIGS. 4a and 4b are a table showing values for a variance futures contract over a variance calculation period;
FIG. 5 is a flow chart of one embodiment of a method of creating and trading a variance option contract;
FIG. 6 is a diagram showing a listing of a variance options contract on a trading facility;
FIG. 7 is a block diagram of one embodiment of a system for creating and trading variance option contracts; and
FIGS. 8a and 8b are a table showing values for a variance option contract over a variance calculation period.



DETAILED DESCRIPTION OF THE DRAWINGS
Variance derivatives are financial instruments such as futures and option contracts that trade on trading facilities, such as exchanges, whose value is based on the volatility of the value of an underlying asset and not on the return of the underlying asset. Variance of an underlying asset is a measure of the statistical dispersion of the value of the underlying asset. Thus, variance indicates the movement in the value of an underlying asset from trading day to trading day. Typically, variance is computed as the average squared deviation of the value of an underlying asset from an expected value, represented by an average (mean) price return value.
Those skilled in the art will recognize that variance derivatives having features similar to those described herein and statistical properties which reflect the variance of an underlying asset, but which are given labels other than variance derivatives, variance futures, or variance options will nonetheless fall within the scope of the present invention.
FIG. 1 is a flow chart of one embodiment of a method for creating and trading a variance futures contract 100. A variance futures contract is a financial instrument in which the realized variance of an underlying asset is calculated at the end of each trading day over a predefined period, known as the variance calculation period. Typically, the realized variance of an underlying asset is calculated using a standardized equation, which is a function of a daily return of the underlying asset. The daily return of an underlying asset is typically the natural log of a final value of the underlying asset over an initial value of the underlying asset.
An investor is generally able to purchase a variance futures contract before a variance calculation period begins, or an investor may trade into or out of a variance futures contract during the variance calculation period. To facilitate the purchase and trading of variance futures contracts, trading facilities such as exchanges like the CBOE Futures Exchange will calculate and disseminate cumulative realized variance and implied realized variance values for a variance futures contract. Cumulative realized variance and implied realized variance provide tools for investors to determine when to trade into and out of a variance futures contract.
The method for creating and trading a variance futures contract begins at step 102 by identifying an underlying asset or a set of underlying assets for the variance futures contract. Typically, an underlying asset or set of assets is selected based on trading volume of a prospective underlying asset, the general level of interest of market participants in a prospective underlying asset, or for any other reason desired by a trading facility. The underlying assets for the variance futures contract may be equity indexes or securities; fixed income indexes or securities; foreign currency exchange rates; interest rates; commodity indexes; commodity or structured products traded on a trading facility or in the over-the-counter (“OTC”) market; or any other type of underlying asset whose value may change from day to day.
Once the underlying asset or assets have been selected at 102, a formula is developed at 104 for generating a value for a statistical property reflecting the realized variance of the underlying asset or assets over the defined variance calculation period. In one embodiment, realized variance is calculated using a standard formula that uses an annualization factor and daily S&P 500 returns over the variance calculation period, assuming a mean daily price return of zero. The annualization factor is normally a number that represents the number of days the underlying asset will trade in a year. Typically, the annualization factor is 252 to represent the number of trading days an underlying asset is traded in a year. However, for underlying assets that trade in international trading facilities or specialized trading facilities, the annualization factor may be a value other than 252.
Realized variance is typically calculated according to the formula:




  
    
      Realized
      ⁢
      
        
      
      ⁢
      Variance
    
    =
    
      A
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Pi is an initial value of the underlying asset used to calculate a daily return, Pi+1 is a final value of the underlying asset used to calculate the daily return, Ne is a number of expected underlying asset values needed to calculate daily returns during the variance calculation period, Na is an actual number of underlying asset values used to calculate daily returns during the variance calculation period, and AF is the annualization factor.

A “daily return” (Ri) is the natural log of a final value (Pi+1) of an underlying asset over an initial value (Pi) of the underlying asset. The initial value (Pi) and the final value (Pi+1) of the underlying asset may be on the same trading day, consecutive trading days, or non-consecutive trading days. For example in one embodiment, the daily return may be the natural log of a closing price of an underlying asset on one day over the closing price of the underling asset on a previous trading day. In another embodiment, the daily return may be the natural log of a closing price of an underlying asset over an opening price of the underlying asset on the same trading day.
The initial value (Pi) and final value (Pi+1) of an underlying asset may be based on a Special Opening Quotation (“SOQ”), closing price, intraday price, or any other price. Similarly, the final value (Pi+1) of an underlying asset may be based on a SOQ, closing price, intraday price quote, or any other price.
After determining a formula for calculating realized variance at 104, specific values are defined at 106 for the variables within the formula for calculating realized variance during the variance calculation period. Typically, specific values will be defined for an initial value for the first daily return, a final value for the first daily return, an initial value for the last daily return, and the final value of the last daily return. In one embodiment, the initial value (Pi) for a first daily return in a variance calculation period is defined to be an initial value of the underlying asset on a first day of the variance calculation period; a final value of the underlying asset for the first daily return is defined to be a closing price value of the underlying asset on a following trading day; an initial value for a last daily return in the variance calculation period is defined to be a closing value of the underlying asset on a trading day immediately prior to the final settlement date; and a final value for the last daily return is defined to be a SOQ on the final settlement date. For all other daily returns, the initial and final values are defined to be the closing values of the underlying asset on consecutive trading days.
Generally, the total number of actual daily returns during the variance calculation period is defined to be Na-1, but if one or more market disruption events occurs during the variance calculation period, the actual number of underlying asset values will be less than the expected number of underlying asset values by an amount equal to the number of market disruption events that occurred during the variance calculation period.
A market disruption event generally occurs on a day on which trading is expected to take place to generate a value for an underlying asset, but for some reason trading is stopped or a value for the underlying asset is not available. In one embodiment, a market disruption event may be defined to be (i) an occurrence or existence, on any trading day during a one-half period that ends at the scheduled close of trading, of any suspension of, or limitation imposed on, trading on the primary trading facility(s) of the companies comprising the underlying asset in one or more securities that comprise 20 percent or more of the level of the asset; or (ii) if on any trading day that one or more primary trading facility(s) determines to change scheduled close of trading by reducing the time for trading on such day, and either no public announcement of such reduction is made by such trading facility or the public announcement of such change is made less than one hour prior to the scheduled close of trading; or (iii) if on any trading day one or more primary trading facility(s) fails to open and if in the case of either (i) or (ii) above, such suspension, limitation, or reduction is deemed material. A scheduled close of trading is the time scheduled by each trading facility, as of the opening for trading in the underlying asset, as the closing time of the trading of such asset on the trading day. Examples of market disruption events include days on which trading is suspended due to a national day of mourning or days on which trading is suspended for national security.
If a trading facility determines that a market disruption event has occurred on a trading day, the daily return of the underlying asset on that day will typically be omitted from the series of daily returns used to calculate the realized variance over the variance calculation period. For each such market disruption event, the actual number of underlying asset values used to calculate daily returns during the settlement calculation, represented by Na, will be reduced by one. Typically, if a market disruption event occurs on a final settlement date of a variance futures contract, the final settlement date may be postponed until the next trading day on which a market disruption event does not occur. Alternatively, any other action may be taken as agreed upon by a trading facility. These actions will typically be listed in the Rules and By-Laws of a clearing agent.
Once the variance calculation period begins for a variance futures contract, the value represented by Ne will not change regardless of the number of market disruption events that occur during the variance calculation period, even if the final settlement date is postponed. Typically, if the final settlement date of the expiring variance futures contract is postponed, the length of the variance calculation period for the next variance futures contract is shortened by the number of market disruption events that occur at the beginning of the variance calculation period. Likewise, the value represented by Ne is reduced by the number of market disruption events that occur at the beginning of the variance calculation period.
Similarly, if a market disruption event occurs at the beginning of the variance calculation period, the first daily return of the shortened variance calculation period for the next variance futures contract will be calculated using the same procedure as described. For example, if the final settlement date for the previous variance calculation period of a variance futures contract is postponed to a Tuesday, the initial value for the first daily return of the variance calculation period of the next variance futures contract would be calculated using the SOQ (or other price designated) of the underlying asset on Tuesday morning and the closing value of the asset the following Wednesday.
Once the underlying asset or assets is chosen at 102, the formula for generating the value of the statistical property reflecting the variance of the underlying asset or assets is determined at 104, and the value of the variables within the variance calculation period are defined at 106, the variance futures contract based on the chosen underlying asset or assets is assigned a unique symbol at 108 and listed on a trading platform at 110. Generally, the variance futures contract may be assigned any unique symbol that serves as a standard identifier for the type of standardized variance futures contract.
Generally, a variance futures contract may be listed on an electronic platform, an open outcry platform, a hybrid environment that combines the electronic platform and open outcry platform, or any other type of platform known in the art. One example of a hybrid exchange environment is disclosed in U.S. patent application Ser. No. 10/423,201, filed Apr. 24, 2003, the entirety of which is herein incorporated by reference. Additionally, a trading facility such as an exchange may transmit variance futures contract quotes of liquidity providers over dissemination networks 114 to other market participants. Liquidity providers may include Designated Primary Market Makers (“DPM”), market makers, locals, specialists, trading privilege holders, registered traders, members, or any other entity that may provide a trading facility with a quote for a variance derivative. Dissemination Networks may include networks such as the Options Price Reporting Authority (“OPRA”), the CBOE Futures Network, an Internet website or email alerts via email communication networks. Market participants may include liquidity providers, brokerage firms, normal investors, or any other entity that subscribes to a dissemination network.
Additionally, in another embodiment, variance futures contracts may be traded through an exchange-operated parimutuel auction and cash-settled based on the variance of daily price returns of the underlying equity. An electronic parimutuel, or Dutch, auction system conducts periodic auctions, with all contracts that settle in-the-money funded by the premiums collected for those that settle out-of-the-money.
As mentioned, in a parimutuel auction, all the contracts that settle in-the-money are funded by those that settle out-of-the-money. Thus, the net exposure of the system is zero once the auction process is completed, and there is no accumulation of open interest over time. Additionally, the pricing of contracts in a parimutuel auction depends on relative demand; the more popular the strike, the greater its value. In other words, a parimutuel action does not depend on market makers to set a price; instead the price is continuously adjusted to reflect the stream of orders coming into the auction. Typically, as each order enters the system, it affects not only the price of the sought-after strike, but also affects all the other strikes available in that auction. In such a scenario, as the price rises for the more sought-after strikes, the system adjusts the prices downward for the less popular strikes. Further, the process does not require the matching of specific buy orders against specific sell orders, as in many traditional markets. Instead, all buy and sell orders enter a single pool of liquidity, and each order can provide liquidity for other orders at different strike prices and the liquidity is maintained such that system exposure remains zero. This format maximizes liquidity, a key feature when there is no tradable underlying instrument.
As seen in FIG. 2, in one embodiment the variance futures contracts are listed on a trading platform by displaying the variance futures contracts on a trading facility display device coupled with the trading platform. The listing 200 displays the variance futures contract for purchase in terms of variance points 204 or a square root of variance points 206. A variance point is an expected realized variance over a variance calculation period multiplied by a scaling factor such as 10,000. In FIG. 2, one of the variance futures contract has a value of 625.00 in terms of variance points 208 and a value of 25.00 in terms of volatility 210. A value of 625.0 is calculated by multiplying a realized variance calculation of 0.0625 by a scaling factor of 10,000. Further, a value of 25.00 is calculated by taking the square root of 625.00 (price in terms of variance points).
In addition to listing variance futures contracts in terms of variance points and the square root of variance points, the prices for variance futures contracts may also be stated in terms of a decimal, fractions, or any other numerical representation of a price. Further, scaling factors for the variance derivatives may be determined on a contract-by-contract basis. Scaling factors are typically adjusted to control the size, and therefore the price of a derivative contract.
Over the course of the variance calculation period, in addition to listing variance futures contracts in terms of variance points, the trading facility may also display and disseminate a cumulative realized variance and an implied realized variance on a daily basis, or in real-time, to facilitate trading within the variance futures contract. Cumulative realized variance is an average rate of the realized variance of a variance futures contract through a specific date of the variance calculation period. Thus, using the formula described above, after Np days in a variance calculation period, the cumulative realized variance may be calculated according to the formula:




  
    Cumulative
    ⁢
    
      
    
    ⁢
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Implied realized variance is a weighted average of both the cumulative realized variance and a most recent closing price of a variance futures contract during the variance calculation period. Specifically, implied realized variance is calculated according to the formula:




  
    
      Implied
      ⁢
      
        
      
      ⁢
      Variance
    
    =
    
      
        TP
        -
        
          RV
          *
          
            
              Day
              Current
            
            
              Day
              Total
            
          
        
      
      
        
          Day
          Left
        
        /
        
          Day
          Total
        
      
    
  
  ,




where TP is the last trading price of the variance futures contract; RV is the cumulative realized variance; Daycurrent is the total number of trading days that have passed in the variance calculation period; DayTotal is the total number of trading days in the variance calculation period; and DayLeft is the number of trading days left in the variance calculation period. Referring to FIG. 1, the cumulative realized variance and implied realized variance values provide investors a tool for determining when to trade into and out of variance futures contracts at 116.

At expiration of the variance calculation period for a variance futures contract, the trading facility will settle a variance futures contract at 118 such that the settlement value is equal to the cumulative realized variance over the specified variance calculation period. Typically, settlement of variance futures contracts will result in the delivery of a cash settlement amount on the business day immediately following the settlement date. The cash settlement amount on the final settlement date shall be the final mark to market amount against the final settlement price of the variance futures contract multiplied by the contract multiplier.
In the parimutuel auction embodiment mentioned above, the payout amount is capped as a result of the parimutuel auction process through which they are traded.
FIG. 3 is a block diagram of a system 300 for creating and trading variance futures contracts. Generally, the system comprises a variance property module 302, a dissemination module 304 coupled with the variance property module 302, and a trading module 306 coupled with the dissemination module 304. Typically, each module 302, 304, 306 is also coupled to a communication network 308 coupled to various trading facilities 322 and liquidity providers 324.
The variance property module 302 comprises a communications interface 310, a processor 312 coupled with the communications interface 310, and a memory 314 coupled with the processor 312. Logic stored in the memory 314 is executed by the processor 312 such that that the variance property module 302 may receive current values for an underlying asset of a variance futures contract through the communications interface 310; calculate realized variance, cumulative realized variance, and implied realized variance, as described above, for the underlying asset; and pass the calculated values to the dissemination module 304.
The dissemination module 304 comprises a communications interface 316, a processor 318 coupled with the communications interface 316, and a memory 320 coupled with the processor 318. Logic stored in the memory 320 is executed by the processor 318 such that the dissemination module 304 may receive the calculated values from the variance property module 302 through the communications interface 316, and disseminate the calculated values over the communications network 308 to various market participants 322, as described above.
The trading module 306 comprises a communications interface 326, a processor 328 coupled with the communications interface 326, and a memory 330 coupled with the processor 328. Logic stored in the memory 330 is executed by the processor 328 such that the trading module 306 may receive buy or sell orders over the communications network 308, as described above, and pass the results of the buy or sell order to the dissemination module 304 to be disseminated over the communications network 308 to the market participants 322.
FIGS. 4a and 4b are a table showing values for a variance futures contract over a variance calculation period. The first column 402 represents the number of days that have passed in the variance calculation period; column 404 shows the daily closing price of the underlying asset; column 406 shows the natural log of the current closing price of the underlying asset over the previous closing price of the underlying asset; column 408 shows the square of the value of column 406; column 410 shows the summation of the values in column 408; column 412 shows the cumulative realized variance on each day; column 414 shows the closing price of the variance futures contract for each day; and column 416 shows the calculated implied realized variance for each day.
As shown in column 402, a variance futures contract with a 90-day variance calculation period typically includes 64 trading days. In the example, on the first trading day 418, the underlying asset closes at a value of 1122.20 (420). To calculate the realized variance for day 1, the natural log is taken of the closing value 420 of the underlying asset on day 1 (1122.20) over the closing value 422 of the underlying asset on the previous trading day (1127.02), resulting in a value of—0.0042859 (424). The value of the natural log is squared, resulting in the value of 1.83693*10−5 (426). The value of the square of the natural log of the current day's closing price over the previous day's closing price is then summed with any previous values in column 408. Due to the fact there are no previous values on the first day, the summation is equal to 1.83693*10−5 (428). The value of the summation is then divided by the number of trading days in the variance calculation period that has passed (1) to obtain an average variance over the variance calculation period, multiplied by an annualization factor to represent the number of trading days in a year (252) and multiplied by a scaling factor (10,000), resulting in a value of 46.29 (430).
The realized variance of the variance futures contract after one day can be used with the closing value of the variance futures contract on the first day 432 (284.00) to calculate the implied variance 434 according to the implied realized variance formula described above. Specifically, the implied realized variance is calculated to be:




  
    Implied
    ⁢
    
      
    
    ⁢
    Variance
  
  =
  
    
      
        284.00
        -
        
          46.29
          *
          
            1
            64
          
        
      
      
        63
        64
      
    
    =
    
      287.77
      .
    
  




This above-described process for the first trading day is repeated for each trading day in the variance calculation period. For example on the 20th day of the variance calculation period 436, the underlying asset closes at 1108.20 (438). The natural log is taken of the closing value of the current day 438 (1108.20) over the closing value of the previous closing day 440 (1103.29). The result of the natural log 441 (0.00444045) is squared 442 (1.97176*10−5), and summed with the previous values 444 (0.000958131) to obtain a value 446 (0.000977849). The resulting value is divided by the current number of trading days that have passed (20) to obtain an average; multiplied by an annualization factor representing the number of trading days in a year (252); and multiplied by a scaling factor (10,000) to obtain the cumulative realized variance 448 of 123.21 on day 20.
The implied variance of the 20th day 452 of the variance calculation period can then be calculated using the closing value of the variance futures contract 450 on the 20th day (203.5) according to the formula described above:




  
    Implied
    ⁢
    
      
    
    ⁢
    Variance
  
  =
  
    
      
        203.5
        -
        
          123.21
          *
          
            20
            64
          
        
      
      
        44
        64
      
    
    =
    
      240.00
      .
    
  




In addition to variance futures contracts, variance derivatives also encompass variance option contracts. A variance option contract is a type of option product that has a strike price set at a cumulative realized variance level for an underlying asset. The strike price to be listed may be any variance level chosen by the trading facility.
As with traditional option contracts, a variance option contract may include both call variance options and put variance options. Typically, the holder of a variance call option receives the right to purchase a cash amount equal to the difference between the current value of the statistical property reflecting the variance of the underlying asset and the strike price multiplied by the multiplier. Similarly, the holder of a variance put option receives the right to sell a cash amount equal to the difference between the current value of the statistical property reflecting the variance of the underlying asset and the strike price multiplied by the multiplier.
Due to the fact the variance option contract is based on a statistical property, in kind settlement is not desired and cash settlement is employed. Typically, the cash settlement will be equal to the value of the statistical property reflecting variance of the underlying asset multiplied by a predefined multiplier. Any predefined multiplier may be chosen by the trading facility.
As shown in FIG. 5, to create and trade a variance option contract an underlying asset is first chosen 502. As with the variance futures contract, the underlying asset may be selected based on trading volume of a prospective underlying asset, a general interest in a prospective underlying asset among market participants, or for any other reason desired by a trading facility. The underlying asset for the variance option contract may be equity indexes or securities; equity fixed income indexes or securities; foreign currency exchange rates; interest rates; commodity indexes; commodity or structured products traded on a trading facility or in the over-the-counter (“OTC”) market; or any other type of underlying asset known in the art.
Once the underlying asset or assets have been selected at 502, a formula is developed at 504 for generating a value of a statistical property reflecting the realized variance of the underlying asset or assets over the defined variance calculation period. Typically, the formula to generate a value of a statistical property reflecting realized variance for a variance option contract is the same formula used to generate a value of a statistical property reflecting realized variance for the variance futures contract. Specifically, variance for a variance option contract is typically calculated according to the formula:




  
    
      Realized
      ⁢
      
        
      
      ⁢
      Variance
    
    =
    
      A
      ⁢
      
        
      
      ⁢
      F
      *
      
        
          ∑
          
            i
            =
            1
          
          
            
              N
              a
            
            -
            1
          
        
        ⁢
        
          
            R
            i
            2
          
          
            
              N
              e
            
            -
            1
          
        
      
    
  
  ,
  
    

  
  ⁢
  
    
      wherein
      :
      
        

      
      ⁢
      
        R
        i
      
    
    =
    
      ln
      ⁢
      
        
      
      ⁢
      
        
          P
          
            i
            +
            1
          
        
        
          P
          i
        
      
    
  
  ,




Pi is an initial value of the underlying asset used to calculate a daily return, Pi+1 is a final value of the underlying asset used to calculate the daily return, Ne is a number of expected underlying asset values needed to calculate daily returns during the variance calculation period, Na is an actual number of underlying asset values used to calculate daily returns during the variance calculation period, and AF is the annualization factor.

As with the variance futures contracts, specific values are defined at 506 for the variables within the formula for calculating realized variance during the variance calculation period. The variance option contract is then assigned a unique symbol at 508 and listed on a trading platform at 510. Generally, the variance option contract may be assigned any unique symbol that serves as a standard identifier for the type of standardized variance options contract.
Generally, a variance option contract may be listed on an electronic platform, an open outcry platform, a hybrid environment that combines the electronic platform and open outcry platform, or any other type of platform known in the art. Additionally, a trading facility may disseminate quotes for variance option contracts over dissemination networks 514 such as the OPRA, the CBOE Network, an internet website or email alerts via email communication networks to market participants.
Additionally, in another embodiment, variance options contracts may be traded through an exchange-operated parimutuel auction and cash-settled based on the variance of daily price returns of the underlying equity. By way of example, on Sep. 16, 2005, an investor may purchase an IBM “Vanilla Call Limited Risk Option (LRO)” contract (European-style call option with a contract payout that is a capped variable amount), with a strike price of 400, which pays $10 for every variance point that the contract finishes in-the-money. One year later, on Sep. 16, 2006, the settlement value is determined to be 625.00 based on the variance of the daily price returns of IBM from Sep. 15, 2005 through Sep. 15, 2006. Therefore, a long LRO call position receives $2,250, or $10×(625−400). The payout from all Vanilla LRO's is capped as a result of the parimutuel auction process through which they are traded.
In addition to Vanilla LRO contracts, other limited risk options contracts may be utilized, including but not limited to: Digital LROs, which are European-style call and put options whose payout is a constant amount when the value of the underlying asset is in a specified strike range at expiration and is zero outside that strike range; and Zero-cost Collar LROs, which combine long (or short) Vanilla LRO call option with a short (or long) Vanilla LRO put option with having the same strike prices and option premium prices.
As seen in FIG. 6, in one embodiment, similar to variance futures contracts, variance option contracts are listed 600 on a trading platform for purchase with a strike price in terms of variance points 604 or a square root of variance points 606. A variance point is an expected realized variance over a variance calculation period multiplied by a scaling factor such as 10,000.
Referring again to FIG. 5, after a variance option contract is listed on a trading facility, an investor may trade into or out of the option contract at 516 as is well known in the art, until the option contract expires at 518.
FIG. 7 is a block diagram of a system 700 for creating and trading variance option contracts. Generally, the system comprises a variance property module 702, a dissemination module 704 coupled with the variance property module 702, and a trading module 706 coupled with the dissemination module 704. Typically, each module 702, 706, 708 is also coupled to a communication network 708 coupled to various market participants 722.
The variance property module 702 comprises a communications interface 710, a processor 712 coupled with the communications interface 710, and a memory 714 coupled with the processor 712. Logic stored in the memory 714 is executed by the processor 712 such that that the variance property module 702 may receive current values for an underlying asset of a variance option contract through the communications interface 710; calculate realized variance, as described above, for the underlying asset; and pass the calculated realized variance the dissemination module 704.
The dissemination module 704 comprises a communications interface 716, a processor 718 coupled with the communications interface 716, and a memory 720 coupled with the processor 718. Logic stored in the memory 720 is executed by the processor 718 such that the dissemination module 704 may receive the calculated realized variance from the variance property module 702 through the communications interface 716, and disseminate the calculated realized variance over the communications network 708 to various market participants 722, as described above.
The trading module 706 comprises a communications interface 726, a processor 728 coupled with the communications interface 726, and a memory 730 coupled with the processor 728. Logic stored in the memory 730 is executed by the processor 728 such that the trading module 706 may receive buy or sell orders over the communications network 708, as described above, and pass the results of the buy or sell order to the dissemination module 704 to be disseminated over the communications network 708 to the market participants 722.
FIGS. 8a and 8b are a table showing values for a variance option contract over a variance calculation period. In one example, a variance call option contract may have a strike price of 135.00 and be exercised at any time during the 90-day calculation period. Therefore, a holder of the variance call option contract could only exercise their option to make a profit during the 90-day variance calculation period when the cumulative realized variance is calculated to be above 135.00 such as on days 3-5 (804, 806, 808), 10 (810), 11 (812), 14 (814), 16 (816), 28 (818), 19 (820), 34-37 (822, 824, 826, 828), and 40 (830). On all other trading days of the variance calculation period, if the holder of the variance call option exercised their option it would result in a loss.
Similarly, in another example, a variance call option contract may have a strike price of 115.00 and only be exercised at the end of the 90-day calculation period. Therefore, due to the fact the cumulative realized variance is calculated to be above 115.00 at the end of the 90-day calculation period 831, the holder of the variance call option may exercise their option for a profit. However, if the cumulative realized variance was calculated to be at or below 115.00 at the end of the 90-day calculation period 831, the holder of the variance option may not exercise their option for a profit.
In yet another example, a variance put option contract may have a strike price of 117.00 and be exercised at any time during the 90-day calculation period. Therefore, a holder of the variance put option contract could only exercise their option to make a profit during the 90-day variance calculation period when the realized variance is calculated to be below 117.00 such as on days 1 (832), 2 (834), 8 (836), 9 (838), 24 (840), and 26 (842). On all other trading days of the variance calculation period, if the holder of the variance put option exercised their option it would result in a loss.
Similarly, in another example, a variance put option contract may have a strike price of 125.00 and only be exercised at the end of the 90-day calculation period 831. Therefore, due to the fact the cumulative realized variance is calculated to be below 125.00 at the end of the 90-day calculation period 844, the holder of the variance option contract may exercise their option for a profit. However, if the cumulative realized variance was calculation to be at or above 125.00 at the end of the cumulative calculation period 831, the holder of the variance option may not exercise their option for a profit.
In one embodiment, a limited risk option may include a combination of a long call option and a short put option. The long call option includes a contract payout that is a capped variable amount and the short put option includes a contract payout that is a capped variable amount. In this instrument, both the long call option and the short put option have the same strike prices and option premium prices. Capping both the payout of the long call option and the short put option limits the amount of risk of the instrument.
In another embodiment, chooser options may be created based on variance options. A chooser option is an option wherein the purchaser of the option buys a call or a put option at some time in the future. The call and the put option will typically share the same expiration date and the same strike price (value), although split chooser options may be crafted wherein the call and the put options have different expirations and/or different strikes.
Chooser options are advantageous in situations in which investors believe that the price of the underlying asset is ripe for a significant move, but the direction of the move is in doubt. For example, some events, such as the approval (disapproval) of a new product, a new earnings report, or the like, may be anticipated such that positive news is likely cause the share price to rise, and negative news will cause the share price to fall. The ability to choose whether an option will be a put or a call with knowledge of the outcome of such an event is a distinct advantage to an investor.
The purchase of a chooser option is akin to purchasing both a put and a call option on the same underlying asset. Typically the chooser option is priced accordingly. In the present case, purchasing a variance chooser option amounts to buying both a put and a call option based on the variance of an underlying asset. Chooser options may be traded on an exchange just like other variance derivative. The only accommodations necessary for adapting an exchange for trading chooser options is that a final date for making the choice between a call option and a put option must be established and maintained. Also, post trade processing on the exchange's systems must be updated to implement and track the choice of the call or a put once the choice has been made. One option for processing the chosen leg of a chooser option is to convert the chooser option into a standard option contract according to the standard series for the same underlying asset and having the same strike price as the chosen leg of the chooser option.
A capped option contract as disclosed herein is advantageous to a buyer since the degree of risk to the buyer is capped to a known amount, unlike the risk amounts associated with a traditional contract. For example, in a traditional contract, where a buyer is short on a call option, the risk is unlimited. Such positions are typically not permitted and, as such, the contract could not be made available to one or more categories of consumer. With the capped option detailed herein, the value of risk is known and limited to the capped amount. As a result, this capped option contract permits trading in positions that have heretofore not been possible for many consumers.
It is therefore intended that the foregoing detailed description be regarded as illustrative rather than limiting, and that it be understood that it is the following claims, including all equivalents, that are intended to define the spirit and scope of this invention.

## Claims

1. A system for creating a limited risk derivative product based on a realized variance of an underlying equity, the system comprising:
a variance property module comprising a processor and a memory coupled with the processor, the processor configured to execute logic stored in the memory to create a limited risk derivative product based on a realized variance of an underlying equity, the limited risk derivative comprising a capped value for a statistical property reflecting the variance of the underlying equity and an average of a summation of each squared daily return of the underlying equity included in the capped value for the statistical property reflecting the variance of the underlying equity;
wherein the capped value for the statistical property reflecting the variance of the underlying equity comprises a value and a cap, the value reflecting an average volatility of price returns of the underlying equity over a predefined time period and the cap reflecting a maximum value of the value reflecting the average volatility of price returns of the underlying equity over the predefined time period; and
wherein the value of the statistical property reflecting the variance of the underlying equity is calculated according to the formula:




  
    
      Realized
      ⁢
      
        
      
      ⁢
      Variance
    
    =
    
      A
      ⁢
      
        
      
      ⁢
      F
      ×
      
        (
        
          
            ∑
            
              i
              =
              1
            
            
              
                N
                a
              
              -
              1
            
          
          ⁢
          
            
              R
              i
              2
            
            /
            
              (
              
                
                  N
                  e
                
                -
                1
              
              )
            
          
        
        )
      
    
  




  
    
      
        wherein
        :
        
          

        
        ⁢
        
          R
          i
        
      
      =
      
        ln
        ⁢
        
          
        
        ⁢
        
          
            P
            
              i
              +
              1
            
          
          
            P
            i
          
        
      
    
    ,
  



Pi is an initial value of the underlying equity used to calculate a daily return, Pi+1 is a final value of the underlying equity used to calculate the daily return, Ne is a number of expected underlying equity values needed to calculate daily returns during a variance calculation period, Na is an actual number of underlying equity values used to calculate daily returns during the variance calculation period; and AF is an annualization factor; and
wherein the limited risk derivative product is settled based on the capped value and a difference between a cumulative realized variance and a strike price set at a fixed second cumulative realized variance, wherein the strike prices is set at the fixed second cumulative realized variance when the limited risk derivative product is created.

2. The system of claim 1, wherein a squared deviation of a daily return of the underlying equity that corresponds to a market disruption event is removed from the average volatility of price returns of the underlying equity over the predefined time period.

3. The system of claim 1, wherein the underlying equity is selected from the group consisting of: commodity or structured products traded on a trading facility or over-the-counter market; equity indexes or securities; fixed income indexes or securities; and foreign currency exchange rates; interest rates; commodity indexes.

4. The system of claim 1, wherein the capped value is calculated from a parimutuel auction.

5. The system of claim 1, wherein the product is a variance futures product.

6. The system of claim 1, wherein the product is a variance options product.

7. The system of claim 6, wherein variance options product is a limited risk option contract.

8. The system of claim 7, wherein the limited risk option contract comprises a contract payout that is a capped variable amount.

9. The system of claim 7, wherein the limited risk option contract comprises a payout that is a constant amount when the value of the underlying asset is in a specified strike range at expiration and zero when the value of the underlying asset is outside the specified strike range.

10. The system of claim 7, wherein the limited risk option contract comprises a combination of a long call option comprising a contract payout that is a capped variable amount with a short put option comprising a contract payout that is a capped variable amount, both the long call option and the short put option having the same strike prices and option premium prices.

11. The system of claim 1, wherein the limited risk variance derivative product is a chooser limited risk variance derivative product where a buyer of the chooser limited risk variance derivative product may choose after purchase of the chooser limited risk derivative product whether the chooser limited risk derivative is a call options or a put option.

12. A computer-implemented method of calculating and disseminating a value of an underlying asset associated with at least one limited risk variance derivative, the method comprising:
calculating, with a processor, from a plurality of pari-mutuel auctions, a capped value for a statistical property reflecting the variance of the underlying equity, the capped value comprising a value and a cap, the value reflecting an average volatility of price returns of the underlying equity over a predefined time period and the cap reflecting a maximum value of the value reflecting the average volatility of price returns of the underlying equity over the predefined time period;
calculating, with the processor, an average of a summation of each squared daily return of the underlying equity included in the value for the statistical property reflecting the variance of the underlying equity;
wherein the value of the statistical property reflecting the variance of the underlying equity is calculated according to the formula:




  
    
      Realized
      ⁢
      
        
      
      ⁢
      Variance
    
    =
    
      A
      ⁢
      
        
      
      ⁢
      F
      ×
      
        (
        
          
            ∑
            
              i
              =
              1
            
            
              
                N
                a
              
              -
              1
            
          
          ⁢
          
            
              R
              i
              2
            
            /
            
              (
              
                
                  N
                  e
                
                -
                1
              
              )
            
          
        
        )
      
    
  




  
    
      
        wherein
        :
        
          

        
        ⁢
        
          R
          i
        
      
      =
      
        ln
        ⁢
        
          
        
        ⁢
        
          
            P
            
              i
              +
              1
            
          
          
            P
            i
          
        
      
    
    ,
  



Pi is an initial value of the underlying equity used to calculate a daily return, Pi+1 is a final value of the underlying equity used to calculate the daily return, Ne is a number of expected underlying equity values needed to calculate daily returns during a variance calculation period, Na is an actual number of underlying equity values used to calculate daily returns during the variance calculation period; and AF is an annualization factor;
displaying, with the processor, at least one limited risk variance derivative based on the statistical property reflecting variance of the underlying equity on a trading facility display device coupled to a trading platform;
transmitting, with the processor, at least one limited risk variance derivative quote of a liquidity provider from the trading facility to at least one market participant; and
settling, with the processor, the at least one limited risk variance derivative based on the capped value and a difference between a cumulative realized variance and a strike price set at a fixed second cumulative realized variance, wherein the strike price is set at the fixed second cumulative realized variance when the at least one limited risk variance derivative is created.

13. The method according to claim 12, wherein calculating the value of the statistical property reflecting the variance of the underlying equity comprises:
removing the squared deviation of a daily return of the underlying equity that corresponds to a market disruption event.

14. The method according to claim 12, further comprising:
executing trades for the limited risk variance derivative by matching bids and offers to buy and sell positions in limited risk variance derivatives.

15. The method according to claim 12, wherein the underlying equity is selected from the group consisting of: commodity or structured products traded on a trading facility or over-the-counter market; equity indexes or securities; fixed income indexes or securities; foreign currency exchange rates; and interest rates; commodity indexes.

16. The method according to claim 12, wherein at least one of the at least one limited risk variance derivative is a limited risk option contract.

17. The method according to claim 12, wherein at least one of the at least one limited risk variance derivative is a limited risk futures contract.

18. The method according to claim 12, wherein the trading platform is an open outcry platform.

19. The method according to claim 12, wherein the trading platform is an electronic platform.

20. The method according to claim 12, wherein the trading platform is a hybrid of an open outcry platform and an electronic platform.

21. The method according to claim 12, further comprising:
transmitting the at least one limited risk variance derivative quote from the trading facility over at least one dissemination network.

22. The method according to claim 21, wherein the dissemination network is the Options Price Reporting Authority.

23. The method according to claim 12, wherein the trading facility is an exchange.

24. The method according to claim 12, wherein the liquidity provider is selected from the group consisting of: Designated Primary Market Makers (“DPM”), market makers, locals, specialists, trading privilege holders, members, and a registered trader.

25. The method according to claim 12, wherein the market participant is selected from the group consisting of: a liquidity provider, a brokerage firm, and an investor.

26. The method of claim 12, wherein the limited risk variance derivative is a chooser limited risk variance derivative where a buyer of the chooser limited risk variance derivative may choose after purchase of the chooser limited risk derivative whether the chooser limited risk derivative is a call options or a put option.