NASDAQ OMX OMS II. Margin methodology guide for Equity and Index derivatives. 8/29/2014 NASDAQ OMX Clearing (NOMX)

Transcription

1 NASDAQ OMX OMS II Margin methodology guide for Equity and Index derivatives 8/29/2014 NASDAQ OMX Clearing (NOMX)

2 DOCUMENT INFORMATION Date Version Comments Initial Margin types, Margin calculations GENERAL READING GUIDELINES The document is divided in two parts; a theoretical part that describes the basic principles and a practical part that contains margin calculation examples. In order to facilitate the reading, many of the mathematical explanations are found in the practical part of this document and in the appendix. The aim with the calculation examples in the guide is to illustrate the basic concepts of the margin calculations. 2 NASDAQ OMX

3 TABLE OF CONTENTS Document information... 2 General reading guidelines... 2 Background... 5 Purpose of document... 5 Introduction... 6 Margin Requirement... 6 Flows between NOMX and the clearing participants OMS II Margin Methodology... 8 Executive summary... 8 Definitions... 8 Liquidation Period... 8 Valuation Interval... 8 Valuation Points... 8 Vector File/ Risk Matrix... 8 Volatility Shifts... 9 Margin Offset... 9 Margin types... 9 Parameters Valuation interval Pricing parameters Window Method (margin offset) Margin calculations Futures contracts Example 1 Index Futures Example 2 Index Futures at expiration Forward contracts Example 3 Single Stock Forwards Example 4 Single Stock Forward at expiration Options Example 5 Equity call option Example 6 Equity put option Example 7 Equity Option at expiry (Delivery Margin) NASDAQ OMX

4 Example 8 Index Option at expiry (Payment Margin) Example 9 Portfolio Appendix I Option valuation formulas Binomial valuation model Black-Scholes Black Binary Options valuation Window method Window size Number of nodes in window NASDAQ OMX

5 BACKGROUND PURPOSE OF DOCUMENT The purpose of this document is to describe how margin calculations are performed and how the NASDAQ OMX s OMS II margin methodology is applied for standardized equity and index derivatives. The first part of the document describes the basic margin principles and the second part presents examples on margin calculations. The margin examples will be performed on both single positions and on hedged positions. 5 NASDAQ OMX

6 INTRODUCTION MAR GIN R EQ UIREMENT The margin requirement is a fundamental part of CCP clearing. In case of a clearing participant s default, it is that participant s margin requirement together with the financial resources of the CCP that ensures that all contracts registered for clearing will be honored. NOMX requires margins from all clearing participants and the margin requirement is calculated with the same risk parameters regardless of the clearing participant s credit rating. The margin requirement shall cover the market risk of the positions in the clearing participant s account. NOMX applies a 99.2% confidence level and assumes a liquidation period of two to five days (depending on the instrument) when determining the risk parameter. FLOW S BET W EEN NOMX AN D T HE CLEARING PARTI CI PAN TS There are two main flows between NOMX and the clearing participants. 1. Margin Collateral; NOMX calculates the margin requirement at the end of each trading day (T). The margin requirement becomes available to the clearing participants approximately at CET 20:00 on day T. The clearing participants have to cover their margin requirement with collateral. The clearing participants must have sufficient collateral in place before CET 10:30 on day T Call back; Participants with excess collateral can request a call back. A call back of collateral can be made between 07:00 and 11:00, e.g. day T. An evaluation of current collateral level is done by NOMX before any call back is approved. Once approved, settlement instructions will be created by NOMX and sent to the CSD/ICSD, and will be matched and settled with the corresponding participant or its settlement agent, T+1. 6 NASDAQ OMX

7 M A R G I N A N D SETTLEMENT FLOWS Figure 1: Margin and call back flows. Note: all time stamps are referring to the margin flow. 7 NASDAQ OMX

8 1. OMS II MARGIN METHODOLOGY EXECUTIVE SUMMARY The NOMX OMS II margin methodology is a scenario based risk model that aims to produce a cost of closeout, given a worst case scenario. For OMS II, the scenarios are defined by the model s input. The model inputs such as individual valuation intervals and volatility shifts are calculated with a minimum confidence level. OMS II is validated by back testing individual risk parameters and portfolio output. The confidence risk intervals are applied on each product using one year of historical price data. To enable a trustworthy clearing service, reasonably conservative margins are required to avoid the risk of the clearing organization incurring a loss in a default. In theory, the margin requirement should equal the market value at the time of the default. However, under normal conditions an account cannot be closed at the instant a participant defaults at the prevailing market prices. It typically takes time to neutralize the account and the value of the portfolio can change during this period, which must be catered for in the margining calculation. DEFINITIONS LIQUIDATIO N PERIO D It can take time to neutralize a position and, as a result, a lead time exists from the moment collateral has been provided until the clearing organization is able to close the participant's portfolio. The length of the lead time depends on the time it takes to discover that the participant has not provided enough collateral and the time required to neutralize the portfolio. VALUATION INT ER VAL OMS II varies the price for the underlying security for each series to calculate the neutralization cost. In this way, OMS II creates a valuation interval for each underlying security. The size of the valuation interval is given by the risk parameters for the instrument in question. VALUATION POINT S The upper and lower limits of the valuation interval represent the extreme movements allowed for the calculation. However, the worst-case scenario for a portfolio with different options and forwards/futures based on the same underlying instrument can occur anywhere in the valuation interval. In order to reflect this, the valuation interval is divided into 31 valuation points for equities. OMS II calculates the neutralization cost for each series with the same underlying security in each valuation point; the actual margin requirement is then based on the valuation point that rendered the highest margin requirement, i.e. the worst-case scenario. VECTOR FI LE/ RI SK M ATRI X By adding position data to a risk matrix, we obtain the neutralizing cost in each valuation point for a single position. The risk matrix is a vector file in which each cell is a valuation point. In one dimension the underlying price is altered and in the 8 NASDAQ OMX

9 other the volatility is altered. If this instrument is not affected by volatility, i.e. futures and forwards, the values for the different volatilities will be the same. Different vector files are created for bought and sold positions. Figure 2: Example of a vector file Point Low volatility Closing volatility High volatility VO LAT ILI TY SHI FTS The price of an option can be strongly affected by changes in volatility. The risk of fluctuations in volatility is taken into account by calculating the value of the account, based not only on the current volatility, but also on a higher and a lower volatility. The amount by which the volatility is increased or decreased is determined and configured by NOMX. The neutralizing cost is calculated at each of the valuation points for three different volatility levels; hence for options a valuation interval consists of 3 X 31 valuation points. MAR GIN OFFSET In case price dependencies are observed, NOMX can provide offsets in margins between different instruments within the same instrument group as well as offsets between different instruments from different instrument groups. Margin offset is only provided in case correlation between products historically has been high and proven stable. In the OMS II model the margin offsets are calculated by portfolio. Offsets are provided when the shift (stress) of the underlying prices of the products demonstrate a stable dependency. For options, the deviation in the shift (stress) of the implied volatilities is also assessed to decide the offset level. Currently OMS II only apply offset for contracts with the same underlying instrument. MAR GIN TY P ES M A R G I N REQUI R E M E N T The Margin Requirement is the collateral that an Account Holder has to deposit to cover the credit risk of his counterparty, taking into account any netting effects allowed in the margin model. It is the expected cost of closing out the Account Holder s positions in a worst case scenario. The Margin Requirement is calculated using vector files. Margin Requirement = Initial Margin + Market Value + Payment Margin + Delivery Margin 9 NASDAQ OMX

10 N AKED M A R G I N The Margin Requirement when the position is held in isolation. Hence, no netting effects are taken into account. I N I T I A L M A R G I N The Initial Margin of a position reflects the market risk of the position during a close-out period in a worst case scenario. R E Q U I R E D INI T I A L M A R G I N The Required Initial Margin of an instrument series is the Initial Margin, taking into account netting effects allowed in the margin model. N A K E D INI T I A L M A R G I N The Initial Margin when the position is held in isolation. Hence, no netting effects are taken into account. P A Y M E N T M A R G I N Payment Margin should cover the risk of a participant failing to fulfill the settlement payment of cash settled instruments. Payment Margin is applied at expiration to index futures and index options and equals the amount to be cash settled. D E L I V E R Y M A R G I N Delivery Margin should cover the risk of a participant failing to deliver the contracted instruments. Delivery Margin is applied to instruments with physical delivery and consists of the position s profit and loss plus the market risk of the position between expiration and the final settlement. PARAMETERS This section covers the configurable parameters utilized in the margin calculations for forwards, futures and options. OMS II uses theoretical formulas for pricing options in each valuation point. The valuation formula differs depending on the option type. The specific valuation formula for each option type is specified in the appendix. VALUATION INT ER VAL Parameter Risk parameter Description Determines the size of the valuation interval. The parameter is a percentage of the underlying price. PRI CING P ARAMET ERS Parameter Risk free interest rate (%) Dividend yield (%) Description Risk-free interest rate used when evaluating options. The simple interest rate is translated to a continuous rate. Dividend yield used when evaluating options. To 10 NASDAQ OMX

11 Adjustment for erosion of time value Adjustment of futures (%) Highest volatility for bought options Lowest volatility for sold options Volatility shift parameter Volatility spread Highest value bought in relation to sold options (%) Adjustment for negative time value Adjustment for negative time value Underlying price Strike price Volatility Time to expiration of the option Dividends properly evaluate American options on futures, the dividend yield is set equal to the risk free interest rate. The number by which the number of days to maturity will be reduced when evaluating held options. Adjustment factor (spread parameter) for futures. Applies only to bought options. Applies only to sold options. Fixed parameter that determines the size of the volatility interval. Defines the spread for options. The spread parameter is a fixed value. Min. spread between the values for bought and sold options; if spread is too small the value of the bought option is decreased. If the theoretical option value is lower than the intrinsic value, the price is adjusted to equal the latter. If the theoretical option value is lower than the intrinsic value, the price is adjusted to equal the latter. The price of the underlying (stock or index etc.) in this valuation point. The strike price of the option. See the following section. Calculated as number of actual days / risk parameter. Days Per Year For Interest Rate Calculations. Known or expected dividends of the underlying affects the value of the option. This can be modeled as a continuous dividend yield or with discrete dividends (time and amounts). V O L A T I L I T Y NOMX uses the implied volatility which is based on the price of the individual option. Two separate volatilities are calculated, one for bought and one for sold options. The volatility for bought options is based on the bid price of the option, and the volatility for sold options is based on the ask price of the option. When calculating implied volatility for exchange traded options, there are two opposing forces to consider: flexibility and stability. Using the individual implied volatility for each series theoretically allows the clearing organization to cover smile effects in volatility. However, the problem with obtaining accurate pricing for less liquid instruments makes this method unstable in such case. 11 NASDAQ OMX

12 To account for smile effects, NOMX applies a volatility surfaces model 1 for the most liquid instruments. For other liquid instruments, an arithmetic average is calculated for the three closest at-the-money series for each expiration and underlying instrument. The mean value is then used as market volatility. For the most illiquid instruments NOMX applies a fixed volatility. The two latter methods do not consider smile effects, but have proven to be very stable. F I N E T U N I N G In order to obtain more appropriate margin requirements the following fine tuning features are used by OMS II. These features are illustrated in the following sections. Adjustment for erosion of time In the case of bought options, the clearing house will have to sell the position in a default situation. This means that the clearing house will probably have to sell an option with a shorter time to delivery because of the lead time. This motivates that the time to expiration used when valuating bought positions is reduced by the number of lead days. Min/max volatility In order not to value bought options too high, the volatility used for bought options has a maximum value. The value is defined in the risk parameter Highest Volatility for bought Options. A similar approach is used for sold positions where the volatility is not allowed to go below a minimum value. Negative time value A bought option is adjusted for negative time value if the volatility is zero. Since this would indicate a negative time value, the following factor is calculated: If this factor is less than 1, the factor is later multiplied to all the bought option values in the vector file. In this way, the theoretical values are scaled to give a better representation of the market. This adjustment is only made for bought positions. The most common case when this occurs is when projected dividends are not used in the margin calculations. Minimum value sold options For a sold option with a theoretical value of less than 0.01 NOMX apply a minimum value of Minimum spread bought/sold options The spread between bought and sold options is not allowed to become too narrow. This is prevented by comparing vector file values for bought and sold options in the same valuation point and adjusting the value of the bought option if needed. The current spread is set to 95% for a bought option in relation to a sold option. Rounding 1 Volatility Surfaces 12 NASDAQ OMX

13 As a last step, the vector file values are multiplied by the contract size and then rounded to two decimal places. In the subsequent equations to two decimals. WINDO W METHO D ( MARGIN O FFSET) means rounding For different instruments that show a high correlation to each other, there is a need for a method that takes this into consideration with respect to margining calculations. The method used in the OMS II methodology is called the window method. In this method, the scanning range limits the individual movement for each series, but there is a maximum allowed difference between the scanning points of the two series. This range can be represented as a window, hence the name. The size of this window is estimated roughly by the same method that is used to estimate scanning ranges. Daily differences between the movements of the series are calculated using one year of data. These values are then used to build a numerical cumulative distribution from which 99.2 % confidence interval is applied. Based on a given covariance, the window can display a spread demonstrating the maximum allowable difference in price variation between two different underlying securities. In a narrow window, prices cannot vary as much as in a broad one. As a result, high covariance causes a narrow window, and vice versa. Currently the window is set to 100% for all contracts with different underlying instruments, i.e. no correlation and 0% for contracts with the same underlying instrument, i.e. full correlation. 2. MARGIN CALCULATIONS In the subsequent sections margin calculations for futures, forwards and options are illustrated. D E F I N I T I O N S Variable NM PM DM P F t CP Q T VOL AP VOL BP VOL AC VOL BC Instrument data CS S Definition Naked margin Payment margin Delivery margin Spot price of underlying stock/index value Fixing price (Margin settlement price) of future/forward on day t Contract price Number of contracts Time to expiration day expressed in years (days/365) Volatility used for a sold put option (based on ask prices) Volatility used for a bought put option (based on bid prices) Volatility used for a sold call option (based on ask prices) Volatility used for a bought call option (based on bid prices) Contract size. The number of instruments that defines one contract for an instrument Strike price of the option 13 NASDAQ OMX

14 DIV I Risk parameters Par V u/d AD r q ER Discrete dividend number that is included in the valuation of the option. Offset days for dividends If the value equals 0, this is dividends with exdate in the time range (current date + 1 : expiration date). If the value equals 1, this is dividends with ex-date in the time range ( current date + 1 : expiration date + 1 ) Risk interval parameter (variable) Volatility shift parameter i.e. the maximum increase/ decrease in volatility Adjustment factor (spread) Risk free interest rate Dividend yield Adjustment for erosion of time value In the following equations means rounding to two decimals. Current up-to-date parameters are found in Appendix 13 of NOMX s rules and regulations. 14 NASDAQ OMX

15 FUTURES CONTRACTS N AKED M A R G I N Bought position 1 Sold position: 2 P A Y M E N T M A R G I N Bought position: 3 Sold position: 4 EXAMP LE 1 IN DEX F UT UR ES P O S I T I O N Consider a position of 50 bought OMXS303A contracts expiring in January P A R A M E T E R S A N D V A R I A B L E S F t P Par 8.5% AD 0.5% Q 50 CS 100 M A R G I N C A L C U L A T I O N Using equation 1: The worst case value of this position is found at the bottom of the vector file, i.e. that the price of the futures contract will decrease. 15 NASDAQ OMX

16 EXAMP LE 2 IN DEX F UT UR ES AT EX PIRATION P O S I T I O N 100 sold OMXS30 futures contracts expiring today. P A R A M E T E R S A N D V A R I A B L E S F t F t Q 100 CS 100 P A Y M E N T M A R G I N C A L C U L A T I O N Using equation 4: The Payment Margin equals the final cash settlement amount of the contract. FORWARD CONTRACTS N AKED M A R G I N Bought position: ) ) 5 Sold position: ) ) 6 D E L I V E R Y M A R G I N Bought position: ) ) 7 Sold position: ) ) 8 16 NASDAQ OMX

17 EXAMP LE 3 SINGLE S TOCK FO RW AR DS P O S I T I O N 100 sold HMB forward contracts with expiry in December P A R A M E T E R S A N D V A R I A B L E S F t P CP Par 7.5% AD 2% Q 100 CS 100 M A R G I N C A L C U L A T I O N Using equation 6: ) ) ) ) The worst case value of this position is found at the top of the vector file, i.e. the price of the forward increases. EXAMP LE 4 SINGLE S TOCK FO RW AR D AT EXP IRATI ON P O S I T I O N 100 bought HMB forwards expiring today. P A R A M E T E R S A N D V A R I A B L E S P CP Par 10% AD 2% Q 100 CS 100 D E L I V E R Y M A R G I N C A L C U L A T I O N Using equation 7: ) ) ) ) The worst case value of this position is found at the bottom of the vector file, i.e. the underlying price of the stock decreases. 17 NASDAQ OMX

18 OPTIONS The option price as a function of the underlying price is non-linear. Genium Risk assumes that two major factors affect option prices: Underlying price Implied volatility For a position consisting of one option, we know that the largest and smallest option value will be at the end points of the interval. E.g. a bought (sold) call will have the smallest value when underlying stock price and volatility are as low (high) as possible. N AKED M A R G I N F O R E Q U I T Y O P T I O N S Equity options are defined as premium paid options with American style expiry. Bought call option: [ ] 9 Bought put option: [ ] 10 Sold call option: [ ] 11 Sold put option: [ ] 12 D E L I V E R Y M A R G I N F O R E Q U I T Y O P T I O N S If the options are to be exercised, the Delivery Margin at exercise is calculated in the following manner: Bought call option or sold put option: ) 13 Sold call option or bought put option: NASDAQ OMX

19 N AKED M A R G I N F O R I N D E X O P T I O N S Standardized index options are European future style options where the margin requirement is calculated similarly to premium paid options. Bought call option: [ ] 15 Bought put option: [ ] 16 Sold call option: [ ] 17 Sold put option: [ ] 18 P A Y M E N T M A R G I N F O R I N D E X O P T I O N S For cash settled options, for example index options, there is no market risk between exercise and settlement (IV is the index value at exercise). If the options are to be exercised, the Payment Margin at expiration is calculated in the following manner: Sold call option: 19 Bought call option: 20 Sold put option: 21 Bought put option: NASDAQ OMX

20 EXAMP LE 5 EQUITY CALL OPTION Consider a position with 10 sold equity call options with 32 days to expiry. P A R A M E T E R S A N D V A R I A B L E S S P DIV none r 3% P 10 Par 7.5% AD 2% V U 10 CS 100 Q 10 T 32/365 VOL AC 18.85% For each scenario point, using equation 11: [ ] The vector file combines the price- and volatility scenarios: V E C T O R F I L E Scenario Vol Down Vol Mid Vol Up NASDAQ OMX

21 M A R G I N REQUI R E M E N T The worst case scenario for a sold call position is to stress the price and volatility up. Hence the margin requirement for this position is: EXAMP LE 6 EQUITY PUT OPTIO N Consider a sold equity put option position with 32 days to expiry. P A R A M E T E R S A N D V A R I A B L E S S P DIV None r 3% Par 7.5% AD 2% V U 10 CS 100 Q 1 T 32/365 VOL AP 17.79% M A R G I N C A L C U L A T I O N Using equation 12: [ ] M A R G I N REQUI R E M E N T By combining each scenario point for price and volatility we create the vector file. The worst-case value for a sold put option is to stress the price down and the volatility up. Thus we will find the margin requirement in the bottom right corner of the vector file. 21 NASDAQ OMX

22 EXAMP LE 7 EQUITY OPTIO N AT EX PIRY (DELIV ER Y MARGI N) Consider a position with 50 sold equity put options with expiration today. P A R A M E T E R S A N D VARI A B L E S S 36 P 18 DIV None r 3% Par 25% AD 2% V U 10 CM 100 T 32/365 Q 50 VOL AP 17.79% D E L I V E R Y M A R G I N C A L C U L A T I O N Using equation 13: ) EXAMP LE 8 IN DEX OPTION AT EXPI RY (PAYMENT M AR GIN) Consider a sold index call option at expiration with OMXS30 as the underlying index. P A R A M E T E R S A N D V A R I A B L E S S 922 IV 1000 Q 10 CS 100 P A Y M E N T M A R G I N C A L C U L A T I O N Using equation Error! Reference source not found.: EXAMP LE 9 P ORT FO LIO Consider a portfolio consisting of 50 bought OMXS30 futures contracts and 70 sold OMXS30 options expiring in January To calculate the margin requirement of this portfolio the window method is used. The window size in this example is 0%, which corresponds to 100% correlation. 22 NASDAQ OMX

23 V E C T O R F I L E S 50 bought OMXS303A 70 sold OMXS303A920 call options futures Point F t Point F t Vol = 14.42% Vol = 24.42% Vol = 34.42% As seen in the vector files the two individual worst case values are located on opposite sides in the valuation interval. These values are referred to as each position s naked margin requirement. If there was no correlation between the instruments the portfolio s total margin requirement would equal the sum of these values. A window is applied to determine the maximum allowable difference in price variation. In this case, the window size is 0% which corresponds to a window that is 1 row wide. The number of rows used in the window method is determined by the following set of calculations: 1. Let x = (1.0 window size/100)*(# points 1) x = (1 0/100) * (31 1) = Round x to nearest integer x = Let x = # points x x = = 1 4. If x is even, increment x by 1 x = = 1 23 NASDAQ OMX

24 S L I D I N G W I N D O W 50 bought OMXS303A futures 70 sold OMXS303A920 call options In the example above the sliding window is centered over row 7. The worst case value in this window is ( ) = The overall worst case value, and the total margin requirement of the portfolio, is found at the bottom of the vector file, and equals S U M M A T R I X Sum matrix As the window slides down the vector files, worst case values are calculated at each point. The naked margin for each contract is found within the window where the overall worst case value is found. The sum of each position s naked margin equals the total margin requirement. M A R G I N REQUI R E M E N T Series Bought Sold Naked margin Margin OMXS300A OMXS300A Total NASDAQ OMX

25 APPENDIX I OPTION VALUATION FORMULAS The following table shows the various NOMX uses for options with and without dividends. The risk free interest rate is denoted r and the dividend yield is denoted as q. Product Method without dividends Method with discrete dividends American call based on spot Black -Scholes Binomial with dividends American put based on spot Binomial if interest rate is non-zero, Black -Scholes if interest rate equals zero. Binomial with dividends European opt based on spot Black -Scholes Discount spot with dividends, then use Black -Scholes European opt based on future Black -76 Black -76 Binary Cash-or-Nothing based on spot Standard formula Discount spot with dividends, then use Black -Scholes Binary Cash-or-Nothing based on future Standard formula with q = r Standard formula with q = r Below sections illustrate the option valuation formulas used by NOMX. These are well known, widely used industry-standard formulas which can be found in the financial literature e.g. Hull BINO MIAL V ALUATION MO DEL The binomial pricing model traces the evolution of the option's underlying variables in time. This is done by means of a binomial tree, for a number of time steps between the valuation and expiration dates. NOMX utilize a tree structure with 30 steps. The following values for the parameters in the method are used: D E F I N I T I O N S Variable C P S 0 K r Q σ T Definition Call price Put price Stock price at time zero Strike price Continuously compounded risk-free rate Dividend yield Stock price volatility Time to maturity 25 NASDAQ OMX

26 u Up movement d Down movement p Probability up movement 1-p Probability down movement Up movement: ) ) Down movement: Probability: where ) and ) The discounting when walking backwards in the tree is done with r (not r-q). Black-Scholes The Black-Scholes formula calculates the price of European put and call options. D E F I N I T I O N S Variable C P S 0 K r σ T Definition Call price Put price Stock price at time zero Strike price Continuously compounded risk-free rate Futures price volatility Option s time to maturity ) ) and ) ) where 26 NASDAQ OMX

27 and ( ) ( ) ( ) ( ) The function N(x) is the cumulative distribution function for a standardized normal distribution. BLACK-76 For options on futures contracts the Black-76 formula is used. D E F I N I T I O N S Variable C P F 0 K r σ T Definition Call price Put price Futures price at time zero Strike price Continuously compounded risk-free rate Stock price volatility Option s time to maturity ) )) and ) )) where ( ) ( ) and ( ) ( ) 27 NASDAQ OMX

28 BINARY OPTIO NS V ALUATION For binary Cash-or-Nothing options the definitions follow the Black-Sholes notation. D E F I N I T I O N S Variable C P S 0 K Q r σ T Definition Call price Put price Stock price at time zero Strike price Fixed payout amount Continuously compounded risk-free rate Stock price volatility Option s time to maturity Cash-or-nothing call option: ) Cash-or-nothing put option: ) WINDOW METHOD Generally the default cross margining may be described as instruments with the same underlying being totally correlated, and instruments with different underlying instruments being non-correlated. However, a more advanced correlation method, called the window method, can be utilized. In this method, the different instruments are sorted into a number of groups called window classes. Each window class has a window size in percent. Price correlation goes up when the window size goes down. A window size of 0% means full price correlation and a window size of 100% means no price correlation. Additionally, each window class may, or may not, use volatility correlation. A window class may also be a member of another window class, thereby creating a tree-structure in order to achieve more complicated correlations. WINDO W SI ZE The window size (WS) roughly corresponds to the inverse of the correlation of two or more instruments. To determine the correlation between instruments, the window size is calculated by using the instruments normalized daily price changes in percentage. Based on one year of historical data the second largest difference is chosen. 1. Price changes in percentage per underlying instrument (i=1 to n) over a time period (t=1 to T). ) 2. Price changes are normalized by each underlying instrument s risk parameter (Par). 28 NASDAQ OMX

29 ) 3. A vector containing the largest price difference between instruments for each t is calculated. ) ) 4. The second largest value from the vector is obtained ) ) ) The obtained parameter, that is the maximum allowable price difference, is the risk interval parameter. Since the size of the risk interval is twice as big as the parameter (both up and down price stress is implied) the parameter needs to be divided by The parameter is then multiplied by liquidation period hence the Window Size equals: ) ) ) NUMBER O F NODES IN WI NDO W The algorithm for converting a window size in percent into points is shown below. Figure 3: Algorithm for converting window size to valuation point 1. Let x = (1.0 window size/100)*(# points 1) x = (1 0.5) * (31 1) =15 2. Round x to nearest integer x = Let x = # points x x = = If x is even, increment x by 1 x = = 17 The sum matrix values the combined position in each valuation point. The margin requirement will equal the sum of the worst case from each individual position within the sliding window. 2 E.g. During a trading day the price move of stock A is +15% while the price moves of stock B is -15%. Both stocks have a parameter of 30%. The normalized values for A and B are 0.5 respectively Hence ) ), i.e. the full length of the valuation interval. It is only possible for the price to move half that distance, up and down from the spot price. The value has to be divided in half to obtain the true interval. 29 NASDAQ OMX

30 Figure 4: Illustration of the window method, where X and Y are individual worst case values As seen in the figure above, the window displays a spread demonstrating the maximum allowable difference in price variation between two different products belonging to the same window class. 30 NASDAQ OMX