Guide to the Volatility Indices of Deutsche Börse

Transcription

1 Volatlty Indces of Deutsche Börse Verson.4

2 Volatlty Indces of Deutschen Börse Page General Informaton In order to ensure the hghest qualty of each of ts ndces, Deutsche Börse AG exercses the greatest care when complng and calculatng ndces on the bass of the rules set out n ths gudelne. However, Deutsche Börse AG cannot guarantee that the varous ndces, or the varous ratos that are requred for ndex complaton and computaton purposes, as set out n ths gudelne, are always calculated free of errors. Deutsche Börse AG accepts no lablty for any drect or ndrect losses arsng from any ncorrect calculaton of such ndces or ratos. Decsons concernng the way ts ndces are calculated are always taken by Deutsche Börse AG to the best of ts knowledge and belef. Deutsche Börse AG shall not be lable for any losses arsng from such decsons. The ndces of Deutsche Börse AG do not represent a recommendaton for nvestment of whatever nature. In partcular, the complaton and calculaton of the varous ndces shall not be construed as a recommendaton of Deutsche Börse AG to buy or sell ndvdual securtes, or the basket of securtes underlyng a gven ndex.

3 Volatlty Indces of Deutschen Börse Page 3 Key Features 5. Concept 5.. Bass 5.. Volatlty Sub-Indces 6..3 VDAX and VDAX-NEW 6. Selecton of Input Data 6.3 Publcaton 7.4 Hstorcal Data 8.4. VDAX 8.4. VDAX-NEW VDAX-NEW Fxed Identfer Sub-Indces 9.5 Lcensng 0 VDAX. Calculaton Method. Extractng Data 3.3 Flterng of Data 3.4 Preparng Data 4.5 Determnng Forward Prces 5.5. Prelmnary Forward Prce 5.5. Fnal Forward Prce 6.6 Calculatng Volatltes of Indvdual Optons 6.7 Determnng the Sub-Indces 7.8 Constructng the Volatlty Index 8 3 VDAX-NEW 0 3. Calculaton Method 0 3. Extractng Data 3.3 Flterng of Data 3.4 Preparng Data 3.5 Calculaton Example Determnng the Forward Prce F and the Exercse Prces K, Determnng the Opton Prce M(K,j ) Determnng the Sub-Indces Constructng the Volatlty Index Calculaton of Settlement Index 5 4 Appendx 6

4 Volatlty Indces of Deutschen Börse Page 4 4. VDAX Master Data 6 4. VDAX-NEW Master Data 7 5 Your Drect Lne to Deutsche Börse 8

5 Volatlty Indces of Deutschen Börse Page 5 Key Features. Concept Volatlty s a measure of the level of uncertanty prevalng n certan markets, or wth respect to ndvdual underlyng nstruments. In prncple, there are two dfferent approaches for the estmaton of volatlty: on the one hand, t s possble to determne hstorcal volatlty by measurng the standard devaton of prces for any partcular securty over a gven perod of tme. On the other hand, volatlty can be derved mplctly from opton prces ( mpled volatlty ); ths knd of volatlty represents the estmates and assumptons of market partcpants nvolved n a trade, on the bass of a gven opton prce. Deutsche Börse calculates volatlty ndces that measure mpled volatlty usng two dfferent models. ) The VDAX computes mpled volatlty,.e. the expected prce fluctuaton n the DAX ndex, as mpled by the prces of DAX optons. A sub-ndex s publshed for each at-the-money DAX opton wth a tme to expraton of up to two years, whch s avalable for tradng at Eurex. In addton, the VDAX Index s determned for a fxed remanng tme to expraton of 45 days (t s therefore not lnked to a specfc tme to expraton). Both the VDAX as well as ts sub-ndces are calculated once at the end of each tradng day at 5:45 p.m.. ) The VDAX-NEW s calculated on the bass of a more recent model. The model has been jontly developed by Goldman Sachs and Deutsche Börse. It has replaced mpled volatltes of at-themoney DAX optons wth the square root of mpled varance across at- & out-of-the-money optons of a gven tme to expraton. Ths model offers great advantages n terms of creatng, tradng and hedgng dervatve products on ths ndex. The man ndex (whch s not lnked to a specfc maturty) has a fxed remanng tme to expraton of 30 days. The VDAX-NEW and ts varous sub-ndces are also updated every mnute... Bass The DAX comprses the 30 largest and most lqud companes havng ther operatng or regstered headquarters n Germany or ther prmary tradng venue at the Frankfurt Stock Exchange (FWB ). Tradng n shares of these companes accounts for more than 80 percent of Germany s exchangetraded equty volumes. Based on ts real-tme concept DAX provdes a comprehensve and up-to-date pcture of the German stock market. The optons contract on ths ndex s one of the most traded products of Eurex, the nternatonal dervatves exchange, and ranks among the top ndex optons contracts worldwde. The VDAX and DAX, Eurex, VDAX, VDAX-NEW, REX and Xetra are regstered trademarks of Deutsche Börse AG. Operatng headquarters s defned as the locaton of management or company admnstraton, n part or n full. Ths must be publcly dentfed by the company. The prmary tradng venue requrement s met f at least 33 percent of aggregate turnover for each of the last three months took place on the Frankfurt Stock Exchange, ncludng Xetra.

6 Volatlty Indces of Deutschen Börse Page 6 VDAX-NEW are calculated each on the bass of eght maturtes wth a maxmum tme to expraton of two years. Volatlty represents the key rsk factor for the prce determnaton n optons tradng. The hgher the estmaton of volatlty, the hgher the prce of an opton... Volatlty Sub-Indces Apart from the man ndces VDAX and VDAX-NEW (whch are rrespectve of a specfc tme to expraton), sub-ndces for each maturty of the DAX optons rangng from one month up to two years are calculated and dstrbuted for both models. For optons wth a longer lfetme, no such subndces are currently avalable. The varous VDAX-NEW sub-ndces are calculated on the bass of a broad strp of optons, whereas those of the VDAX are restrcted to only four at-the-money seres. The calculatons are based on the best bd and best ask prces avalable for these optons n the Eurex system...3 VDAX and VDAX-NEW Each man ndex s determned by way of nterpolaton usng the two nearest sub-ndces to the remanng tme to expraton of 45 days (VDAX ) or 30 days (VDAX-NEW ). Both man ndces are therefore calculated on a constant tme to expraton. Ths helps to elmnate effects that typcally result n strong volatlty fluctuatons close to expraton.. Selecton of Input Data Durng the calculaton hours for the VDAX-NEW and the sub-ndces (9:5 a.m. to 5:30 p.m. CET), the followng data s recorded every mnute, usng snapshots: DAX - DAX Index, calculated on the bass of Xetra prces FDAX - Best bd and best ask of the futures contracts on the DAX (only relevant to the VDAX ) ODAX EONIA - Best bd and best ask of all DAX optons - Euro Overnght Index Average overnght nterest rate EURIBOR - Euro Interbank Offered Rates money market reference rates for,, months (calculated once a day, :00 a.m. CET, by the European Bankng Federaton) REX - Yeld of the -year REX (calculated from exchange-traded prces) as the longer-term nterest rate

7 Volatlty Indces of Deutschen Börse Page 7 Index name Perod Code ISIN EONIA day EUD EU EURIBOR month month EUM EU EURIBOR months months EUM EU EURIBOR 3 months 3 months EU3M EU EURIBOR 4 months 4 months EU4M EU EURIBOR 5 months 5 months EU5M EU EURIBOR 6 months 6 months EU6M EU EURIBOR 7 months 7 months EU7M EU EURIBOR 8 months 8 months EU8M EU EURIBOR 9 months 9 months EU9M EU EURIBOR 0 months 0 months EU0 EU EURIBOR months months EU EU EURIBOR months months EU EU REX -YEAR (PRICE INDEX) years REX DE Publcaton VDAX-NEW and the varous volatlty sub-ndces are calculated on every Eurex exchange tradng day, durng the perod from 9:5 a.m. to 5:30 p.m. CET 3. VDAX and the correspondng sub-ndces are calculated once at the end of each tradng day at 5:45 p.m. CET. The calculaton of a sub-ndex does, however, only commence as soon as all requred nput data s avalable. The precse scope of data that s actually requred depends on the respectve calculaton model, and s therefore descrbed n the relevant chapters for calculaton (VDAX, cf. chapter ; VDAX-NEW, cf. chapter 3). The dssemnaton of each man ndex begns as soon as two sub-ndces are avalable for an nterpolaton. However, the VDAX-NEW can start wth data from the prevous tradng day (settlement prces) as long as no data from the current day s at hand. Therefore, t may happen that the VDAX and ts subndces do not start smultaneously wth the VDAX-NEW. In lne wth the expraton structure of DAX optons, each of the VDAX and VDAX-NEW sub-ndces s assgned to a specfc expraton, whch can be drectly dentfed from the respectve code. There s a system of 0 codes and ISINs, of whch are only eght n smultaneous use at any tme (cf. chapter 4). 3 VDAX-NEW and the correspondng sub-ndces have been calculated snce 8:50 a.m. untl 0 October 006.

8 Volatlty Indces of Deutschen Börse Page 8.4 Hstorcal Data The followng tme seres are avalable for the ndces of Deutsche Börse:.4. VDAX Index Code ISIN Daly closng prces snce VDAX VDAX DE Jan. 99 VDAX sub-ndex ( mth) VXmj cf Jan. 99 VDAX sub-ndex ( mth) VXmj cf Jan. 99 VDAX sub-ndex 3 (3 mth) VXmj cf Jan. 99 VDAX sub-ndex 4 (6 mth) VXmj cf Jan. 99 VDAX sub-ndex 5 (9 mth) VXmj cf Jan. 99 VDAX sub-ndex 6 ( mth) VXmj cf Mar. 996 VDAX sub-ndex 7 (8 mth) VXmj cf Mar. 996 VDAX sub-ndex 8 (4 mth) VXmj cf Mar. 996 m represents the respectve expry month ( A=Jan,..., L=Dec); j represents the respectve year (0, ; 9) Deutsche Börse has been publshng the DAX volatlty ndex VDAX once a day, snce 5 December 994. Snce January 99, a hstory of daly values has been exstng for both the VDAX and ts varous sub-ndces. Long-term DAX optons (wth tme to expratons of, 8 and 4 months) have only been avalable snce 8 March 996. Therefore, pror to that date, only fve sub-ndces and the VDAX tself were calculated. Wth the start of calculatng all ndces on an ntraday bass 4 on December 997, the calculaton model was also modfed. For ths reason, the hstorcal tme seres had been recalculated up untl that date, usng the varous FDAX and ODAX settlement prces determned by Eurex. For the DAX ndex, the respectve closng prces from electronc tradng (IBIS or Xetra ) were ncluded. The REUTERS overnght rate, the -month to -month LIBOR (London Interbank Offered Rates) rates, and the yeld of the -year REX were used as nterest rates. Snce the begnnng of 999, EURIBOR rates have been used n leu of LIBOR and REUTERS rates. 4 VDAX and the correspondng sub-ndces are calculated once at the end of each tradng day as of 4 July 006.

9 Volatlty Indces of Deutschen Börse Page 9.4. VDAX-NEW Index Code ISIN Daly closng prces snce VDAX-NEW VX DE000A0DMX99 0 Jan. 99 VDAX-NEW sub-ndex ( mth) Vmj cf Jan. 99 VDAX-NEW sub-ndex ( mth) Vmj cf Jan. 99 VDAX-NEW sub-ndex 3 (3 mth) Vmj cf Jan. 99 VDAX-NEW sub-ndex 4 (6 mth) Vmj cf Jan. 99 VDAX-NEW sub-ndex 5 (9 mth) Vmj cf Jan. 99 VDAX-NEW sub-ndex 6 ( mth) Vmj cf Mar. 996 VDAX-NEW sub-ndex 7 (8 mth) Vmj cf Mar. 996 VDAX-NEW sub-ndex 8 (4 mth) Vmj cf Mar. 996 m represents the respectve expry month ( A=Jan,..., L=Dec); j represents the respectve year (0, ; 9) The VDAX-NEW and ts varous sub-ndces have been calculated on a contnuous bass wth effect from 8 Aprl 005. Hstorcal tme seres for the man ndex and the frst fve sub-ndces, based on daly settlement prces, date back to January 99. Long-term DAX optons (wth tme to expratons of, 8 and 4 months) and the correspondng VDAX-NEW sub-ndces have only been avalable snce 8 March 996. The REUTERS overnght rate, the - to -month LIBOR rates, and the yeld of the -year REX were used as nterest rates. Snce the begnnng of 999, EURIBOR rates have been used n leu of LIBOR and REUTERS overnght rates..4.3 VDAX-NEW Fxed Identfer Sub-Indces As of 3 October 006 Deutsche Börse addtonally calculates eght sub-ndces wth a fxed ISIN. Contrary to the sub-ndces wth varable ISIN classfcaton specfed n chapter.4. the ISIN n ths procedure refers to the remanng tme to expraton of the opton. Over a perod of tme the optons move nto a sub-ndex wth the adequate tme to expraton (compare followng table).

10 Volatlty Indces of Deutschen Börse Page 0 Index Code ISIN Daly closng prces snce VDAX-NEW VX DE000A0DMX99 0 Jan. 99 VDAX-NEW sub-ndex ( mth) V4F DE000A0G83V9 3 Oct. 006 VDAX-NEW sub-ndex ( mth) V4F DE000A0G83W7 3 Oct. 006 VDAX-NEW sub-ndex 3 (3 mth) V4F3 DE000A0G83X5 3 Oct. 006 VDAX-NEW sub-ndex 4 (6 mth) V4F4 DE000A0G83Y3 3 Oct. 006 VDAX-NEW sub-ndex 5 (9 mth) V4F5 DE000A0G83Z0 3 Oct. 006 VDAX-NEW sub-ndex 6 ( mth) V4F6 DE000A0G Oct. 006 VDAX-NEW sub-ndex 7 (8 mth) V4F7 DE000A0G838 3 Oct. 006 VDAX-NEW sub-ndex 8 (4 mth) V4F8 DE000A0G836 3 Oct Lcensng The ndces of Deutsche Börse are regstered trademarks of Deutsche Börse AG, and protected as such aganst any unauthorzed use both n Germany and abroad. Exchanges, banks and nvestment companes may, however, apply to Deutsche Börse for lcenses to use these ndces as underlyng nstruments for dervatve nstruments. The standardzed lcensng agreement grants the lcensee the rght to use all ndces for any number of nstruments, wth the lcense fee set accordng to the actual usage. Any nqures regardng the lcensng of ndces should be drected to Deutsche Börse. Contact detals are provded on the last page of ths gude.

11 Volatlty Indces of Deutschen Börse Page VDAX. Calculaton Method Accordng to the Black & Scholes opton prcng model (973) or, more precsely, accordng to the modfcaton developed by Black n 976 the prce for a European-style vanlla opton s determned as follows (compared to the orgnal verson, the forward prce of the respectve underlyng s used nstead of the underlyng tself): () C = e r t (F N(d ) K N(d )) () P = e r t (K N(-d ) F N(-d )) whereby: (3) d = F ln( ) K σ t + σ t and (4) = d σ t d whereby: C = Call prce P = Put prce K = Exercse prce of the opton F = Forward prce of the underlyng ndex t = Annualzed remanng tme to expraton r = Rsk-free nterest rate for tme to expraton t, assumng a constant nterest rate σ = Volatlty of the opton N(...) = Cumulatve normal dstrbuton

12 Volatlty Indces of Deutschen Börse Page R s defned as the refnancng factor: (5) r R = e t Formulae () and () are now rewrtten, usng the followng standardzatons: (6) v = σ t (generalzed volatlty) (7) c = C R F K (generalzed call prce) (8) p = P R F K (generalzed put prce) (9) f = F F K (generalzed forward prce) (0) u = ln (f) (lognormal, generalzed forward prce) The followng opton prce relatons apply to these generalzed values: + u u u u () c = e N( + + v) e N( + v) v v u u + u u () p = e N( + v) e N( v) v v

13 Volatlty Indces of Deutschen Börse Page 3 Usng these transformatons, t s now possble to express call and put prces as functons of forward prce and volatlty: (3) c = f (u, v) (4) p = g (u, v) Furthermore, the transformaton also llustrates the symmetry between calls and puts. The standardzed call and put prces become dentcal, subject to the mrror functon u -u. Ths s a result of the symmetrcal normal dstrbuton for fluctuatons of the underlyng nstrument, whch s one of the fundamental assumptons n the Black model. Usng ths symmetry, we ntroduce a common opton prce o: (5) o (u, v) = f (u, v) (6) o (u, v) = g (-u, v) The opton prcng formula on whch the varous volatlty ndces are based s then wrtten as follows: + u(t) u(t) u(t) u(t) (7) o(u(t),v) = e N( + + v) e N( + v) v v. Extractng Data For the calculaton the respectve best bd and best ask of all DAX optons and futures contracts lsted on Eurex are extracted from the stream of data generated by the Eurex system. To ths end, a snapshot s taken at one mnute ntervals. The DAX calculated on the bass of Xetra prces and the varous nterest rates mentoned under. are recorded smultaneously..3 Flterng of Data Opton prces are subject to flterng of data. All optons wth a one-sded market.e. wth ether a bd or an ask only are dsregarded. The same apples to optons wthout any prce data.

14 Volatlty Indces of Deutschen Börse Page 4 Another flter verfes whether the bd/ask spread of each of the remanng optons s n lne, measured aganst the maxmum quotaton spreads establshed for Eurex market-makers. Accordngly, the maxmum spread must not exceed 0 percent of the bd quote, subject to a mnmum of.40 ponts and a maxmum of 3.40 ponts. If Eurex actvates Fast Market status, permttng market-makers to double ther quotaton spreads under very turbulent tradng condtons, each of the specfed spreads s doubled accordngly..4 Preparng Data The md prce s calculated for fltered opton and futures prces, usng the respectve best bd and best ask. Therefore, the calculaton process contnues as follows: (8) C j = C a j + C b j (9) (0) P F j a j P = a F = + P + F b j b whereby represents the expraton, j the exercse prce, b the bd and a the ask prce. The correspondng nterest rates whch match the tme to expraton of the ndex opton are derved through the use of lnear nterpolaton. The two nearest nterest rates r(t k ) and r(t k+ ) (e.g. EONIA and month EURIBOR rates) to the tme to expraton T of the opton under consderaton and ther respectve tme to expratons T k and T k+ are nterpolated to derve an approxmaton of the nterest rate to be used n the calculaton of the ndex. T T T T r r T r Tk k+ T T T T k+ k () ( ) = ( ) + r( T ); whereby: k+ () T k T < Tk + k k+ The refnancng factor R s calculated usng relaton (5). k

15 Volatlty Indces of Deutschen Börse Page 5.5 Determnng Forward Prces.5. Prelmnary Forward Prce The next step sees the determnaton of forward prces for the DAX ndex usng the optons remanng tme to expraton. Where a DAX future wth a matchng remanng tme to expraton s quoted wthn a gven ODAX expry month, F the md prce of the future s used as the (fnal) forward prce. Thngs are more complex f there s no such future wth a matchng tme to expraton, as no forward prce s then drectly avalable at Eurex for a gven ODAX expry month. Ths occurs for ODAX seres whch do not expre n March, June, September or December. In such a case, a forward prce wll be calculated n two steps: At frst, a prelmnary forward prce F s determned by way of lnear nterpolaton, usng those futures that have not been fltered out and are quoted around the tme to expraton concerned. In ths context, the DAX (cash) ndex level s assumed as the prce of a theoretcal futures contract wth a remanng tme to expraton of 0. T T T T + F' T = F + F+ T+ T T+ T (3) ( ) whereby: (4) T T < T + If such nterpolaton s not possble because no future wth a longer remanng tme to expraton has been quoted, the prelmnary forward prce wll be determned usng extrapolaton, based on the longest avalable futures contract: F' T = F e r t (5) ( ) F descrbes the md prce of the longest avalable futures contract, t the tme between the maturty date of ths future and the opton s expraton, and r the forward rate of the opton, calculated usng the followng formula: + r (6) O r = + r r F O whereby r O and r F are defned as the nterest rate for the tme to expraton of the opton and future, respectvely, both as prevously determned n (). The prelmnary forward prce determned that way defnes the prelmnary at-the-money pont. Only those opton seres j wthn a gven expry month, the exercse prces of whch are close to the forward prce, are taken nto account n the next step of the calculaton process.

16 Volatlty Indces of Deutschen Börse Page 6.5. Fnal Forward Prce For expry months, where a prelmnary forward prce was calculated by means of nterpolaton or extrapolaton, the fnal forward prce s now determned from the opton prces, usng the put-call party method. For ths purpose, pars of call and put seres wth the same exercse prce are created. Around the prelmnary at-the-money pont, a range of sxteen optons s determned.e. the pars of puts and calls of each of the four exercse prces above and below ths pont. If no two pars are smultaneously quoted wthn ths range, the fnal forward prce and therefore a current sub-ndex value cannot be determned. In such a case, the exstng sub-ndex (f there has been one already) wll contnue to be used. If there are two or more pars, every vald par wll be consdered n formula (7). The purpose of ths restrcton to only eght exercse prces at ths pont s to exclude any seres from the forward prce calculaton (usng the put-call party) whch are ether quoted not frequently enough, or wth too wde a bd/ask spread. (7) F = (Cj P j)r + K N Pars C,P j Now that the refnancng factor R and the forward prce F have been establshed for every expry month, the general, emprcal opton prces o j are calculated from the adjusted call and put prces accordng to relatons (7), (8) and (5), (6), usng the exercse prces K j. Standardzed, lognormal forward prces u j result from defnton (0)..6 Calculatng Volatltes of Indvdual Optons As soon as the fnal forward prce s avalable for a gven tme to expraton, mpled volatltes are calculated for all ndvdual optons whch are relevant to ths tme to expraton and have not been fltered out. Snce the Black model or the opton prcng formula set forth n (7) cannot be solved for volatlty, an teraton method s used to determne the requred value. The startng value for v (defned as standardzed volatlty) s set to The theoretcal standardzed opton prce, calculated from (7) usng ths value, s compared to the market prce of the opton. Applyng the Gauss-Newton method, a new standardzed volatlty v s gradually determned and used as the startng value for each successve teraton step. Upon reachng a gven degree of accuracy (.e. when v and v + only dffer from each other by ), the process s stopped, yeldng the opton s standardzed mpled volatlty. Ths s the value where the theoretcal opton prce, calculated on the bass of that value, s almost dentcal to the market prce of the opton.

17 Volatlty Indces of Deutschen Börse Page 7.7 Determnng the Sub-Indces The calculaton of a sub-ndex V frst requres the determnaton of a range of eght optons, ths tme around the fnal forward prce (calculated usng the call-put party), or the futures prce. Impled volatltes of each of four optons are weghted accordng to the dstance of the relevant exercse prces from the forward or futures prce (8). Those four optons selected have to be two pars each wth the same exercse prce, one above and one below the calculated fnal forward prce. Furthermore, the up-to-dateness of volatlty data s gven prorty over the dstance from the forward prce. Accordngly, f there are current volatltes for both pars above the forward prce, then the par closer to the at-the-money pont wll be used. However, f the more dstant par's volatlty represents the more current nformaton, then ths par wll be chosen. The reason for ths procedure s that opton seres wth even exercse prces usually have more lqudty. Hence, these exercse prces are used for calculaton, whenever seres whch mght be closer are no longer actvely quoted durng the course of a tradng day. For the varous nterest rates, calculaton s based on the values of the respectve prevous tradng day untl these nterest rates are updated. Wth the FDAX and ODAX, ths s dfferent: n order to avod volatlty fluctuatons, whch are caused by changes n the ndex level from one day to the other, no prevous day s values are used. The dssemnaton of a sub-ndex requres the avalablty of certan data: frst, a forward prce for the DAX ndex wth the same tme to expraton as the sub-ndex; ths value results drectly from the FDAX prces, or t s calculated. Around ths forward prce, defnng the at-the-money pont, the four ndvdual volatltes used to calculate the ndex must be avalable. For ths purpose, these volatltes do not need to have been traded smultaneously. As soon as the forward prce as well as best bd and best ask, whch are not beng fltered out, are avalable for an opton, ts volatlty can be determned. If the current data stuaton does not allow for a recalculaton of ths opton s volatlty n the subsequent calculaton process, the last calculated value wll contnue to be used. (8) V = (K h F) (v Put l Call + vl ) + (F K l) (v (K K ) h l Put h + v Call h ) whereby: V = Sub-ndex,.e. generalzed volatlty of tme to expraton F = Forward or futures prce of the correspondng tme to expraton v = Volatlty of an ndvdual opton K = Exercse prce of the opton In formula (8), h and l ndcate whether reference s made to the hgher or lower neghborng exercse prce.

18 Volatlty Indces of Deutschen Börse Page 8 If the fnal forward prce moves across an exercse prce from one calculaton to the next, and f the volatltes actually requred for the calculaton of the sub-ndex are not avalable there, the ndex s recalculated all the same provded that both volatltes are avalable for at least one par of the new neghborng exercse prces. In case there s only one volatlty for a gven exercse prce, ths volatlty s also used to estmate the mssng fourth. If there are no such volatltes at all, the sub-ndex results as the average of the two exstng volatltes. Agan, the recentness of data takes precedence over closeness to the forward prce. Usng relaton (6), t s possble to ndcate a sub-ndex as well as ndvdual volatltes n the commonly used form of an annualzed value σ. In the cases referred to above, where a current sub-ndex cannot be calculated, the respectve prevous sub-ndex value wll be mantaned nstead..8 Constructng the Volatlty Index The prevous chapter has llustrated the calculaton process for volatlty sub-ndces. These subndces typcally have no constant remanng tme to expraton, and eventually expre. The objectve of the VDAX s to construct a volatlty ndex wth a rollng fxed perod of tme. The nterpolaton procedure requred for ths purpose s derved from the followng reflectons: There are four ponts n tme t (snapshot tme), t a, t b and t c and addtonal two perods of tme T and T +, whereby t < t a < t b < t c Assume t a to be the expraton of tme to expraton and t c to be the expraton of tme to expraton +, respectvely. t b represents the notonal expraton of the volatlty ndex. Therefore, the followng apples accordngly: T = [t, t a ] T = [t, t b ] (= 45 days) T + = [t, t c ] Both expratons have been selected from those eght avalable, for ther remanng tme to expraton exactly ncludes the fxed tme to expraton T of the volatlty ndex. The related sub-ndces V and V + have been determned on the assumpton that volatlty s constant and the Black model s applcable. However, f volatltes σ and σ + are not dentcal, and f, for example, σ + s greater than σ, market partcpants are obvously assumng hgher average prce fluctuatons n the DAX for perod t a to t c than for perod t to t a. The latter, however, s ncluded n T + such that n the absence of any addtonal nformaton assumptons have to be

19 Volatlty Indces of Deutschen Börse Page 9 made as to how and when volatlty changes. For the sake of smplcty, constant volatlty s assumed for each perod, wth the change n volatlty exactly occurrng at tme t a. Of course, n such a case, the statstcal dstrbuton of the DAX prce fluctuatons s no longer Gaussshaped and, strctly speakng, the Black model s therefore no longer approprate to reflect ths ssue. However, the constructon of the volatlty ndex ams at stckng to the Black model all the same. Based on the varous assumptons stated above, volatlty V over the remanng tme to expraton T s equal to the followng: T + T T T (9) V = V + V +. T T T T + + In addton, ths approach s subject to the assumpton that the prce fluctuatons n the DAX durng perod [t a, t c ] are statstcally ndependent of fluctuatons durng perod [t, t a ], and that t s therefore sensble to add ndvdual varances up to the varance of the overall dstrbuton. The form of equaton (9) corresponds to a lnear nterpolaton of varances. The nterpolated volatlty V (n the usual annualzed form) represents the volatlty ndex VDAX to be constructed. (30) σ = V T The tme scale approprate to the ssue presented has to be ncluded not before nterpolaton step (9). The tme used there s the physcal tme (calendar tme). Tme to expraton T, whch s covered by the ndex, should be as short as possble, snce experence shows that most of the lqudty s avalable n short-term optons tradng. On the other hand, the necessary rollover of the varous expratons must not necesstate an extrapolaton. The selecton of T=45 calendar days reconcles both requrements. However, n order to be able to dssemnate a VDAX ndex as early as possble durng tradng hours, the nterpolaton does not necessarly have to take place between those two sub-ndces whch are drectly ncluded n the 45-day perod. Therefore, the VDAX s dssemnated as soon as a sub-ndex wth a remanng tme to expraton of less than 45 days and a sub-ndex wth one of more than 45 days can be calculated for the frst tme durng the course of a tradng day. If, at a later stage, addtonal sub-ndces become avalable whch are closer to T, these sub-ndces wll be used nstead for the purpose of nterpolaton. Due to the weghtng of ntervals n (9) above, any gaps that mght occur durng the transton to another sub-ndex are mnmzed.

20 Volatlty Indces of Deutschen Börse Page 0 3 VDAX-NEW 3. Calculaton Method The model for VDAX-NEW ams at makng pure volatlty tradable.e. the ndex should be trackable by a portfolo whch does not react to prce fluctuatons, but only to changes n volatlty. Ths s not drectly acheved through volatlty, but rather through varances or squared volatltes. A portfolo of DAX optons wth dfferent exercse prces wth a gven weghtng, as descrbed below, meets ths requrement. So, nstead of usng mpled volatltes of the at-the-money optons, n lne wth the old VDAX model, mpled varances of at-the-money as well as out-of-the-money optons of a gven tme to expraton are consdered. The sub-ndces are calculated accordng to the formula shown below: (3) VDAX - NEW = 00 σ whereby: (3) K,j F σ = R M(K,j) T j K,j T, =,,..8 K,0 and: T = Tme to expraton of the th ODAX F = Forward prce derved from the prces of the th ODAX, for whch the absolute dfference between call and put prces (C and P) s smallest. Therefore: (33) F K + R (C P) = mn C P (Note: If a clear mnmum does not exst, the average value of the relevant forward prces wll be used nstead.) K,j = Exercse prce of the j th out-of-the-money opton of the th ODAX expry month both n ascendng order K,j = Interval between the relevant exercse prces or half the nterval between the one hgher and one lower exercse prce. On the boundares, the smple nterval between the hghest and second hghest exercse prce (or lowest and second lowest exercse prce) s used: (34) K,j+ K,j K,j =

21 Volatlty Indces of Deutschen Börse Page K,0 = Hghest exercse prce below forward prce F R = Refnancng factor of the th ODAX (35) R = e r T r = Rsk-free nterest rate to expraton of the th ODAX M(K,j ) = Prce of the opton K,j, whereby K,j K,0 M(K,0 ) = Average of the put and call prces at exercse prce K,0 The sub-ndces are calculated up untl two days pror to expraton. Each new sub-ndex s dssemnated for the frst tme on the second tradng day 5 of the relevant DAX optons. The ndvdual steps wth regard to data extracton and flterng are explaned n the followng chapters, sometmes wth examples, as s the calculaton process for the varous factors used. 3. Extractng Data Durng the calculaton hours from 9:5 a.m. to 5:30 p.m. CET, the respectve best bd and best ask prces of all DAX optons contracts lsted on Eurex along wth the varous nterest rates mentoned under. are extracted from the stream of data generated by the Eurex system. To ths end, a snapshot s taken at one mnute ntervals. 3.3 Flterng of Data a) Opton prce data s subject to flterng. All opton prces that are one-sded.e. wth ether a bd or an ask prce only are dsregarded. Naturally, the same apples to optons wthout any prce data. b) Another flter verfes whether the remanng optons are quoted wthn the establshed maxmum spreads for Eurex market-makers. The maxmum spread s derved from bd prces as shown n the table below: Bd (ndex ponts) Maxmum Spread % > Generally the second tradng day after the opton seres expry day s a Tuesday (Excepton: Bank holday).

22 Volatlty Indces of Deutschen Börse Page Example: Bd = 45.3 and ask = 54.3 Max. spread: = 4.53 => both prces (bd and ask) are rejected. If Eurex actvates Fast Market status, permttng market-makers to ncrease ther quotaton spreads under very turbulent tradng condtons, maxmum spreads are set hgher accordngly. Ths s also taken nto account for the calculaton of the VDAX-NEW, wth the applcable flter crtera beng adjusted accordngly. 3.4 Preparng Data a) Determnng the prces used The md prce s calculated for fltered opton prces, usng the respectve best bd and best ask. The most recent of each of the followng peces of nformaton s used subsequently: Settlement prce (prevous day) Md prce Last traded prce Example (Call optons): Underlyng Settlement Bd (tme) Ask (tme) Md (tme) Last-traded (tme) Prce 4, , (09:05) ,00 4, (09:04) 90.0 (09:05) (09:05) 37. (09:03) 40. (09:05) (09:05) (09:0) b) Cuttng the wngs Yet another flter ensures that the varous prces used (settlement, md and last traded prce) do not fall short of a mnmum value of 0.5 ndex ponts. If there are two or more optons wth dfferent exercse prces and md prces exactly equal the mnmum value of 0.5 just the one nearest to the atthe-money pont s taken nto consderaton. Wth ths, optons that are far out-of-the money and that do not have much nfluence on the result of the calculaton are fltered out and do not need to be consdered. c) Determnng the tme to expraton T (36) T = T Settlement-Calculaton /T Year T Settlement-Calculaton = Seconds between ndex calculaton and settlement T Year = Seconds per annum

23 Volatlty Indces of Deutschen Börse Page 3 Example: Index calculaton: Expraton ( = ): 5 November 004 at :00 a.m. CET 7 December 004 at :00 p.m. CET,908,000 T = = d) Determnng rsk-free nterest rates Lnear nterpolaton s used as wthn the calculaton of the VDAX to determne nterest rates, the terms of whch match the tme to expraton of the ODAX. T T T T r r T r Tk k+ T T T T k+ k (37) ( ) = ( ) + r( T ); T k T < Tk + k+ k k+ k Example: r(t k ) =.05% (EONIA) r(t k+ ) =.8% (EURIBOR, month) r(t ) =,4% e) The refnancng factor R s determned accordng to equaton (35) Example: R = e r t = Calculaton Example 3.5. Determnng the Forward Prce F and the Exercse Prces K,0 The forward prce of the th expry month s derved from ODAX prces, for whch the dfference (n absolute terms) between call and put prces s smallest. The forward prce F of the st expry month s subject to the followng: F = K mn C P + R ( Call Put ) Example: R =.0098 K mn C-P = 4,50 F = 4, Where there are several pars of calls and puts wth dentcal dfferences, a forward prce wll be calculated for each of the correspondng exercse prces. K,0 s accordngly defned as the closest exercse prce below the smple average of these forward prces.

24 Volatlty Indces of Deutschen Börse Page Determnng the Opton Prce M(K,j ) The prce M(K,j ), whch s used for the j th out-of-the-money opton of the th expry month, s determned as follows: Put Put + Call M(K,j) = Call : K : K : K,j,j,j < K = K > K,0,0, Determnng the Sub-Indces VDAX - NEW = 00 σ σ K F = ) - -,j R M(K,j T j K T,j K, 0 Exercse Prce K,j K,j Call Put Call Put M(K.j ) 3, K,j R M(K,j) K,j 3, , , , , , , , , , , , , , , , , , , , , , , , Σ

25 Volatlty Indces of Deutschen Börse Page 5 σ = = VDAX - NEW = = Constructng the Volatlty Index Apart from the sub-ndces for the varous ndvdual tme to expraton, the VDAX-NEW s determned as the man ndex wth a constant remanng tme to expraton of 30 days (ths ndex s not lnked to a specfc tme to expraton). It s calculated n the same way as the VDAX. The VDAX- NEW s determned by nterpolaton of the sub-ndces whch are nearest to a remanng tme to expraton of 30 days. If there are no such surroundng sub-ndces, the VDAX-NEW s calculated usng extrapolaton. In ths case, the two nearest avalable ndces are used, whch are as close to the tme to expraton of 30 calendar days as possble. VDAX- NEW = 00 = T σ N N T + T + T VDAX- NEW N T + T NT N N T + T + + σ + N T + T NT N N + T T + N T N NT N T VDAX- NEW N N T T + N T N NT N T 365 N T N T+ N T N 365 = Tme to expraton of the th ODAX = Tme to expraton of the = Tme for next x days = Tme for a standard year th + ODAX 3.7 Calculaton of Settlement Index VDAX-NEW future settlement prce s calculated 30 calendar days before the maturty date of the DAX opton. For ths purpose the equally weghted mean of all ndex values of VDAX-NEW between :30 p.m. and :00 p.m. s determned.

26 Volatlty Indces of Deutschen Börse Page 6 4 Appendx 4. VDAX Master Data 0(4) DE VXA4 DE VXB4 DE VXC4 DE VXD4 DE VXE4 DE VXF4 DE VXG4 DE VXH4 DE VXI4 DE VXJ4 DE VXK4 DE VXL4 0(3) DE VXA3 DE VXB3 DE VXC3 DE VXD3 DE VXE3 DE VXF3 DE VXG3 DE VXH3 DE VXI3 DE VXJ3 DE VXK3 DE VXL3 0() DE VXA DE VXB DE VXC DE VXD DE VXE DE VXF DE VXG DE VXH DE VXI DE VXJ DE VXK DE VXL 0() DE VXA DE VXB DE VXC DE VXD DE VXE DE VXF DE VXG DE VXH DE VXI DE VXJ DE VXK DE VXL 0(0) DE VXA0 DE VXB0 DE VXC0 DE VXD0 DE VXE0 DE VXF0 DE VXG0 DE VXH0 DE VXI0 DE VXJ0 DE VXK0 DE VXL0 00(9) DE VXA9 DE VXB9 DE VXC9 DE VXD9 DE VXE9 DE VXF9 DE VXG9 DE VXH9 DE VXI9 DE VXJ9 DE VXK9 DE VXL9 00(8) DE VXA8 DE VXB8 DE VXC8 DE VXD8 DE VXE8 DE VXF8 DE VXG8 DE VXH8 DE VXI8 DE VXJ8 DE VXK8 DE VXL8 00(7) DE VXA7 DE VXB7 DE VXC7 DE VXD7 DE VXE7 DE VXF7 DE VXG7 DE VXH7 DE VXI7 DE VXJ7 DE VXK7 DE VXL7 00(6) DE VXA6 DE VXB6 DE VXC6 DE VXD6 DE VXE6 DE VXF6 DE VXG6 DE VXH6 DE VXI6 DE VXJ6 DE VXK6 DE VXL6 00(5) DE VXA5 DE VXB5 DE VXC5 DE VXD5 DE VXE5 DE VXF5 DE VXG5 DE VXH5 DE VXI5 DE VXJ5 DE VXK5 DE VXL5 VX Jan (A) Feb (B) Mar (C) Apr (D) May (E) Jun (F) Jul (G) Aug (H) Sep (I) Oct (J) Nov (K) Dec (L)

27 Volatlty Indces of Deutschen Börse Page 7 4. VDAX-NEW Master Data 0(4) DE000A0DMZQ0 VA4 DE000A0DMZR8 VB4 DE000A0DMZS6 VC4 DE000A0DMZT4 VD4 DE000A0DMZU VE4 DE000A0DMZV0 VF4 DE000A0DMZW8 VG4 DE000A0DMZX6 VH4 DE000A0DMZY4 VI4 DE000A0DMZZ VJ4 DE000A0DMZ06 VK4 DE000A0DMZ4 VL4 0(3) DE000A0DMZC0 VA3 DE000A0DMZD8 VB3 DE000A0DMZE6 VC3 DE000A0DMZF3 VD3 DE000A0DMZG VE3 DE000A0DMZH9 VF3 DE000A0DMZJ5 VG3 DE000A0DMZK3 VH3 DE000A0DMZL VI3 DE000A0DMZM9 VJ3 DE000A0DMZN7 VK3 DE000A0DMZP VL3 0() DE000A0DMY07 VA DE000A0DMY5 VB DE000A0DMY3 VC DE000A0DMY3 VD DE000A0DMY49 VE DE000A0DMY56 VF DE000A0DMY64 VG DE000A0DMY7 VH DE000A0DMY80 VI DE000A0DMY98 VJ DE000A0DMZA4 VK DE000A0DMZB VL 0() DE000A0DMYN0 VA DE000A0DMYP5 VB DE000A0DMYQ3 VC DE000A0DMYR VD DE000A0DMYS9 VE DE000A0DMYT7 VF DE000A0DMYU5 VG DE000A0DMYV3 VH DE000A0DMYW VI DE000A0DMYX9 VJ DE000A0DMYY7 VK DE000A0DMYZ4 VL 0(0) DE000A0DMYA7 VA0 DE000A0DMYB5 VB0 DE000A0DMYC3 VC0 DE000A0DMYD VD0 DE000A0DMYE9 VE0 DE000A0DMYF6 VF0 DE000A0DMYG4 VG0 DE000A0DMYH VH0 DE000A0DMYJ8 VI0 DE000A0DMYK6 VJ0 DE000A0DMYL4 VK0 DE000A0DMYM VL0 00(9) DE000A0DMG VA9 DE000A0DMH9 VB9 DE000A0DMJ5 VC9 DE000A0DMK3 VD9 DE000A0DML VE9 DE000A0DMM9 VF9 DE000A0DMN7 VG9 DE000A0DMP VH9 DE000A0DMQ0 VI9 DE000A0DMR8 VJ9 DE000A0DMS6 VK9 DE000A0DMT4 VL9 00(8) DE000A0DM04 VA8 DE000A0DM058 VB8 DE000A0DM066 VC8 DE000A0DM074 VD8 DE000A0DM08 VE8 DE000A0DM090 VF8 DE000A0DMA4 VG8 DE000A0DMB VH8 DE000A0DMC0 VI8 DE000A0DMD8 VJ8 DE000A0DME6 VK8 DE000A0DMF3 VL8 00(7) DE000A0DM0S8 VA7 DE000A0DM0T6 VB7 DE000A0DM0U4 VC7 DE000A0DM0V VD7 DE000A0DM0W0 VE7 DE000A0DM0X8 VF7 DE000A0DM0Y6 VG7 DE000A0DM0Z3 VH7 DE000A0DM009 VI7 DE000A0DM07 VJ7 DE000A0DM05 VK7 DE000A0DM033 VL7 00(6) DE000A0DM0E8 VA6 DE000A0DM0F5 VB6 DE000A0DM0G3 VC6 DE000A0DM0H VD6 DE000A0DM0J7 VE6 DE000A0DM0K5 VF6 DE000A0DM0L3 VG6 DE000A0DM0M VH6 DE000A0DM0N9 VI6 DE000A0DM0P4 VJ6 DE000A0DM0Q VK6 DE000A0DM0R0 VL6 00(5) DE000A0DMZ VA5 DE000A0DMZ30 VB5 DE000A0DMZ48 VC5 DE000A0DMZ55 VD5 DE000A0DMZ63 VE5 DE000A0DMZ7 VF5 DE000A0DMZ89 VG5 DE000A0DMZ97 VH5 DE000A0DM0A6 VI5 DE000A0DM0B4 VJ5 DE000A0DM0C VK5 DE000A0DM0D0 VL5 V Jan (A) Feb (B) Mar (C) Apr (D) May (E) Jun (F) Jul (G) Aug (H) Sep (I) Oct (J) Nov (K) Dec (L)

28 Volatlty Indces of Deutschen Börse Page 8 5 Your Drect Lne to Deutsche Börse Informaton on Prces and Index Concepts Market Data & Analytcs Customer Servce Phone: Fax: E-mal: [email protected] Lcenses for Prce Data and Indces Market Data & Analytcs Phone: Fax: E-mal: [email protected] Publcatons Publcaton Hotlne Phone: Fax: E-mal: [email protected] Internet /mda_e Malng Address Deutsche Börse AG Frankfurt/Man Germany