The VIX Volatility Index

Transcription

1 U.U.D.M. Project Report :7 he VIX Volatlty Index Mao Xn Examensarbete matematk, 3 hp Handledare och examnator: Macej lmek Maj Department of Mathematcs Uppsala Unversty

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3 Abstract. VIX plays a very mportant role n the n fnancal dervatves prcng, tradng, rsk control strategy. It could be sad t would not be a fnacal market wthout the fnancal market volatlty. If t s the lack of rsk management tools and the market volatlty s too large, the nvestors may be worred about the rsk and gve up tradng, then the market s less attractve. hs s why I dscuss about t. In ths paper, I want to research more about ts dervaton and calculaton process, and to know how the VIX volatlty ndex changes, n accordng to &P 5 Index opton prces.

4 able of Contents Abstract.... able of Contents... Chapter. Introducton.... he Orgn of VIX.... he Development of VIX... Chapter. he General Informaton About VIX and Optons waps he ypes of Volatlty Realzed volatlty and he hstorcal volatlty he mpled volatlty he volatlty ndex ntraday volatlty Prncples of Volatlty Index Varance waps... 6 Chapter 3. EstmatngVolatlty rom Opton Data [][4]... 8 Chapter 4. Calculaton of VIX Parameter And elect he Optons o Calculate VIX Parameter : Parameter R: Parameter : Parameter and elect optons Parameter Parameter Calculate volatlty of both near-term and next-term optons Calculate the 3-day weghted average of both σ and σ hen take the square root of the value and multply by to get VIX, accordng to the formula 4. and formula Chapter 5. Concluson... 3 References Appendx Codes he Calculaton Results... 39

5 Chapter. Introducton. he Orgn of VIX VIX plays a more and more mportant role n the n fnancal dervatves prcng, tradng, rsk control strategy. After the global stock market crash n 987, t s to stablze the stock market and protect the nvestors, the New York tock Exchange NYE n 99 ntroduced a crcut breaker mechansm Crcut-breakers. When the stock prce changes unusually, t occurs a temporary suspenson of tradng, and t s helpful to try to reduce the market volatlty n order to restore the nvestor confdence on the stock market. However, due to the ntroducton of crcut breaker mechansm, there s many new nsghts for how to measure market volatlty, and t s gradually produced a dynamc dsplay of market volatlty requrements. herefore, not long after the New York tock Exchange NYE used Crcut-Breakers to solve the problem of excessve volatlty n the market, the Chcago Board Optons Exchange began to ntroduce the CBOE market volatlty ndex VIX n 993,whch s used to measure market volatlty mpled by at the money &P Index OEX opton prces [].. he Development of VIX When the stock opton transactons began n Aprl 973, Chcago Board Optons Exchange CBOE has envsaged that the market volatlty ndex can be constructed by the opton prce, whch could be shown that the expectaton of the future volatlty n the opton market. nce then there were gradually many varous calculaton methods proposed by some scholars, Whaley 993 proposed a calculaton approach whch s the preparaton of market volatlty ndex as a measure of future stock market prce volatlty. In the same year, Chcago Board Optons Exchange CBOE started to research the complaton of the CBOE Volatlty Index VIX, whch s based on the mpled volatlty of the &P Index optons, and at the same tme also calculate mpled volatlty of call opton and put opton n order to take nto account the rght of traders to buy or sell opton under the preferences [8]. It s shown the nvestors expectaton of the further stock market volatlty by VIX. he hgher the volatlty ndex s, the larger the nvestors expect the volatlty of stock ndex n the future s; the lower the VIX ndex s, the more moderate t shows that the nvestors beleve

6 that the future stock prce volatlty wll be. nce the ndex VIX can reflect the nvestors expectatons of the further stock prce volatlty, and t can be observed the mental performance of the opton partcpants, also known as "nvestor sentment gauge "he nvestor fear gauge. After ten years of development and mprovement, the VIX ndex gradually was agreed by the stock market, CBOE calculated several other volatlty ndexes ncludng, n NADAQ ndex as the underlyng volatlty ndex NADAQ Volatlty Index, VXN, n 3 the VIX ndex based on the &P 5 Index, whch s much closer to the actual stock market than the &P Index, CBOE DJIA volatlty Index VXD, CBOE Russell Volatlty Index RVX,n 4 the frst volatlty futures Volatlty Index utures VIX utures, and at the same year a second volatlty commercalzaton futures, that s the varance futures Varance utures, subject to threemonth the &P 5 Index of realzed Varance Realzed Varance. In 6, the VIX Index optons began to trade n the Chcago Board Optons Exchange. In 8, CBOE poneered the used of the VIX methodology to estmate the expected volatlty of some commodtes and foregn currences []. here are many developments. or example, n Inda, VIX was launched n Aprl, 8 by Natonal stock exchange NE. he VIX ndex of Inda s based on the Nfty 5 Index Opton prces. he methodology of calculatng the VIX ndex s same as that for CBOE VIX ndex. he current focus on the VIX s due to ts nherent property of negatve correlaton wth the underlyng prce ndex, and ts usefulness for predctng the drecton of the prce ndex [9]. And n Hongong, Hong ong got ts own volatlty ndex for fnancal products that allow nvestors to hedge aganst excessve market movements. Hang eng Indexes Company, the company whch owns and manages the benchmark ndexes n Hong ong ncludng the Hang eng ndex HI, launched the HI Volatlty ndex or "VHI" on eb.. he ndex s modeled on the lnes of the Chcago Board of Exchanges VIX ndex.vix n that t measures the 3-calendar-day expected volatlty of the Hang eng ndex usng prces of optons traded on the ndex []. In ths paper, I plan to dscuss more about ts dervaton and calculaton process n the paper, and to know how the VIX changes.

7 Chapter. he General Informaton About VIX and Optons waps. he ypes of Volatlty.. Realzed volatlty and he hstorcal volatlty In order to apply most fnancal models n practce, t s necessary to be able to use emprcal data to measure the degree of varablty of asset prces or market ndexes. uppose that t s the prce of an asset or a market ndex at tme t. he realzed volatlty of ths asset n a perod t,t based on n+ daly observatons,,, s defned by the formula σ 5 n r n Where r ln,,,3, n, n. And 5 s an annualzaton factor correspondng to the typcal number of tradng days n a year [4]. mlarly the hstorcal volatlty s defned by a smlar formula: ~σ 5 n n r r Where r n n r, s the mean return. If the returns are supposed to be drawn ndependently from the same probablty dstrbuton, then r s the sample mean and the hstorcal volatlty s smply the annualzed sample standard devaton. he realzed volatlty s then the annualzed sample second moment. Note that n r r n r nr, and hence 3

8 ~ 5nr σ σ n hs means that the realzed volatlty s approxmately equal to the hstorcal volatlty f the sample mean s very close to zero. he quanttes and are called respectvely the realzed varance and the hstorcal varance. V σ σ In both cases the factor 5 annualzes the result. In general, f a tme nterval between two observatons s t expressed n years, then the annualzaton factor s / t.. In our case t /5. Both the realzed volatlty and the hstorcal volatlty measure varablty of exstng fnancal data and n the fnancal context n many cases gve smlar results. Both types of volatlty can be used as predctors of future volatlty. Also t s sometmes mportant to try to forecast ther future values... he mpled volatlty he mpled volatlty s always about market belefs about the future volatlty when the opton nvestors traded the opton, and ths awareness has been reflected n the opton prcng process. In theory, t s not dffcult to obtan the mpled volatlty value. In the Black-choles opton prcng model there s the fve basc parameters, X, r, t t, and σ quanttatvely related to the opton prce, as long as the frst four basc parameters and the actual market opton prce are known n the opton prcng model, the only unknown parameter σ wll be solved, whch s the mpled volatlty. herefore, the mpled volatlty also can be regarded as an expectaton of the actual market volatlty [4]..3 he volatlty ndex he volatlty ndex s a weghted average of mpled volatltes for optons on a partcular ndex. As we can calculate a stock's volatlty or the mpled volatlty based on ts optons, we aslo can calculate the volatlty for an ndex such as the &P 5. hs concept can be taken one step further. A volatlty ndex has been orgnated and s usually quoted n the fnancal meda for many ndces,.he followng s three most common volatlty ndex: &P Volatlty Index VXO 4

9 &P 5 Volatlty Index VIX_ Nasdaq Volatlty Index VXN he above volatlty ndexes are a weghted average of the mpled volatltes for several seres of puts and calls optons. Many market partcpants and observers wll look these ndexes as an ascertanment of market sentment. here are many nterestng and mportant nformaton about the VIX, other volatlty ndexes and related products on the CBOE Web ste []...4 ntraday volatlty he ntraday volatlty s the prce change n a stock or ndex on or durng a defned tradng day. We can also say that t shows the market swngs s the most notceable and readly avalable defnton of volatlty durng the lfe of a tradng day. Intraday volatlty s the Justce Potter tewart type of volatlty, as t s dffcult to defne but you know t s the ntraday volatlty when you see t. A general mstake s people thnk the ntraday volatlty s equated wth the mpled volatlty ndex. Both of these types of volatlty are not nterchangeable, but do carry ther own partcular mportance on a certan extent n measurng nvestor sentment and expectatons [].. Prncples of Volatlty Index he mpled volatlty s a core data for calculatng volatlty ndex VIX whch s calculated through the latest deals prce n the optons market. It s reflected the nvestors expectatons of future market prces. he concept s smlar to bond yeld to maturty Yeld o Maturty: As the market prce changes, through the approprate nterest rate, the bond prncpal and coupon nterest dscount to the present value, then when the present value of bond s equal to the dscount rate of the market prce, t s the bonds yeld to maturty, that s, the mpled return rate of bonds. In the calculaton process based on the bond evaluaton model, the yeld to maturty can be obtaned accordng to the market prce, whch s the mpled yeld to maturty. here are many ways to estmate the mpled volatlty. rstly, you must determne the optons evaluaton model, the requred other parameters values and the present opton prce, when computng the mpled volatlty of the optons. or example, n Black-choles opton 5

10 prcng model 973, we can get the opton theoretcal prce, as long as theunderlyng prce, strke prce, rsk-free nterest rate, tme to maturty, the volatlty of stock returns and other data are put nto the opton prce model formula. If the underlyng assets and the opton market are effcent, the opton theoretcal prce has fully reflected ts true opton value, and at the same tme the opton prce model s also correct, then we can get the mpled volatlty, wth takng the opton market prce nto the Black-choles opton model based on the concept of the nverse functon. nce t shows nvestors expectatons of changes n future market prces, so t s called mpled volatlty. CBOE launched the frst VIX ndex VXO n 993, whch s based on the opton model whch s proposed by Black, choles and Merton. In ths model, except the volatlty, we also requre other many parameters such as the current stock prce, the opton prce, the strke prce, duraton, rsk-free nterest rate, tme to expected payment of cash dvdends and amount of expected payment of cash dvdends. However, the &P optons of CBOE are Amercan optons and are related to the cash dvdends of the underlyng stocks, so when CBOE calculated the VIX ndex, they used the mpled opton volatlty of the bnomal model proposed by Cox, Ross and Rubnsten 797 [8]. In 3, CBOE launched another new VIX ndex, compared to the old ndex VXO, the new VIX ndex s more accurate. However, the old VIX ndex s stll sustaned released, so n order to dstngushng easly between the old and new VIX ndex, the old VIX ndex was renamed as VXO ndex..3 Varance waps Among many volatlty products, the varance swaps have become wdely used. he varance swaps, compared to the volatlty swaps, have more convenent mathematcal characterstcs. In short, a varance swap s a forward contract whose payoff s based on the dfference between the realzed varance of the underlyng asset durng the lfetme of the contract and the value of var, whch s the varance delvery prce n the contract. he payment to maturty can be expressed as payoff σ R M var 6

11 Where M s the notonal amount of the swap n dollars per annualzed volatlty pont squared [3], and σ s the actual varance, that s the square of realzed volatlty [4]. R σ R 5 N ln N hrough a seres of bref ntroductons of varances products and the above explanaton about the varance swaps, t s not dffcult to fnd many varance products are more smlar wth specal futures contracts, whch means the goods varance must be exchanged n accordng to the varance delvery prce. Because these products we are dscussng have the smlar propertes wth futures optons and forward contracts, compared to the value of the contract tself, we pay more attenton to the varance delvery prce var n the contract. However, t s drectly related to the volatlty arbtrage or the hedgng effect of ths contract product on expraton. o at present many research reports and lteratures are nvolved t. In the early perod of volatlty tradng market, a normal way, whch s the statstcal arbtrage method, s used to compute the dfference between the actual varance and the the varance delvery prce. As the development of the varance research, a standardzed method gradually appeared whch s buldng a mathematcal model to determne a more reasonable and accurate exchange value of var. Now, there are two methods to solve t: one s only through bonds and a combnaton of a varety of standard European optons to smulate a portfolo of varance nvestment products, not through a accurate stochastc model, whch s based on the paper named as owards a heory of Volatlty radng publshed by Peter Carr and Dlp Madan n 998 [3]. he other way s to calculate the value based on a defnte stochastc volatlty model, drectly regardng the volatlty as the trade standard, through the GARCH prce model. 7

12 Chapter 3. EstmatngVolatlty rom Opton Data [][4] We can assume that the stock prce satsfes the followng stochastc dfferental equaton d r q dt + σ dz Where r, q, σ are constant denotng the rsk free rate, the contnuous dvded yeld and volatlty respectvely, and z s a Wener process wth respect to a rsk-neutral probablty measure. 3. At the same tme, from the Ito's lemma σ d ln r q dt + σ dz hen from the equatons 3.-3., we can obtan d σ dt d ln 3. σ σ d ln r q dt + σ dz r q dt σ dz dt d By ntegratng from tme to tme, t can be got And so σ dt σ σ d d ln d ln ln d ln Because the varance s the square of the volatlty V σ, 3.3 we get or V V d ln d ln 3.4 8

13 9 hen take the expectatons of the equaton 3.4 wth respect to the rsk-neutral probablty measure 3.5 ln E d E V E Because we can take the expectaton of the equaton 3., the followng equaton can be got q r d E dz E dt q r E d E + σ o puttng the above equaton nto 3.5, we get ln E q r V E We know that X exp and the random s Gaussan wth hen q r e X Var X E E + ] [ ] [ exp ] [ herefore 3.6 ln ln E V E Where, whch s the forward prce of the asset for a contract maturng at tme. Now, frstly we consder d, max Where s some value of.. If, max d X Var q r X E ] [ and ] [ σ σ z q r X σ σ + E < *

14 . If econdly, we consder next. If,. If, * > * > * < hen we can get that or all value of so that max max, d ln + ln * + * max -, d * max -, d * -, d * * ln ln ln + - d + + ln d max, + max -, ln + d d ln max, d max -, d 3.7 We can take the expectatons wth respect to the rsk-neutral probablty measure n equaton 3.7 Eln E E max, d E max -, d

15 Note that E R [ max,] e p and then we get E R [ max, ] e c E ln R R E e p d e c d Where c and p are the prces of the call and put opton wth the strke prce, s the maturty tme and R s the rsk-free nterest rate for a maturty of. 3.8 Accordng to ln ln ln ln ln ln + ln + ln ln akng the expectaton of the above the equaton, we can get Eln Eln ln + Eln + Eln 3.9 Now combnng the equatons 3.6, 3.8 and 3.9,

16 and Δ Δ n n n d c e d p e d c e d p e d c e d p e d c e d p e E E E V E R R R R R R R R ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln In concluson, we get the 3., the expected value of the average varance from tme to tme, 3. ln + + d c e d p e V E R R where s the forward prce of the optons for a contract maturng tme, c s the prces of the call opton wth the strke prce and tme to maturty, p s the prces of the put opton wth the strke prce and tme to maturty, R s the rsk-free nterest rate for a maturty of, and s some value of []. We can assume that the prce of the optons wth strke prces n are known, and and we set equal to the frst strke prce below, then approxmate the ntegrals as 3. n r R R Q e d c e d p e Δ + Where n and Δ + n partcular, the functon Q s the prce of a put opton wth strke prce. If, the functon Q s the prce of a call opton wth strke prce. If, n n < < < < < 3 L * < * > *

17 3 V E If, the functon Q s the average of the prce of a call and a put opton wth strke prce Puttng the equaton 3. nto equaton 3., we can get Δ + ln n r Q e E V or 3. ln n Δ + r Q e E V Accordng to ln N and ts the followng Maclaurn polynomal ln + N N N N ο We can get ln + ο In the functon t can be approxmated as follow ln or 3.3 ln We can use equaton 3.3 to replace the part of equaton ln n n Δ + Δ + r r Q e Q e V E he process used on any gven day s to calculate for optons that trade n the market and have maturtes mmedately above and below 3 days. he 3-day rsk-neutral expected cumulatve varance s calculated from these two numbers usng nterpolaton. hs s then multpled by 365/3 and the ndex s set equal to the square root of the result. However, we can get the formula used n the above calculaton s

18 4 Δ + Δ + R R Q e Q e 3.5 σ σ

19 Chapter 4. Calculaton of VIX In eptember, 3, CBOE launched the new VIX volatlty ndex, whch s calculated on the bass to the &P 5 optons, whle there are also many mprovements n the algorthm, and the ndex s much closer to the actual market stuaton. VIX s a volatlty ndex of optons rather than stocks, and each opton prce shows the market s expectaton of the future volatlty []. mlar to the conventonal ndexes, VIX also chooses the component opton and a formula to compute ndex values. he new VIX used the varance and volatlty swaps varance & volatlty swaps method to update the calculaton formula, whch reflects better the overall market dynamcs, and ts formula s as follows: VIX σ 4. Where [] me to expraton orward ndex level derved from ndex opton prces rst strke below the forward ndex level, trke prce of th out-of the-money opton; a call f > and a put f < l both put and call Δ Interval between strke prces half the dfference between the strke on ether sde of : Δ + Note: for the lower 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. R Rsk-free nterest rate to expraton 5

20 Q he mdpont of the bd-ask spread for each opton wth strke 4. Parameter And elect he Optons o Calculate VIX 4.. Parameter : VIX s used to measure the 3 days expected volatlty of the &P 5 Index, whch s dvded to near- and next term put and call opton, regularly n the frst and second &P 5 Index PX contract months. Calculaton of the tme to maturty s based on mnutes,, n calendar days and separates every day nto mnutes for replcatng the precson, whch s used by opton and volatlty traders. me to expraton s calculated as follow: mnutes from the current tme to 8 : 3 hrs on the settlement day mnutes n a year 4. Here, we assume today s uesday 5, Aprl, n the hypothetcal, the near-term and nextterm optons have days and 46 days to expraton respectvely. Near-term optons exprng Aprl 6, and next-term optons exprng May, and reflect prces observed at the open tme of tradng, 8:3 a.m. Chcago tme the source of data from CBOE. hen we can get the for the near-term and next-term optons, and respectvely, s: { 4 6} { } Notes: there must be more than one week for Near-term optons to expraton. hs s because there are prcng anomales when t s very closed to expraton. We make a rule for the Near-term optons n order to reduce these anomales. If the near-term optons would 6

21 expre after less than one week, the VIX wll roll to the second and thrd PX contract month. or example, for, June, the second rday n June, the near-term opton and nextterm opton wll expre n June and July. But f t would change to 4, June, the second Monday n June, the near-term opton and next-term opton wll expre n July and August []. 4.. Parameter R: R, the rsk-free nterest rate, s the U.. reasury yelds whch has the same maturng as the relevant PX opton. We also can say, wth dfferent maturtes, the rsk-free nterest rates of near-term optons and next-term optons are dfferent. However, PX optons are of the European type wth no dvdends so ths works. he put-call party formula can be solved for R as follows: Where t P t C t R ln t t + P C he strke prce t t he underlyng &P5 prce Put opton prce Call opton prce 4.3 Here, we can get the rsk-free nterest rate of the near-term and the next-term respectvelydata from CBOE as follow, R R 33 ln ln R R + R / Parameter : We can use the followng formula to calculate : trke Prce + e R Call Prce Put Prce 4. 4 In the formula, the Call Prce mnus Put Prce means, the absolute smallest dfference between the call and put prces for the same strke prce, then trke Prce s the above 7

22 strke prce wth the absolute smallest dfference between the call and put prces. Now, we use my computer program based on the Yahoo fnancal data to calculate the smallest dfference, then we get the absolute dfference of near-term opton and next-term opton n the followng table, Near-erm Opton trke prce Call Put Bd Ask Bd Ask Absolute Dfference able 4.:he Dfference of Near-erm Optons rom able 4., we can see. the strke prce s 33 s the smallest dfference of the near-term optons, so 33 s the related strke prce, then we get 8

23 33 + e 33 + e Next-erm Opton trke prce Call Put Bd Ask Bd Ask Absolute Dfference able 4.:he Dfference of Next-erm Optons rom able 4. we can see. s the smallest dfference of the next-term optons, so 33 s the related strke prce, then we get 9

24 33 + e 33 + e Parameter and. If there are N optons wth dfferent strke prces n the near-term &P 5 optons ncludng N call opton and N put opton, a total of N,wth the order of the N opton by strke prce, n fact, the strke prce whch s frst lower than the value s defned as the, the rest of the opton wth the order from small strke prce to large strke prce are defned as,,,, N, N. whle we defned the set φ. rom the above calculaton, we know 33., 33.4, then 33, 33.,, 4..5 elect optons It s shown VIX ndex s compled, accordng to certan crtera of all the near-term optons and next-term optons,we can flter these optons nto a standard opton set. hen the VIX ndex s calculated by the strke prces and call/put prces of the optons of set. However, now n the followng select process, I wll use the near-term optons as the example to flter the sutable optons nto set. If <, we choose the strke prce of put optons, set + In fact, selectng set starts wth the strke prce of put optons lower than and move to the followng lower strke prces, at the same tme, reject any opton of whch a put bd prce s equal to zero. If t occurs that the two consecutve strke prces have zero bd prces, the lower strke are not consdered. trke prce Bd Near-erm Opton Put Ask Include? Not Consdered

25 If Not Consdered.5 NO 5. NO.5. YE 5.5. YE YE YE 4.5. YE YE YE YE YE able 4.3:electon of Near-erm Put Optons >, we choose the strke prce of call optons, set + electng out-of- the-money call optons s the same as the above method, there s only one dfference whch s based on call optons. tart wth the call opton strke prce hgher than and move to the followng hgher strke prces, at the same tme, we must reject any opton of whch a call bd prce s equal to zero.however, f the two consecutve strke prces whch both have zero bd prces t occur, the hgher strke prce are not consdered. We wll stop to select the optons. hen the optons wth hgher strke prce are not useful. trke prce Bd Near-erm Opton Call Ask Include? YE YE 4.5. YE YE

26 4.5. YE 45. YE YE 435. NO 44. NO Not Consdered able 4.4:electon of Near-erm Call Optons If, we put both call opton and put opton of the strke prce nto set. In concluson, from the above flter process we can see, when <, only choose the put opton wth strke prce ; when >, only choose the call opton wth strke prce ; when, choose both put and call opton wth strke prce Parameter If Note: If, we set Δ + s the lowest strke prce of the put optons n set, d If s the hghest strke prce of the call optons n set, u Δ d d + Δ u u d u or example, from the above able 4.3, we can see the strke prce s the lowest prce of set, and the strke prce 5 s the neghbour of the strke prce n the set, so Δ strke prce 5 5

27 4..7 Parameter s the average of quoted bd and ask opton prces, or md-quote prces. Bd prce + Ask prce Q rom our data, we can get the followng tables: Near-erm Opton trke prce Opton ype Bd Ask Md-quote Prce Put Put Put Put Put Put Put Put Put/Call Average Call Call Call Call Call Call Call Call Call Call Call able 4.5: Md-quote Prce of Near-erm Optons 3

28 Next-erm Opton trke prce Opton ype Bd Ask Md-quote Prce Put Put Put Put Put Put Put/Call Average Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call Call able 4.6: Md-quote Prce of Next-erm Optons 4

29 4. Calculate volatlty of both near-term and next-term optons rom the formula 3.5, we can get the volatlty formula of near-term and next-term opton wth tme to expraton of and respectvely. σ σ Δ Δ e e R R Q Q,, Now, we want to compute the contrbuton values of both near-term and next-next optons. or example, the contrbuton of the near-term put opton s gven by: Δ Put R e Q Put e Put he contrbuton of the next-term 45 call opton s gven by: Δ45 Call R e Q 45 Call e Call A smlar calculaton s used for every opton, then we get the followng table: Near-erm Opton Next-erm Opton Mdquotquote Md- trke Opton Contrbuton trke Opton Contrbuton prce ype by trke prce ype by trke Prce Prce Put e-7 88 Put e-6 5 Put e-7 89 Put e-6 3 Put e-7 9 Put e-6 35 Put e-7 95 Put e-7 4 Put e-7 9 Put e Put e-5 3 Put e-5 35 Put e-5 35 Put e-5 5

30 3 Put e-5 3 Put e-5 35 Put e-5 35 Put e-5 33 Put/Call Average e-5 33 Put/Call Average e Call e Call e-5 34 Call e-5 34 Call e Call e Call e-5 35 Call e-6 35 Call e Call e Call e-5 36 Call e-6 36 Call e Call e Call e-5 37 Call e-6 37 Call e Call e Call e Call e-7 43 Call e-6 4 Call e Call e-6 45 Call e-7 44 Call e-6 4 Call e Call e-7 43 Call e-7 45 Call e-7 Δ e R Q Next, we calculate Δ R.644 e Q able 4.7: Contrbuton of Near-erm and Next-erm Optons,, of both near-term and next-term optons: Now put the values of the parameters nto the formula 3.5. we get 6

31 σ Δ.43 e R Q , σ Δ R e Q , 4.3 Calculate the 3-day weghted average of both σ and σ hen take the square root of the value and multply by to get VIX, accordng to the formula 4. and formula 4.. Where [], N N N N + N σ N 3 3 VIX σ N N N N Number of mnutes to settlement of the near-term optons Number of mnutes to settlement of the next-term optons N N N 3 Number of mnutes n 3 days 3*4*643 N 365 Number of mnutes n 365-day year 365*4*6556 In the above formula, f the near-term optons have less than 3 days to expraton and the next-term optons have more than 3 days to expraton, from the value of VIX we know t have been shown an nterpolaton of σ and σ.on the other hand, f both the near-term and next-term optons have more than 3 days to expraton, the VIX would roll. he same above formula wll be used to compute the 3-day weghted average, but ths result s an extrapolaton of σ and σ. In concluson, we get the value of VIX VIX N N 3 N3 N N σ + σ N N N N N

32 o we get the fnal value of VIX of the Aprl PX optons, VIX As the same method, we can get the VIX values of other month PX optons, whch s shown n the followng table. Current day Apr 5 ettlement day Days radng days Opton R σ Aprl May able 4.8: VIX of Aprl and May VIX 6.94 Current day Apr 5 ettlement day Days radng days Opton R σ May June able 4.9: VIX of May and June VIX 5.56 Current day Apr 5 ettlement day Days radng days Opton R σ June July able 4.: VIX of June and July VIX

33 hen accordng to able 4.8,, able 4.9, and able 4. we get the followng fgure May June vx July gure 4. he VIX Index 9

34 Chapter 5. Concluson rom the above descrpton, we know that the orgnal ntenton of nvolvng VIX ndex reflects nvestors expectatons for the future of the market. However, wth the deepenng of the study, people gradually found that, except the mentoned earler pont, the VIX ndex also has a wder range of applcatons. or example, after years of emprcal testng, t was dscovered that the VIX ndex often has a negatve correlaton wth returns of the stock. However, ths s what prompted people to thnk about that the VIX ndex may be used to hedge the equty portfolo rsk, so CBOE launched the frst VIX ndex of futures contracts n 4, two years later, t launched VIX ndex optons contracts. he futures and optons contracts, are a powerful tool to prevent the rsk, and helpful to get stable returns for nvestors [8]. Now we can see the followng fgure s the comparson between VIX, &P5, and 3-day Hstorcal Volatlty from March, 5 to December. gure 5. VIX, &P5, and 3-day Hstorcal Volatlty rom the gure 5., we found that VHN and VIX ndex over the same perod showed a hgh degree of negatve correlaton wth &P5; when the VIX and VXN ndex s relatvely hgh, that s the hgher volatlty, there s a bg change on the &P5. In partcular, there s a rapd 3

35 ncrease on VIX and VHX ndex n October 8, at the same tme, &P5 decreased swftly. It s the famous event that Lehman Brothers went bankrupt n October 8. In short, f the stock declnes, usually VIX would contnue to rse. It s shown the nvestors expect the volatlty ndex wll ncrease n future. When the VIX ndex hts the hgh pont, the nvestors because of ncreasng panc, would buy a lot of put optons wthout consder, whch s a huge effect for market to quckly reach the bottom; when the ndex rses, VIX s down, then nvestors are blnd optmsm, wthout any hedgng acton, and the stock prces wll tend to reverse. we beleve that the calculaton of mpled volatlty ndex, compared to the other estmated methods, contans most nformaton, and has a good ncreasng ablty to predct the future actual volatlty. Now, t has been appled to many countres such as Indan, Chna, and many European countres. Inda has an Inda VIX, Germany has the VDAX, Chna Hongkong has the HI Volatlty Index VHI and some other countres n partcular largely n Europe have ther own country volatlty ndces. But Outsde of North Amerca and Europe, the volatlty ndex of the other countres pckngs are slm. In short, fear and anxety may be rsng slowly n U.. markets, but n the crtcal economes of Brazl, Russa, Inda, and Chna, sgns of panc are much more wdespread. In ndan, NE has sad that there s no ntenton to ntroduce tradeable products based on the Inda VIX n the mmedate future, as t s mportant that the market nvestors get used to understandng and trackng the value of the Inda VIX and what t sgnfes. he exchange sad n a release. "Once market partcpants are comfortable, Inda VIX futures and optons contracts can be ntroduced n the Indan markets, based on regulatory approvals, to enable nvestors to buy and sell volatlty and take postons based on the movement of Inda VIX." [5] In Chna Hongkong, we can say that the VHI s a measure of market perceved volatlty n drecton. Hence VHI readngs mean nvestors expect that the market wll move sgnfcantly, regardless of drecton. Real-tme dsplay of HI Volatlty Index VHI s now avalable on HEx webste homepage for publc reference. VHI measures expectatons of volatlty or fluctuatons n prce of the Hang eng Index and s publshed by Hang eng Indexes Company Lmted HIL [6]. 3

36 VHI shows the market s expectatons of stock market volatlty n the next 3-day perod. It s calculated real-tme usng prces of the Hang eng Index Optons whch s lsted on HEx. he VHI s always quoted n percentage ponts. If the VHI s hgher value, t shows that nvestors expect the value of the HI to change sharply - up, down, or both - n the next 3 days [6]. In concluson, VIX s known as 'nvestor fear gauge' and 'fear ndex', reflects nvestors' best predcton of near-term market volatlty. If emergng markets are the buffer the contnued growth of whch s supposed to buttress developed markets wth slowdown of ths economc, then emergng markets need to fnd ther own frm ground and balance anxous nvestors before they can be expected to lubrcate the wheels of the global economy. 3

37 References. []. he CBOE Volatlty Index-VIX, Whte paper, CBOE, 3. []. John Hull, Optons, utures, and Other Dervatves, 7th Edton, Pearson-Prentce Hall 9. [3]. resmr Demeterf, Emanuel Derman, Mchael amal, and Joseph Zou, More han You Ever Wanted o now About Volatlty waps, Quanttatve trateges Research Notes, March, 999. [4]. ebasten Bossu, Eva trasser and Regs Guchard, Just Want You Need o now About Varance waps, Equty Dervatves Investor, Quanttatve Research & Development, JPMorgan, ebruary, 5. [5]. Grant V. arnsworth, Econometrcs n R,October 6, 8. [6]. Allen and Harrs, VolatltyVehcles, JPMorgan Equty Dervatves trategy Product Note. [7]. Peter Carrand Luren Wu, A ale of wo Indces, he Journal Of Dervatve, prng, 6. [8]. Brenner, M., and M. ubrahmanyam, A mple ormula to Compute theimpled tandard Devaton,nancal AnalystsJournal, 44. [9]. 'Natonal tock Exchange of Inda', Last accessed on 4 October, 9. []. Hang eng ndexes to launch Hong ong's "VIX", Mon Jan 3, 4:56am E [] Understandng the our Measures of Volatlty, 3 December,7. [] Alreza Javaher, Paul Wlmott and Espen G. Haug, GARCH and Volatlty waps, Publshed on wlmott.com, January. [3] Peter Carr and Dlp Madan, owards a heory of Volatlty radng, New Estmaton echnques for Prcng Dervatves, London:Rsk Publcatons,998. [4] J.Hull, echncal Note. 33

38 [5]. How to Create Your Own Portable VXV, Wednesday, Aprl, 9. [6]. HI Volatlty Index VHI, HEx, Aprl,. 34

39 Appendx.. Codes ## Data neardata <- read.csv"c:/documents and ettngs/owner/ 桌 面 / 毕 设 / 新 数 据 /4.csv",header ALE nextdata <- read.csv"c:/documents and ettngs/owner/ 桌 面 / 毕 设 / 新 数 据 /5.csv",header ALE #neardata <- read.csv"c:/data.csv",header ALE #nextdata <- read.csv"c:/data.csv",header ALE cvx<-functonnedata,r,t { ## tep ## nd Z numodmnedata[] Z<-numercnumo for n :numo {Z[]absnedata[,]+nedata[,3]-nedata[,4]-nedata[,5]/ } zmn<-mnz #trke Prce sn #number of trke Prce for n :numo {fz[]zmn {nedata[,] sn<-} } ## Gve values to r and t #r.38 #t4/365 ##nd <-+expr*t*zmn 35

40 ##tep ##Orderng ##nd k k for n :numo {fnedata[,]< knedata[,] } k ##fnd call j j repeat { fnedata[j,]>k&&nedata[j,]! {jj+} fjnumo nedata[j+,]&&nedata[j+,] {break} jj+ } c<-matrx,j,5 j j repeat { fnedata[j,]>k&&nedata[j,]! { for n :5 { c[j,]nedata[j,] } jj+ } fnedata[j,]&&nedata[j+,] jdmnedata[] 36

41 {break} jj+ } ## Check calls c ## Puts j j repeat { fnedata[j,4]! {jj+} fnedata[j,]>k {break} jj+ } p<-matrx,j,5 j j repeat { fnedata[j,]<k&&nedata[j,4]! { for n :5 { p[j,]nedata[j,] } jj+ } fnedata[j,]>k 37

42 {break} jj+ } ##Check that f puts s full element p ##nal data fdata<-matrx,dmp[]+dmc[]-,5 for n :dmp[] { fdata[,]<-p[,] } for n :dmc[]- { fdata[+dmp[],]<-c[+,] } ##Calculate sgma ##.Calculate delta k dknumercdmfdata[] dk[]<-fdata[,]-fdata[,] dk[dmfdata[]]<-fdata[dmfdata[],]-fdata[dmfdata[]-,] for n :dmfdata[]- {dk[]fdata[+,]-fdata[-,]/ } ##.Calculate Qk qknumercdmfdata[] for n :dmp[]- {qk[]fdata[,4]+fdata[,5]/ } qk[dmp[]]fdata[dmp[],]+fdata[dmp[],3]+fdata[dmp[],4]+fdata[dm p[],5]/4 for n dmp[]+:dmfdata[] {qk[]fdata[,]+fdata[,3]/ } ##3.Calculate s equals the summaton delta-k tmes Qk over k 38

43 s for n :dmfdata[] {ss+dk[]*qk[]/fdata[,]^ } cont<-numercdmfdata[] for n :dmfdata[] {cont[]expr*t*dk[]*qk[]/fdata[,]^ } ##4.gma sgma<-*expr*t*s/t-/k-^/t E<-*expr*t*s/t lstuse.datafdata,zz,trke.prce,kk,,q.kqk,contrbutoncont,um E,sgmasgma } cvxneardata,r,/365 cvxnextdata,r,46/365 vx<-functonneardata,nextdata,r,t,t {vx*sqrtt*cvxneardata,r,t$sgma*t*365*4*6-3*4*6/t*365*4*6-t*365*4*6+t*cvxnextdata,r,t$sgma*3*4*6- t*365*4*6/t*365*4*6-t*365*4*6*365/3 vx} vxneardata,nextdata,r,/365,46/365. he Calculaton Results > r-/8/5*log33/ > r [] > r-/33/5*log33/ > r [] > rr+r/ 39

44 > r [].8763 > cvxneardata,r,/365 $Use.Data [,] [,] [,3] [,4] [,5] [,] [,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [,] [,] [,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [,] [,] [,] [3,] [4,] [5,] [6,] [7,] [8,] [9,]

45 [3,] [3,] [3,] [33,] [34,] [35,] [36,] [37,] [38,] [39,] [4,] [4,] [4,] [43,] [44,] [45,] [46,] [47,] [48,] [49,] [5,] [5,] [5,] [53,] [54,] [55,] [56,] [57,] [58,] [59,] [6,] [6,] [6,] [63,]

46 [64,] [65,] [66,] [67,] [68,] [69,] [7,] [7,] [7,] [73,] [74,] [75,] [76,] [77,] [78,] [79,] [8,] [8,] [8,] $Z [] [] [9] [8] [37] [46] [55] [64] [73] [8] [9] [] [9]

47 [8] [7] [36] [45] $trke.prce [] 33 $k [] 33 $ [] 33. $Q.k [] [] [] [3] [4] [5] [6] [7] [8] $Contrbuton [] e e e e e-7 [6] 4.586e e e e e-7 [] e e e e e-7 [6] 5.789e e e e e-7 [] e e e e e-7 [6] e e-6.539e e e-6 [3].884e e e e-6.59e-6 [36] e e e e e-6 43

48 [4].85446e e e e e-6 [46] e e e e e-6 [5] e e e-6 7.e e-6 [56] e-6.44e e e-5.94e-5 [6].68954e e e e e-5 [66].58674e e e e e-6 [7].6648e e e e e-7 [76] 3.446e e e e e-7 [8].7944e e-7 $um [].644 $sgma [].43 > cvxnextdata,r,46/365 $Use.Data [,] [,] [,3] [,4] [,5] [,] [,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [,] [,] [,] [3,] [4,] [5,]

49 [6,] [7,] [8,] [9,] [,] [,] [,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [3,] [3,] [3,] [33,] [34,] [35,] [36,] [37,] [38,] [39,] [4,] [4,] [4,] [43,] [44,] [45,] [46,] [47,] [48,] [49,]

50 [5,] [5,] [5,] [53,] [54,] [55,] [56,] [57,] [58,] [59,] [6,] [6,] [6,] [63,] [64,] [65,] [66,] [67,] [68,] [69,] [7,] [7,] [7,] [73,] [74,] [75,] [76,] [77,] [78,] [79,] [8,] [8,] [8,] [83,]

51 [84,] [85,] [86,] [87,] [88,] [89,] [9,] [9,] [9,] [93,] [94,] [95,] [96,] [97,] [98,] [99,] [,] [,] [,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [,] [,] [,] [3,] [4,] [5,] $Z 47

52 [] [] [9] [8] [37] [46] [55] [64] [73] [8] [9] [] [9] [8] [7] [36] $trke.prce [] 33 $k [] 33 $ [] 33.4 $Q.k [] [] [] [3] [4] [5] [6]

53 [7] [8] [9] [] [] $Contrbuton [].763e e e-6.734e e-6 [6] e e e e e-6 [].455e e e e-6.857e-6 [6].886e e e e e-6 [] e e e e e-6 [6] e e e e e-6 [3] 4.58e e e e e-6 [36] e e e e e-6 [4] e e e e e-6 [46] e e e e e-6 [5] e e e e e-6 [56].5367e-5.86e e-5.975e-5.67e-5 [6].35486e e e e e-5 [66].77986e e e e-5.77e-5 [7].3558e-5.453e e e-5.866e-5 [76] 3.649e e e e e-5 [8] 4.766e e e e e-5 [86] e e e e e-5 [9] e e e e e-5 [96] e e e e-5.76e-5 [].6337e-5.38e e e e-6 [6] e e e e e-6 [].4739e e-6.495e e e-7 $um []

54 $sgma [].3475 > > vx<-functonneardata,nextdata,r,t,t + {vx*sqrt t*cvxneardata,r,t$sgma*t*365*4*6-3*4*6/t*365*4*6- t*365*4*6+t*cvxnextdata,r,t$sgma*3*4*6- t*365*4*6/t*365*4*6-t*365*4*6*365/3 + vx} > vxneardata,nextdata,r,/365,46/365 []