INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS

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1 INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in the cleaing pocess opeated by LCH.Cleanet. It is a commecial pesentation of one of the sevice to be povided by and not a binding commecial offe. Although all easonable cae has been taken in the pepaation of this document LCH.Cleanet disclaims liability of the infomation and fo the uses to which it is put. The document might be upgaded along the poject implementation peiod. Copyight All the intellectual popety ights of the technical pesentation and the diagams included in this document ae vested in. This wok is issued in confidence fo the pupose fo which it is supplied. It must not be epoduced in whole o in pat o used fo othe puposes except with the consent in witing of and then only on the condition that this notice is included in any such epoduction. The infomation that is pat of the document is solely fo infomation pupose and is not to be constued as technical specification.

2 CONTENTS CONTENTS INTRODUCTION... 3 SUMMARY OF CHANGES... 4 CHAPTER I CALCULATION OF RISK ARRAY VALUES... 5 PRINCIPLE USED: MARKET PREMIUM AND THEORETICAL PREMIUM VALUATION... 6 ) Option value:... 6 ) Risk aay theoetical option value ) Risk aay value... 6 CHAPTER II VALUATION FORMULA FOR OPTIONS ON FUTURES AND INDICES... 7 ALGORITHM USED: THEORETICAL PREMIUM AND DELTA Notation Theoetical pemium Delta... 9 CHAPTER III VALUATION FORMULA FOR EQUITY OPTIONS... 0 ALGORITHM USED: THEORETICAL PREMIUM AND DELTA.... Notation... Theoetical pemium Delta... 3 CHAPTER IV VALUATION FORMULA FOR CURRENCY OPTIONS... 4 ALGORITHM USED: THEORETICAL PREMIUM AND DELTA Notation Theoetical pemium Delta... 6

3 INTRODUCTION INTRODUCTION The pupose of the SPAN appendix is to povide infomation about how to pocess isk aay calculation fo end of day and inta-day calculation. In addition, this document descibes the diffeences between the new elease (fom the 30 th mach 0) coesponding to UCS V9 & SPAN RMC and the pevious one (befoe the 30 th Mach 0) coesponding to UCSV5 & SPAN. Fistly, you will find a summay of the changes, then, in the fomulas, the detailed of the changes highlighted in yellow. Please note that the main pinciples of the picing models emain unchanged and that the diffeences ae elying on how inputs data ae used within these fomulas.

4 SUMMARY OF CHANGES SUMMARY OF CHANGES Lot size Fo calculation pupose, the lot size used in the calculation was the one of the contact in the pevious elease wheeas in the new elease, it is the lot size of the seies. Inteest ate The inteest ate (including foeign inteest ate when exist) is no longe conveted to natual logaithm (Napieian logaithm) of the calculation fo continuous models as Black 76 model and Gaman Kohlhagen model. Fo discete models such as Cox Ross Rubinstein (Euopean and Ameican style), thee is no impact. Time to matuity Fo each scenaio and to calculate the delta, one day in faction of yea is deducted fom the time to matuity of the option. So, in compaison to the pevious vesion, it is euied to decement life time by (please efe to the fomulas below). Numbe of iteation fo Cox Ross modeling Fo discete models such as Cox Ross Rubinstein (Euopean and Ameican style), thee is a numbe of iteation which is fixed to 30 whateve the model. This is a significant modification compaed to the pevious situation whee accoding to the type of picing model (Ameican: n=7; Euopean: n=30), an aveage was done between n and (n+) iteations. In this new vesion thee is no moe aveage. The 30 iteations have just to be taken into account alone. UCS V9 & SPAN RMC UCSV5 & SPAN - Contact value facto (standing fo lot size) The Contact Value Facto is collected at seies level a. Thee is no Contact Value Facto at contact level in SPAN RMC. One contact value facto is selected fom SPAN file ecod 83 and set as seies contact value facto. b. So fo the whole seies, the Contact Value Facto will be uniue. The time to matuity of the option seies is decemented by one calenda day (/365 = ) fo the scenaio picing. So, T = Time to matuity Time to matuity 3 - Inteest ate The Contact Value Facto is collected at contact level a. The Contact Value Facto is detemined at contact level even if it exists at seies level. b. Fo calculation pupose, it is the value at contact level which is used. The time to matuity of each option is taken into account fo the scenaio picing. So, T = Time to matuity Fo option on futues contacts, options on index Fo calculation : option on futues contacts, and cuency options: options on index and cuency options: The value is loaded diectly fom UCS (5 fists - Rd : Yealy continuous financing domestic ate digits) an used without any convesion on the time to matuity of the option ( d = Ln(+ - Rd : Yealy continuous financing domestic ate Rd) whee Rd is the yealy financing domestic on the time to matuity of the option ate) - Rf : Yealy continuous financing foeign ate on - Rf : Yealy continuous financing foeign ate on the time to matuity of the option the time to matuity of the option ( f = Ln(+ Rf) Fo option on stocks: no gap. whee Rf is the yealy financing foeign ate) Fo option on stocks: no gap. 4 - Numbe of iteation The numbe of iteation is a Cox Ross Rubinstein input paamete The numbe of iteation is no longe depending on the contact. A paamete is set as default whateve the contact: Euopean and Ameican type Cox Ross Rubinstein iteation = 30 The numbe of iteation depends on the contact: a. Euopean type Cox Ross Rubinstein iteation = 30 b. Ameican type Cox Ross Rubinstein iteation = 7

5 CALCULATION OF RISK ARRAY VALUES CHAPTER I CALCULATION OF RISK ARRAY VALUES

6 CALCULATION OF RISK ARRAY VALUES A isk aay can be defined as a set of numbes: Fo a paticula contact At a paticula point in time To be magined fo a paticula business function Fo a paticula account. Each isk aay value specifies how a single long o shot position will lose o gain value if the coesponding isk scenaio occus ove the specified look-ahead time. The look-ahead time is the aveage time pe day. So it eflects the unit of time into the futue fom the cuent time. By convention, losses fo long positions ae expessed as positive numbes, and gains as negative numbes. PRINCIPLE USED: MARKET PREMIUM AND THEORETICAL PREMIUM VALUATION ) Option value:.) Fo end-of-day: Option is valued using as inputs: a. Maket pice (o pemium) value of the option poduct (conveted to option cuency if needed and scaled to the option size): (Maket pice * option CVF)..) Fo inta-day: Theoetical option value is calculated using as inputs: a. Value of the undelying poduct (conveted to option cuency if need be and scaled to the option size): (undelying pice * undelying CVF) * Cuency convesion ate * option scaling facto. (Option scaling facto is eual to option CVF/undelying CVF) b. Option implied volatility c. Option stike value: (stike * option CVF) d. Time to matuity e. Inteest ate f. Dividends ) Risk aay theoetical option value Fo each isk aay point new theoetical value is calculated using as inputs: a. Value of the undelying poduct (conveted to option cuency if need be and scaled appopiately if CVF fo undelying and CVF fo options ae diffeent): (undelying pice * undelying CVF + pice scan ange * faction )* cuency convesion ate * option scaling facto. b. Option implied volatility + volatility scan ange * faction c. Option stike value: (stike * option CVF) d. Time to matuity move ahead time e. Inteest ate f. Dividends 3) Risk aay value 3.) Fo end-of-day: Fo each isk aay point diffeence between valued maket pice and theoetical option value is taken as isk aay value. 3.) Fo inta-day: Fo each isk aay point diffeence between theoetical value calculated fo this point and theoetical option value is taken as isk aay value. Fom this desciption it is clea that: Risk aay values will always come out appopiately scaled (in the option value tems). Pice scan ange as input in this algoithm has to be specified in undelying value tems.

7 VALUATION FORMULA FOR OPTIONS ON FUTURES AND INDICES CHAPTER II VALUATION FORMULA FOR OPTIONS ON FUTURES AND INDICES

8 VALUATION FORMULA FOR OPTIONS ON FUTURES AND INDICES Fo detemining the initial magin euied to cove the positions on options on futues contacts (commodity fo instance) and options on index, uses a theoetical pemium valuation fomula of the type Black 76. ALGORITHM USED: THEORETICAL PREMIUM AND DELTA. Notation The algoithm euies the following data: U E T 0 T R N(x) Ln e : Pice of the undelying futues contact of the option : Stike pice of the option : Time to matuity of the option (in faction of yea (base: 365 d o 366 d)). : A time to matuity minus the look-ahead time of the option. The look-ahead time is aveage time pe day (base: (/365) d o ). So it eflects the unit of time (in faction of yea) into the futue fom the cuent time ( T T0 moveahead o T ) : Yealy volatility of the undelying (expessed in %). : Yealy continuous financing ate on the life span of the option : Distibution function of the nomal law : Napieian logaithm : exponential function T 0. Theoetical pemium The call pemium and the put pemium on the commodities futues contact and index ae expessed as follows: C e R* T * U * N( d) E * N( d) P e R* T * U * ( N( d) ) E * ( N( d) ) Whee d U * Ln * v T v * T E * d d v * T In ode to simplify the fomula, the nomal law is appoximated with a fifth degee polynomial such as: P( d) * e whee: d * ( bx cx fx 3 gx 4 ix 5 ) x a * d and a = b = c = f = g = i =

9 VALUATION FORMULA FOR OPTIONS ON FUTURES AND INDICES Then: if d > 0 N(d) = P(d) if d 0 N(d) = P(d) Note: Option theoetical pemium Fo inta-day calculation, theoetical pemiums ae pefomed befoe isk aay calculation. Theefoe the fomula is the same except that T 0 is used instead of T. Contolling via the intinsic value of the option If the Black 76 fomula gives a theoetical pemium that is infeio to the intinsic value, the pemium selected fo detemining the initial magin shall coespond to: The intinsic value if the execise style of the option is Ameican style The theoetical pemium if the execise style of the option is Euopean style. Rounding off The calculated isk aays values elated to SPAN scenaios ae ounded off to two decimals. 3. Delta The call delta and the put delta fo the long ate and commodities futues contact and index ae expessed as follows: Call * N( d) Put * N( d) e RT e RT Rounding off The calculated delta is ounded off to fou decimals.

10 VALUATION FORMULA FOR EQUITY OPTIONS CHAPTER III VALUATION FORMULA FOR EQUITY OPTIONS

11 VALUATION FORMULA FOR EQUITY OPTIONS As pat of detemining the initial magin euied to cove the positions on euity stock options contacts, uses a theoetical pemium valuation fomula of the type Cox Ross Rubinstein. ALGORITHM USED: THEORETICAL PREMIUM AND DELTA. Notation S : Index o settlement pices of the undelying contact of the option E : Stike pice of the option T 0 : Time to matuity of the option (in faction of yea (base: 365 d o 366 d)). t : Time to matuity of the option minus the look-ahead time of the option. The look-ahead time is aveage time pe day (base: (/365) d o ). So it eflects the unit of time (in faction of yea) into the futue fom the cuent time ( T T0 moveahead o T ) R t n h s T 0s t s : Yealy volatility of the undelying (expessed in %). : Financing ate coesponding to the life span of the options. : Numbe of iteations used in the model. n =30 iteations. : Amount of the dividend distibuted to the iteation s (the model accepts 8 payments of dividends pe yea) : numbe of days left befoe the payment of the dividend h s. The numbe is expessed in faction of yeas (base: 365 d o 366 d) : numbe of days left afte applying the look-ahead time and befoe the payment of the dividend h s. The numbe is expessed in faction of yeas (base: 365 d o 366 d) ( t T moveahead s 0 s o t s T 0s ) Definition of intemediay data R t t n is the continuous inteest ate Definition of the neutal isk pobability If no dividend is paid duing the life span of the option, the neutal isk pobability is: T 0 u d d whee v e t n u, and d u If the occuence s fo the payment of the dividend h s coesponds to the iteation i, the neutal isk pobability becomes: h i i j u * S i, j, i n et j i u d whee i = s h s is the amount of the dividend distibuted at the iteation s. t The iteation s is the intege pat of n* s. t Note: if the iteation i = s, no dividend is paid (h i=s =0). In that case, the neutal isk pobability is eual to.

12 VALUATION FORMULA FOR EQUITY OPTIONS Dividend management fo option valuation: in case thee is not a full distibution of dividends detached ove the complete option lifetime, the last known dividend value eceived fo an option instument will be used and duplicated until the matuity date of the option as often as possible fom the last known detachment date accoding to a cetain feuency type. Thee types of feuency can be applied and ae expessed in numbe of days: yealy (365 days), half-yealy (8 days), uately (9 days).this feuency type is defined at the combined commodity level in the magin paametes and follows the maket ules (e.g. Pais secuities geneally detach a yealy dividend). C i,j (espectively P i,j ) is the pice of the call (esp. put) at the iteation i in the hypothesis j consideing the cost evolution of the undelying instument. Theoetical pemium The theoetical pemium fo each scenaio is pefomed with 30 iteations. As the n+ C n,j pices (espectively P n,j ) at the call month date (esp. put) coesponding to the iteation n and to each hypotheses j consideing the cost evolution of the undelying instument ae known, the theoetical pemium is calculated fom the following ecuence fomula afte n iteations, and diffeentiating the Euopean and the Ameican chaacte of the option.. Euopean options (30 iteations) At expiy, Call At expiy, Put n j Cn, j max 0, u * S fo j=,,n+ E n j Pn, j max 0, E u * S fo j=,,n+ C i, j i, j * C i, j i, j * C i, j P i, j i, j * P i, j i, j * P i, j fo i = n-, n-,,, 0 et j =,,, i, i+ Theoetical pemium at the valuation moment fo i = n-, n-,,, 0 et j =,,, i, i+ Theoetical pemium at the valuation moment C 0, 0, * C, 0, * C, P 0, 0, * P, 0, * P, Note: Option theoetical pemium Fo inta-day calculation, theoetical pemiums ae pefomed befoe isk aay calculation. Theefoe the fomula is the same except that T 0 is used instead of t. Rounding off The calculated isk aays values elated to SPAN scenaios ae ounded off to two decimals.. Ameican options (30 iteations) At expiy, Call At expiy, Put

13 VALUATION FORMULA FOR EQUITY OPTIONS n j Cn, j max 0, u * S fo j=,,n+ E n j Pn, j max 0, E u * S fo j=,,n+ i, j * Ci, j i, j * Ci, j n j Ci, j max, u * S E fo i = n-, n-,,, 0 et j =,,, i, i+ Theoetical pemium at the valuation moment i, j * Ci, j i, j * Pi, j n j Pi, j max, E u * S fo i = n-, n-,,, 0 et j =,,, i, i+ Theoetical pemium at the valuation moment 0, * C, 0, * C, C 0, max, S E 0, * C, 0, * P, P 0, max, E S Note: Option theoetical pemium Fo inta-day calculation, theoetical pemiums ae pefomed befoe isk aay calculation. Theefoe the fomula is the same except that T 0 is used instead of t. Rounding off The calculated isk aays values elated to SPAN scenaios ae ounded off to two decimals. 3. Delta The call delta and the put delta ae calculated as follows: Call Put Call C S x C S x P S x P S Put * x * x x Contolling the pice size The following check value is pefomed on the undelying pice in ode to adapt the accuacy of the delta calculation: By default, the value of the paamete x is set to 0. without valuation. To be consistent, this paamete must be valued with the option CVF (as it is done fo pemiums and maket pice). y* S x, then x y * S Else x is kept unchanged. If By default, the value of y is set to 0. without valuation. To be consistent, this paamete must be valued with the option CVF (as it is done fo pemiums and maket pice). Rounding off The calculated delta is ounded off to fou decimals.

14 VALUATION FORMULA FOR CURRENCY OPTIONS CHAPTER IV VALUATION FORMULA FOR CURRENCY OPTIONS

15 VALUATION FORMULA FOR CURRENCY OPTIONS Fo detemining the initial magin euied to cove the positions on options on cuency futues contacts, uses a theoetical pemium valuation fomula of the type Gaman kohlhagen. ALGORITHM USED: THEORETICAL PREMIUM AND DELTA. Notation The algoithm euies the following data: U E T 0 T R d R f N(x) Ln e : Pice of the undelying futues contact of the option : Stike pice of the option : Time to matuity of the option (in faction of yea (base: 365 d o 366 d)). : A time to matuity minus the look-ahead time of the option. The look-ahead time is aveage time pe day (base: (/365) d o ). So it eflects the unit of time (in faction of yea) into the futue fom the cuent time ( T T0 moveahead o T ) : Yealy volatility of the undelying (expessed in %). : Yealy continuous financing domestic ate on the life span of the option : Yealy continuous financing foeign ate on the life span of the option : Distibution function of the nomal law : Napieian logaithm : exponential function T 0. Theoetical pemium The call pemium and the put pemium on the cuency futues contact ae expessed as follows: C R * T * T f d U * e * N( d) E * e * N( d) R P R * T * T f d U * e * N( d) E * e * N( d) R Whee R R * T d f U * e d * Ln * v * T d d v * T v * T E In ode to simplify the fomula, the nomal law is appoximated with a fifth degee polynomial such as: P( d) * e d * ( bx cx fx 3 gx 4 ix 5 ) whee:

16 VALUATION FORMULA FOR CURRENCY OPTIONS x a * d and a = b = c = f = g = i = Then: if d > 0 N(d) = P(d) if d 0 N(d) = P(d) Note: Option theoetical pemium Fo inta-day calculation, theoetical pemiums ae pefomed befoe isk aay calculation. Theefoe the fomula is the same except that T 0 is used instead of T. Rounding off The calculated isk aays values elated to SPAN scenaios ae ounded off to two decimals. 3. Delta The call delta and the put delta fo the cuency futues contact ae expessed as follows: Call e R T * R T * f f N( d ) e N( d ) Put Rounding off The calculated delta is ounded off to fou decimals.

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