THE CBOE VOLATILITY INDEX - VIX

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1 he powerful and flexble tradng and rsk management tool from the Chcago Board Optons Exchange HE CBOE VOLAILIY INDEX - VIX ACCEP NO SUBSIUE.

2 HE CBOE VOLAILIY INDEX HE CBOE NDEX - VIX In 993, the Chcago Board Optons Exchange (CBOE ) ntroduced the CBOE Volatlty Index, VIX, whch was orgnally desgned to measure the market s expectaton of 30- day volatlty mpled by at-the-money S&P 00 Index (OEX ) opton prces. VIX soon became the premer benchmark for U.S. stock market volatlty. It s regularly featured n the Wall Street Journal, Barron s and other leadng fnancal publcatons, as well as busness news shows on CNBC, Bloomberg V and CNN/Money, where VIX s often referred to as the fear ndex. en years later n 003, CBOE together wth Goldman Sachs, updated the VIX to reflect a new way to measure expected volatlty, one that contnues to be wdely used by fnancal theorsts, rsk managers and volatlty traders alke. he new VIX s based on the S&P 500 Index (SPX SM ), the core ndex for U.S. equtes, and estmates expected volatlty by averagng the weghted prces of SPX puts and calls over a wde range of strke prces. By supplyng a scrpt for replcatng volatlty exposure wth a portfolo of SPX optons, ths new methodology transformed VIX from an abstract concept nto a practcal standard for tradng and hedgng volatlty. VOLAILIY AS A RADABLE ASSE VIX FUURES & OPIONS On March 4, 004, CBOE ntroduced the frst exchange-traded VIX futures contract on ts new, all-electronc CBOE Futures Exchange SM (CFE ). wo years later n February 006, CBOE launched VIX optons, the most successful new product n Exchange hstory. In less than fve years, the combned tradng actvty n VIX optons and futures has grown to more than 00,000 contracts per day. he negatve correlaton of volatlty to stock market returns s well documented and suggests a dversfcaton beneft to ncludng volatlty n an nvestment portfolo. VIX futures and optons are desgned to delver pure volatlty exposure n a sngle, effcent package. CBOE/CFE provdes a contnuous, lqud and transparent market for VIX products that are avalable to all nvestors from the smallest retal trader to the largest nsttutonal money managers and hedge funds. BEYOND VIX VIX In addton to VIX, CBOE calculates several other volatlty ndexes ncludng the CBOE Nasdaq-00 Volatlty Index (VXN SM ), CBOE DJIA Volatlty Index (VXD SM ), CBOE Russell 000 Volatlty Index (RVX SM ) and CBOE S&P Month Volatlty Index (VXV SM ). Currently, VXD and RVX futures are lsted on CFE; RVX optons trade on CBOE. In 008, CBOE poneered the use of the VIX methodology to estmate expected volatlty of certan commodtes and foregn currences. he CBOE Crude Ol Volatlty Index (OVX SM ), CBOE Gold Volatlty Index (GVZ SM ) and CBOE EuroCurrency Volatlty Index CBOE Propretary Informaton Copyrght 009, Chcago Board Optons Exchange, Incorporated. All rghts reserved.

3 (EVZ SM ) use exchange-traded fund optons based on the Unted States Ol Fund, LP (USO), SPDR Gold Shares (GLD) and CurrencyShares Euro rust (FXE), respectvely. HISORICAL PRICES FOR VIX AND OHER VOLAILIY INDEXES VIX Perhaps one of the most valuable features of VIX s the exstence of more than 0 years of hstorcal prces. hs extensve data set provdes nvestors wth a useful perspectve of how opton prces have behaved n response to a varety of market condtons. Prce hstory for the orgnal CBOE Volatlty Index (cker VXO ) based on OEX optons s avalable from 986 to the present. CBOE has created a smlar hstorcal record for the new VIX datng back to 990 so that nvestors can compare the new VIX wth VXO, whch reflects nformaton about the volatlty skew or smle. Hstorcal prces for VIX, VXO and CBOE s other volatlty ndexes may be found on the CBOE webste at under CBOE Volatlty Indexes. CBOE Propretary Informaton Copyrght 009 Chcago Board Optons Exchange, Incorporated. All rghts reserved.

4 3 HE VIX CALCULAION SEP-BY BY-SEP VIX Stock ndexes, such as the S&P 500, are calculated usng the prces of ther component stocks. Each ndex employs rules that govern the selecton of component securtes and a formula to calculate ndex values. VIX s a volatlty ndex comprsed of optons rather than stocks, wth the prce of each opton reflectng the market s expectaton of future volatlty. Lke conventonal ndexes, VIX employs rules for selectng component optons and a formula to calculate ndex values. he generalzed formula used n the VIX calculaton s: WHERE HERE σ = R e Q( ) 0 F () σ s F 0 VIX VIX = σ me to expraton Forward ndex level derved from ndex opton prces Frst strke below the forward ndex level, F Strke prce of th out-of-the-money opton; a call f > 0 and a put f < 0 ; both put and call f = 0. Interval between strke prces half the dfference between the strke on ether sde of : = + R (Note: for the lowest strke s smply the dfference between the lowest strke and the next hgher strke. Lkewse, for the hghest strke s the dfference between the hghest strke and the next lower strke.) Rsk-free nterest rate to expraton Q( ) he mdpont of the bd-ask spread for each opton wth strke. Please see More than you ever wanted to know about volatlty swaps by resmr Demeterf, Emanuel Derman, Mchael amal and Joseph Zou, Goldman Sachs Quanttatve Strateges Research Notes, March 999. CBOE Propretary Informaton Copyrght 009 Chcago Board Optons Exchange, Incorporated. All rghts reserved.

5 4 GEING SARED VIX measures 30-day expected volatlty of the S&P 500 Index. he components of VIX are near- and next-term put and call optons, usually n the frst and second SPX contract months. Near-term optons must have at least one week to expraton; a requrement ntended to mnmze prcng anomales that mght occur close to expraton. When the near-term optons have less than a week to expraton, VIX rolls to the second and thrd SPX contract months. For example, on the second Frday n June, VIX would be calculated usng SPX optons exprng n June and July. On the followng Monday, July would replace June as the near-term and August would replace July as the next-term. In ths hypothetcal example, the near-term and next-term optons have 9 days and 37 days to expraton, respectvely, and reflect prces observed at the open of tradng 8:30 a.m. Chcago tme. For the purpose of calculatng tme to expraton, SPX optons are deemed to expre at the open of tradng on SPX settlement day - the thrd Frday of the month. he VIX calculaton measures tme to expraton,, n calendar days and dvdes each day nto mnutes n order to replcate the precson that s commonly used by professonal opton and volatlty traders. he tme to expraton s gven by the followng expresson: WHERE HERE = {M Current day + M Settlement day + M Other days }/ Mnutes n a year M Current day = mnutes remanng untl mdnght of the current day M Settlement day = mnutes from mdnght untl 8:30 a.m. on SPX settlement day M Other days = total mnutes n the days between current day and settlement day Usng 8:30 a.m. as the tme of the calculaton, for the near-term and next-term optons, and, respectvely, s: = { ,50) / 55,600 = = { ,840) / 55,600 = he rsk-free nterest rate, R, s the bond-equvalent yeld of the U.S. -bll maturng closest to the expraton dates of relevant SPX optons. As such, the VIX calculaton may use dfferent rsk-free nterest rates for near- and next-term optons. In ths example, however, assume that R = 0.38% for both sets of optons. Snce many of the nterm calculatons are repettve, only representatve samples appear below. he complete set of SPX opton data and calculatons may be found n Appendx. echncally, the expraton date for SPX optons s the Saturday followng the thrd Frday of the expraton month. In ths example, however, expraton s deemed to take place at the determnaton of the exercse settlement value of the SPX, whch s based on the openng prces of SPX component securtes. CBOE Propretary Informaton Copyrght 009 Chcago Board Optons Exchange, Incorporated. All rghts reserved.

6 5 SEP Select the optons to be used n the VIX calculaton he selected optons are out-of-the-money SPX calls and out-of-the-money SPX puts centered around an at-the-money strke prce, 0. Only SPX optons quoted wth non-zero bd prces are used n the VIX calculaton. One mportant note: as volatlty rses and falls, the strke prce range of optons wth nonzero bds tends to expand and contract. As a result, the number of optons used n the VIX calculaton may vary from month-to-month, day-to-day and possbly, even mnute-tomnute. For each contract month: Determne the forward SPX level, F, by dentfyng the strke prce at whch the absolute dfference between the call and put prces s smallest. he call and put prces n the followng table reflect the average of each opton s bd / ask quotaton. As shown below, the dfference between the call and put prces s smallest at the 90 strke for both the near- and next-term optons. Strke Prce Near-erm Optons Call Put Absolute Dfference Strke Prce Next-erm Optons Call Put Absolute Dfference Usng the 90 call and put n each contract month and the formula, F = Strke Prce + e R (Call Prce Put Prce), the forward ndex prces, F and F, for the near- and next-term optons, respectvely, are: F = 90 + e ( ) ( ) = F = 90 + e ( ) ( ) = Next, determne 0 - the strke prce mmedately below the forward ndex level, F - for the near- and next-term optons. In ths example, 0, = 90 and 0, = 90. CBOE Propretary Informaton Copyrght 009 Chcago Board Optons Exchange, Incorporated. All rghts reserved.

7 6 Select out-of-the-money put optons wth strke prces < 0. Start wth the put strke mmedately lower than 0 and move to successvely lower strke prces. Exclude any put opton that has a bd prce equal to zero (.e., no bd). As shown below, once two puts wth consecutve strke prces are found to have zero bd prces, no puts wth lower strkes are consdered for ncluson. Put Strke Bd Ask Include? Not consdered followng two zero bds No No Yes Yes Yes.... Next, select out-of-the-money call optons wth strke prces > 0. Start wth the call strke mmedately hgher than 0 and move to successvely hgher strke prces, excludng call optons that have a bd prce of zero. As wth the puts, once two consecutve call optons are found to have zero bd prces, no calls wth hgher strkes are consdered. (Note that the 50 call opton s not ncluded despte havng a nonzero bd prce.) Call Strke Bd Ask Include? Yes Yes No No Not consdered followng two zero bds Fnally, select both the put and call wth strke prce 0. Notce that two optons are selected at 0, whle a sngle opton, ether a put or a call, s used for every other strke prce. he followng table contans the optons used to calculate the VIX n ths example. VIX uses the average of quoted bd and ask, or md-quote, prces for each opton selected. he 0 put and call prces are averaged to produce a sngle value. he prce used for the 90 CBOE Propretary Informaton Copyrght 009 Chcago Board Optons Exchange, Incorporated. All rghts reserved.

8 7 strke n the near-term s, therefore, ( )/ = 36.90; and the prce used n the next-term s ( )/ = Near-term Strke Opton ype Md-quote Prce Next-term Strke Opton ype Md-quote Prce 400 Put Put Put Put Put Put Put Put Put Put Put/Call Average Put/Call Average Call Call Call Call Call Call Call Call Call Call 0.60 SEP Calculate volatlty for both near-term and next-term optons Applyng the VIX formula () to the near-term and next-term optons wth tme to expraton of and, respectvely, yelds: σ = R e Q( ) F 0 σ = R e Q( ) F 0 VIX s an amalgam of the nformaton reflected n the prces of all of the selected optons. he contrbuton of a sngle opton to the VIX value s proportonal to and the prce of that opton, and nversely proportonal to the square of the opton s strke prce. Generally, s half the dfference between the strke prces on ether sde of. For example, the for the next-term 300 Put s 75: 300 Put = (350 00)/. At the upper and lower edges of any gven strp of optons, s smply the dfference between and the adjacent strke prce. In ths example, the 400 Put s the lowest strke n the strp of near-term optons and 45 s the adjacent strke. herefore, 400 Put = 5 (.e., ). CBOE Propretary Informaton Copyrght 009 Chcago Board Optons Exchange, Incorporated. All rghts reserved.

9 8 he contrbuton of the near-term 400 Put s gven by: 400 Put 400 Put e R 400 Put 400 Put e R Q(400 Put) ( ) Q(400 Put) = (0.5) 400 e = A smlar calculaton s performed for each opton. he resultng values for the near-term optons are then summed and multpled by. Lkewse, the resultng values for the next-term optons are summed and multpled by. he table below summarzes the results for each strp of optons. Near-term Strke Opton ype Md-quote Prce Contrbuton by Strke Next-term Strke Opton ype Md-quote Prce Contrbuton by Strke 400 Put Put Put Put Put Put Put Put Put Put Put/Call Average Put/Call Average Call Call Call Call Call Call Call Call Call Call e Q( ) R Next, calculate F 0 for the near-term ( ) and next-term ( ): F 0 = = F 0 = = CBOE Propretary Informaton Copyrght 009 Chcago Board Optons Exchange, Incorporated. All rghts reserved.

10 9 Now calculate σ and σ : σ = R e Q( ) F 0 = = σ = R e Q( ) F 0 = = SEP 3 Calculate the 30-day weghted average of σ and σ. hen take the square root of that value and multply by 00 to get VIX. VIX = 00 N σ N N N 30 N + σ N 30 N N N N When the near-term optons have less than 30 days to expraton and the next-term optons have more than 30 days to expraton, the resultng VIX value reflects an nterpolaton of σ and σ ;.e., each ndvdual weght s less than or equal to and the sum of the weghts equals. At the tme of the VIX roll, both the near-term and next-term optons have more than 30 days to expraton. he same formula s used to calculate the 30-day weghted average, but the result s an extrapolaton of σ and σ ;.e., the sum of the weghts s stll, but the near-term weght s greater than and the next-term weght s negatve (e.g.,.5 and 0.5). Returnng to the example N = number of mnutes to settlement of the near-term optons (,960) N = number of mnutes to settlement of the next-term optons (53,80) N 30 = number of mnutes n 30 days (30,440 = 43,00) N 365 = number of mnutes n a 365-day year (365,440 = 55,600) VIX = 00 53,80 43,00 43,00,960 55, ,80,960 53,80,960 43,00 VIX = = 6. CBOE Propretary Informaton Copyrght 009 Chcago Board Optons Exchange, Incorporated. All rghts reserved.

11 0 NOES ON CALCULAING OHER CBOE VOLAILIY INDEXES CBOE BROAD-BASED ASED VOLAILIY INDEXES CBOE calculates volatlty ndexes on three other broad-based ndexes representng dfferent segments of the U.S. stock market: CBOE DJIA Volatlty Index (VXD) based on optons on the Dow Jones Industral Average (DJX); CBOE Nasdaq-00 Volatlty Index (VXN) based on Nasdaq-00 Index (NDX) optons; and CBOE Russell 000 Volatlty Index (RVX) based on Russell 000 Index (RU) optons. For each of these ndexes, the selecton of component optons and calculaton are dentcal to the method detaled n the prevous example. he CBOE S&P Month Volatlty Index (VXV) measures the market's expectaton of 3-month volatlty mpled by SPX optons that bracket a 93-day maturty. Comparng VIX and VXV provdes nvestors wth nformaton about the SPX volatlty term structure n the four near-term contract months. COMMODIY & CURRENCY VOLAILIY IY INDEXES CBOE began calculatng two commodty volatlty ndexes and one currency volatlty ndex n 008: CBOE Crude Ol Volatlty Index (OVX) based on Unted States Ol Fund, LP (USO) optons; CBOE Gold Volatlty Index (GVZ) based on the, SPDR Gold Shares (GLD) optons; and CBOE EuroCurrency Volatlty Index (EVZ) based on CurrencyShares Euro rust (FXE) optons Each of these non-equty volatlty ndexes are calculated usng exchange traded fund, or EF, optons that trade lke optons on ndvdual stocks - they may be exercsed pror to ther expraton date; exercse results n the delvery of EF shares rather than cash; and they settle at the close of tradng rather than at the open. For each of the non-equty volatlty ndexes, the method of selectng component optons and the formula are dentcal to that used for VIX and other broad-based volatlty ndexes. However, there s a slght dfference n the methodology that accounts for the fact that USO, GLD and FXE optons expre at the close rather than at the open. As before, the tme to expraton s gven by the followng expresson: = {M Current day + M Settlement day + M Other days }/ Mnutes n a year CBOE Propretary Informaton Copyrght 009 Chcago Board Optons Exchange, Incorporated. All rghts reserved.

12 WHERE HERE M Current day = mnutes remanng untl mdnght of the current day M Other days = total mnutes n the days between current day and settlement day But now, adjustng for p.m. settlement M Settlement day = mnutes from mdnght untl 3:00 p.m. on expraton day = 900 mnutes As wth the prevous example, assumng near- and next-term optons wth 9 and 37 days to expraton and 8:30 a.m. as the tme of the calculaton, for the near-term and next-term optons, and, respectvely, s: = { ,50) / 55,600 = = { ,840) / 55,600 = 0.08 Specal Note: All CBOE Volatlty Indexes VIX, VXD, VXN, RVX, VXV, OVX, GVZ and EVZ are calculated usng opton prce quotes from CBOE exclusvely. Optons nvolve rsk and are not sutable for all nvestors. Pror to buyng or sellng optons, a person must receve a copy of Characterstcs and Rsks of Standardzed Optons, whch s avalable from your broker, by callng -888-OPIONS, or from he Optons Clearng Corporaton, One North Wacker Drve, Sute 500, Chcago, Illnos he nformaton n ths document s provded solely for general educaton and nformaton purposes and therefore should not be consdered complete, precse, or current. Many of the matters dscussed are subject to detaled rules, regulatons, and statutory provsons whch should be referred to for addtonal detal and are subject to changes that may not be reflected n ths document. he nformaton n ths document s not ntended and should not be construed to consttute nvestment advce or recommendatons to purchase or sell securtes. CBOE, Chcago Board Optons Exchange, CBOE Volatlty Index, OEX and VIX are regstered trademarks, and EVZ, GVZ, OVX, SPX, VXD, VXN and VXV are servcemarks of CBOE. CFE s a regstered trademark and CBOE Futures Exchange s a servcemark of CBOE Futures Exchange, LLC. he methodologes of the CBOE volatlty ndexes are owned by CBOE and may be covered by one or more patents or pendng patent applcatons. S&P 00 and S&P 500 are regstered trademarks of the McGraw-Hll Companes, Inc., and are lcensed for use by CBOE. DJIA and Dow Jones Industral Average are trademarks of Dow Jones & Company, Inc. and have been lcensed for use for certan purposes by CBOE. CBOE's fnancal products based on the Dow Jones ndexes are not sponsored, endorsed, sold or promoted by Dow Jones, and Dow Jones makes no representatons regardng the advsablty of nvestng n such products. Nasdaq-00 Index, Nasdaq-00 and Nasdaq are trademark or servce marks of he Nasdaq Stock Market, Inc. (wth whch ts afflates are the "Corporatons"). hese marks are lcensed for use by CBOE n connecton wth the tradng of products based on the Nasdaq-00 Index. he products have not been passed on by the Corporatons as to ther legalty or sutablty. he products are not ssued, endorsed, sold or promoted by the Corporatons. HE CORPORAIONS MAE NO WARRANIES AND BEAR NO LIABILIY WIH RESPEC O HE PRODUC(S). Russell 000 s a regstered trademark of the Frank Russell Company, used under lcense. CBOE Propretary Informaton Copyrght 009 Chcago Board Optons Exchange, Incorporated. All rghts reserved.

13 APPENDIX - COMPLEE SPX OPION DAA USED IN SAMPLE VIX CALCULAION Opton Seres ncluded n the VIX calculaton are hghlghted. Near-erm Optons Next-erm Optons Strke Calls Puts Calls Puts Strke Bd Ask Bd Ask Bd Ask Bd Ask CBOE Propretary Informaton Copyrght 009, Chcago Board Optons Exchange, Incorporated. All rghts reserved.

14 CBOE Propretary Informaton Copyrght 009 Chcago Board Optons Exchange, Incorporated. All rghts reserved.

15 CBOE Propretary Informaton Copyrght 009 Chcago Board Optons Exchange, Incorporated. All rghts reserved.

16 CBOE Propretary Informaton Copyrght 009 Chcago Board Optons Exchange, Incorporated. All rghts reserved.

17 INDIVIDUAL OPION CONRIBUIONS 0 = 90 Near term Strke Opton ype Md-quote Prce Delta- Contrbuton by Strke Next term Strke Opton ype Md-quote Prce Delta- Contrbuton by Strke 400 Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put CBOE Propretary Informaton Copyrght 009, Chcago Board Optons Exchange, Incorporated. All rghts reserved.

18 7 740 Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put Put/Call Put Average Put Call Put Call Put Call Put Call Put Call Put Call Put Call Put Call Put Call Put Call Put Call Put Call Put/Call 985 Call Average 990 Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call CBOE Propretary Informaton Copyrght 009 Chcago Board Optons Exchange, Incorporated. All rghts reserved.

19 8 00 Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Sum of Indvdual Contrbutons Sum of Indvdual Contrbutons R e Q( ) R e Q( ) CBOE Propretary Informaton Copyrght 009 Chcago Board Optons Exchange, Incorporated. All rghts reserved.