NASDAQ OMX OMS II · 1. Margin – Collateral; NOMX calculates the margin requirement at the end of each trading day (T). The margin requirement becomes available to the clearing

NASDAQ OMX OMS II

Margin methodology guide for Equity and Index derivatives

8/29/2014 NASDAQ OMX Clearing (NOMX)

NASDAQ OMX OMS II 2014

2 NASDAQ OMX

DOCUMENT INFORMATION

Date Version Comments

2013-07-31 1.0 Initial

2014-08-29 1.1 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.


3 NASDAQ OMX

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 ...................... 6

1. 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 .................................................................................................. 10

Valuation interval .......................................................................... 10

Pricing parameters ........................................................................ 10

Window Method (margin offset) .................................................. 13

2. Margin calculations ..................................................................................... 13

Futures contracts ......................................................................................... 15

Example 1 – Index Futures ............................................................ 15

Example 2 – Index Futures at expiration ....................................... 16

Forward contracts ....................................................................................... 16

Example 3 – Single Stock Forwards ............................................... 17

Example 4 – Single Stock Forward at expiration ........................... 17

Options ........................................................................................................ 18

Example 5 – Equity call option ...................................................... 20

Example 6 – Equity put option ...................................................... 21

Example 7 – Equity Option at expiry (Delivery Margin) ................ 22


4 NASDAQ OMX

Example 8 – Index Option at expiry (Payment Margin) ................ 22

Example 9 – Portfolio .................................................................... 22

Appendix I .................................................................................................................. 25

Option valuation formulas ........................................................................... 25

Binomial valuation model ............................................................. 25

Black-Scholes ................................................................................. 26

Black-76 ......................................................................................... 27

Binary Options valuation ............................................................... 28

Window method .......................................................................................... 28

Window size .................................................................................. 28

Number of nodes in window ......................................................... 29


5 NASDAQ OMX

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.


6 NASDAQ OMX

INTRODUCTION

MAR GIN REQ UIR EMENT 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 CLEARIN G P ARTI 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+1.

2. 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.


7 NASDAQ OMX

MA R G I N A N D S E T T L E M E N T F L O W S

Figure 1: Margin and call back flows.

Note: all time stamps are referring to the margin flow.


8 NASDAQ OMX

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

L IQUIDATIO 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 F I LE/ R I SK MATRI 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


9 NASDAQ OMX

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

1

2

… … … …

30

31

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 OFFS ET

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

MA R G I N RE Q U I 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


10 NASDAQ OMX

NA K E D MA R G I N

The Margin Requirement when the position is held in isolation. Hence, no netting

effects are taken into account.

IN I T I A L MA 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.

RE Q U I R E D IN I T I A L MA 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.

NA K E D IN I T I A L MA R G I N

The Initial Margin when the position is held in isolation. Hence, no netting effects

are taken into account.

PA Y M E N T MA 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.

DE L I V E R Y MA 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 Description

Risk parameter Determines the size of the valuation interval. The

parameter is a percentage of the underlying price.

PRI CING P ARAMET ERS

Parameter Description

Risk free interest rate (%) Risk-free interest rate used when evaluating options.

The simple interest rate is translated to a continuous

rate.

Dividend yield (%) Dividend yield used when evaluating options. To


11 NASDAQ OMX

properly evaluate American options on futures, the

dividend yield is set equal to the risk free interest

rate.

Adjustment for erosion of time

value

The number by which the number of days to

maturity will be reduced when evaluating held

options.

Adjustment of futures (%) Adjustment factor (spread parameter) for futures.

Highest volatility for bought

options

Applies only to bought options.

Lowest volatility for sold options Applies only to sold options.

Volatility shift parameter Fixed parameter that determines the size of the

volatility interval.

Volatility spread Defines the spread for options. The spread

parameter is a fixed value.

Highest value bought in relation to

sold options (%)

Min. spread between the values for bought and sold

options; if spread is too small the value of the

bought option is decreased. Adjustment for negative time value

If the theoretical option value is lower than the intrinsic value, the price is adjusted to equal the latter.

Adjustment for negative time value

If the theoretical option value is lower than the intrinsic value, the price is adjusted to equal the latter.

Underlying price The price of the underlying (stock or index etc.) in this valuation point.

Strike price The strike price of the option.

Volatility See the following section.

Time to expiration of the option Calculated as number of actual days / risk parameter. Days Per Year For Interest Rate Calculations.

Dividends 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.


12 NASDAQ OMX

To account for smile effects, NOMX applies a volatility surfaces model1 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 0.01.

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

http://www.nasdaqomx.com/europeanclearing/nordicclearingtoday/riskanddefaultmanagement/


13 NASDAQ OMX

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 means rounding

to two decimals.

W INDO W METHO D (MARGIN O FFSET)

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.

DE F I N I T I O N S

Variable Definition

NM Naked margin

PM Payment margin

DM Delivery margin

P Spot price of underlying stock/Index value

Ft Fixing price (Margin settlement price) of future/forward on day t

CP Contract price

Q Number of contracts

T Time to expiration day expressed in years (days/365)

VOLAP Volatility used for a sold put option (based on ask prices)

VOLBP Volatility used for a bought put option (based on bid prices)

VOLAC Volatility used for a sold call option (based on ask prices)

VOLBC Volatility used for a bought call option (based on bid prices)

Instrument data

CS Contract size. The number of instruments that defines one contract for an instrument

S Strike price of the option


14 NASDAQ OMX

DIVI

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 ex-date 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 parameters

Par Risk interval parameter (variable)

Vu/d Volatility shift parameter i.e. the maximum increase/ decrease in

volatility

AD Adjustment factor (spread)

r Risk free interest rate

q Dividend yield

ER 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.

http://www.nasdaqomx.com/digitalAssets/94/94807_app_13_140917.pdf


15 NASDAQ OMX

FUTURES CONTRACTS

NA K E D MA R G I N

Bought position

1

Sold position:

2

PA Y M E N T MA R G I N

Bought position:

3

Sold position:

4

EXAMP LE 1 – IN DEX FUT UR ES

PO S I T I O N

Consider a position of 50 bought OMXS303A contracts expiring in January 2013.

PA R A M E T E R S A N D V A R I A B L E S

Ft 926.48

P 927.47

Par 8.5%

AD 0.5%

Q 50

CS 100

MA 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.


16 NASDAQ OMX

EXAMP LE 2 – IN DEX FUT UR ES AT EX PIR ATION

PO S I T I O N

100 sold OMXS30 futures contracts expiring today.


Ft 950.00

Ft-1 952.50

Q 100

CS 100

PA Y M E N T MA 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

NA K E D MA R G I N

Bought position:

) ) 5

Sold position:

) ) 6

DE L I V E R Y MA R G I N

Bought position:

) ) 7

Sold position:

) )

8


17 NASDAQ OMX

EXAMP LE 3 – S IN GLE STOCK FO RW AR DS

PO S I T I O N

100 sold HMB forward contracts with expiry in December 2013.


Ft 245.89

P 243.80

CP 244.80

Par 7.5%

AD 2%

Q 100

CS 100


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 – S IN GLE STOCK FO RW AR D AT EXP IRATI ON

PO S I T I O N

100 bought HMB forwards expiring today.


P 410.00

CP 390.00

Par 10%

AD 2%

Q 100

CS 100

DE L I V E R Y MA 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.


18 NASDAQ OMX

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.

NA K E D MA 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

DE L I V E R Y MA 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:

14


19 NASDAQ OMX

NA K E D MA 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

PA Y M E N T MA 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:

22


20 NASDAQ OMX

EXAMP LE 5 – EQ UIT Y CALL OP TION

Consider a position with 10 sold equity call options with 32 days to expiry.


S 220.00

P 237.20

DIV none

r 3%

P 10

Par 7.5%

AD 2%

VU 10

CS 100

Q 10

T 32/365

VOLAC 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

1 -35 380 -35 390 -35 700

2 -34 190 -34 200 -34 570

3 -33 000 -33 030 -33 430

4 -31 820 -31 850 -32 300

5 -30 630 -30 670 -31 160

6 -29 450 -29 490 -30 030

7 -28 260 -28 320 -28 950

8 -27 070 -27 160 -27 870

9 -25 890 -26 000 -26 800

10 -24 700 -24 830 -25 720

11 -23 520 -23 680 -24 640

12 -22 330 -22 550 -23 560

13 -21 140 -21 410 -22 540

14 -19 960 -20 280 -21 550

15 -18 770 -19 160 -20 560

16 -17 590 -18 080 -19 570

17 -16 400 -17 000 -18 570

18 -15 220 -15 920 -17 580

19 -14 040 -14 860 -16 650

20 -12 870 -13 870 -15 780

21 -11 710 -12 880 -14 920

22 -10 560 -11 890 -14 050


21 NASDAQ OMX

23 -9 420 -10 930 -13 190

24 -8 320 -10 060 -12 320

25 -7 240 -9 200 -11 520

26 -6 240 -8 340 -10 810

27 -5 250 -7 520 -10 100

28 -4 380 -6 820 -9 390

29 -3 550 -6 110 -8 690

30 -2 840 -5 400 -7 980

31 -2 210 -4 780 -7 350


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 – EQ UIT Y P UT OPTIO N

Consider a sold equity put option position with 32 days to expiry.


S 230.00

P 237.20

DIV None

r 3%

Par 7.5%

AD 2%

VU 10

CS 100

Q 1

T 32/365

VOLAP 17.79%


Using equation 12:

[

]


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.


22 NASDAQ OMX

EXAMP LE 7 – EQ UIT Y OPTIO N AT EX PIR Y (DELIV ER Y MARGI N)

Consider a position with 50 sold equity put options with expiration today.


S 36

P 18

DIV None

r 3%

Par 25%

AD 2%

VU 10

CM 100

T 32/365

Q 50

VOLAP 17.79%

DE L I V E R Y MA 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 (PAY MENT MAR GIN)

Consider a sold index call option at expiration with OMXS30 as the underlying index.


S 922

IV 1000

Q 10

CS 100

PA Y M E N T MA 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 – PORT FO LIO

Consider a portfolio consisting of 50 bought OMXS30 futures contracts and 70 sold

OMXS30 options expiring in January 2013. 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.


23 NASDAQ OMX

V E C T O R F I L E S

50 bought OMXS303A

futures

70 sold OMXS303A920 call options

Point Ft Point Ft Vol = 14.42% Vol = 24.42% Vol = 34.42%

1 1005.31 371 000 1 1005.31 -621 110 -708 120 -819 490

2 1000.06 344 750 2 1000.06 -588 490 -679 560 -793 030

3 994.80 318 450 3 994.80 -556 430 -651 560 -766 990

4 989.55 292 200 4 989.55 -524 860 -624 050 -741 370

5 984.29 265 900 5 984.29 -493 990 -597 030 -716 100

6 979.04 239 600 6 979.04 -463 750 -570 500 -691 250

7 973.78 213 350 7 973.78 -434 210 -544 530 -666 820

… … … … … … … …

25 879.18 -259 650 25 879.18 -73 290 -185 150 -304 570

26 873.92 -285 950 26 873.92 -63 630 -171 570 -289 030

27 868.67 -312 200 27 868.67 -54 950 -158 690 -273 980

28 863.41 -338 500 28 863.41 -47 110 -146 510 -259 490

29 858.16 -364 800 29 858.16 -40 250 -134 960 -245 420

30 852.90 -391 050 30 852.90 -34 090 -124 040 -231 910

31 847.65 -417 350 31 847.65 -28 770 -113 750 -218 820

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) = 30

2. Round x to nearest integer x = 30

3. Let x = # points – x x = 31 – 30 = 1

4. If x is even, increment x by 1 x = 1 + 0 = 1


24 NASDAQ OMX

S L I D I N G W I N D O W

50 bought OMXS303A futures 70 sold OMXS303A920 call options

1 371 000 371 000 371 000 -621 110 -708 120 -819 490

2 344 750 344 750 344 750 -588 490 -679 560 -793 030

3 318 450 318 450 318 450 -556 430 -651 560 -766 990

4 292 200 292 200 292 200 -524 860 -624 050 -741 370

5 265 900 265 900 265 900 -493 990 -597 030 -716 100

6 239 600 239 600 239 600 -463 750 -570 500 -691 250

7 213 350 213 350 213 350 -434 210 -544 530 -666 820

… … … … … … …

25 -259 650 -259 650 -259 650 -73 290 -185 150 -304 570

26 -285 950 -285 950 -285 950 -63 630 -171 570 -289 030

27 -312 200 -312 200 -312 200 -54 950 -158 690 -273 980

28 -338 500 -338 500 -338 500 -47 110 -146 510 -259 490

29 -364 800 -364 800 -364 800 -40 250 -134 960 -245 420

30 -391 050 -391 050 -391 050 -34 090 -124 040 -231 910

31 -417 350 -417 350 -417 350 -28 770 -113 750 -218 820

In the example above the sliding window is centered over row 7. The worst case

value in this window is 213 350 + (-666 820) = -453 470. The overall worst case

value, and the total margin requirement of the portfolio, is found at the bottom of

the vector file, and equals -636 170.

S U M M A T R I X

Sum matrix

1 -250 110 -337 120 -448 490

2 -243 740 -334 810 -448 280

3 -237 980 -333 110 -448 540

4 -232 660 -331 850 -449 170

5 -228 090 -331 130 -450 200

6 -224 150 -330 900 -451 650

7 -220 860 -331 180 -453 470

… … … …

25 -332 940 -444 800 -564 220

26 -349 580 -457 520 -574 980

27 -367 150 -470 890 -586 180

28 -385 610 -485 010 -597 990

29 -405 050 -499 760 -610 220

30 -425 140 -515 090 -622 960

31 -446 120 -531 100 -636 170

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.


Series Bought Sold Naked margin Margin

OMXS300A 50 -417 350 -417 350

OMXS300A920 70 -819 490 -218 820

Total -636 170


25 NASDAQ OMX

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 2011.

B INO 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:


Variable Definition

C Call price

P Put price

S0 Stock price at time zero

K Strike price

r Continuously compounded risk-free rate

Q Dividend yield

σ Stock price volatility

T Time to maturity


26 NASDAQ OMX

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.


Variable Definition

C Call price

P Put price


K Strike price


σ Futures price volatility

T Option’s time to maturity

) )

and

) )

where


27 NASDAQ OMX

(

) (

)

√

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.


Variable Definition

C Call price

P Put price

F0 Futures price at time zero

K Strike price




) ))

and

) ))

where

(

) (

)

√

and

(

) (

)

√ √


28 NASDAQ OMX

B INARY OPTIO NS V ALUATION

For binary Cash-or-Nothing options the definitions follow the Black-Sholes notation.


Variable Definition

C Call price

P Put price


K Strike price

Q Fixed payout amount




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.

W INDO 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).


29 NASDAQ OMX

∑

)

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 22

5. 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 = 15

3. Let x = # points – x x = 31 – 15 = 16

4. If x is even, increment x by 1 x = 16 + 1 = 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 -0.5. 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.


30 NASDAQ OMX

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.

Documents

NASDAQ OMX OMS II · 1. Margin – Collateral; NOMX calculates the margin requirement at the end of each trading day (T). The margin requirement becomes available to the clearing