Range Volailiy Models and Their Applicaions in Finance Ray Yeuien Chou * Insiue of Economics, Academia Sinica & Insiue of Business Managemen, Naional Chiao Tung Universiy Hengchih Chou Deparmen of Shipping & Transporaion Managemen, Naional Taiwan Ocean Universiy Nahan Liu Insiue of Finance, Naional Chiao Tung Universiy This version: January 6, 009 Absrac There has been a rapid growh of range volailiy due o he demand of empirical finance. This paper conains a review of he imporan developmen of range volailiy, including various range esimaors and range-based volailiy models. In addiion, oher alernaive models developed recenly, such as range-based mulivariae volailiy models and realized ranges, are also considered here. A las, his paper provides some relevan financial applicaions for range volailiy. Keywords: CARR, DCC, GARCH, high/low range, realized volailiy, mulivariae volailiy, volailiy. * Corresponding auhor. Conac address: Insiue of Economics, Academia Sinica, #18, Yen-Jio-Yuan Road, Sec, Nankang, Taipei, Taiwan. Telephone: 886--78791 ex 31, fax: 886--7853946, email: rchou@econ.sinica.edu.w. 1
I. Inroducion Wih he coninual developmen of new financial insrumens, here is a growing demand for heoreical and empirical knowledge of he financial volailiy. I is well-known ha financial volailiy has played such a cenral role in derivaive pricing, asse allocaion, and risk managemen. Following Barndorff-Nielsen and Shephard (003) or Andersen e al. (003), financial volailiy is a laen facor and hence is no direcly observable, however. Financial volailiy can only be esimaed using is signaure on cerain known marke price process; when he underlying process is more sophisicaed or when observed marke prices suffer from marke microsrucure noise effecs, he resuls are less clear. I is well known ha many financial ime series exhibi volailiy clusering or auocorrelaion. In incorporaing he characerisics ino he dynamic process, he generalized auoregressive condiional heeroskedasiciy (GARCH) family of models proposed by Engle (198) and Bollerslev (1986), and he sochasic volailiy (SV) models advocaed by Taylor (1986) are wo popular and useful alernaives for esimaing and modeling ime-varying condiional financial volailiy. However, as poined by Alizadeh, Brand, and Diebold (00), Brand and Diebold (006), Chou (005) and oher auhors, boh GARCH and SV models are inaccurae and inefficien, because hey are based on he closing prices, of he reference period, failing o used he informaion conens inside he reference. In oher words, he pah of he price inside he reference period is oally ignored when volailiy is esimaed by hese models. Especially in urbulen days wih drops and recoveries of he markes, he radiional close-o-close volailiy indicaes a low level while he daily price range shows correcly ha he volailiy is high.
The price range, defined as he difference beween he highes and lowes marke prices over a fixed sampling inerval, has been known for a long ime and recenly experienced renewed ineres as an esimaor of he laen volailiy. The informaion conained in he opening, highes, lowes, and closing prices of an asse is widely used in Japanese candlesick charing echniques and oher echnical indicaors (Nisson, 1991). Early applicaion of range in he field of finance can be raced o Mandelbro (1971), and he academic work on he range-based volailiy esimaor sared from he early 1980s. Several auhors, back o Parkinson (1980), developed from i several volailiy measures far more efficien han he classical reurn-based volailiy esimaors. Building on he earlier resuls of Parkinson (1980), many sudies 1 show ha one can use he price range informaion o improve volailiy esimaion. In addiion o being significanly more efficien han he squared daily reurn, Alizadeh, Brand, and Diebold (00) also demonsrae ha he condiional disribuion of he log range is approximaely Gaussian, hus grealy faciliaing maximum likelihood esimaion of sochasic volailiy models. Moreover, as poined by Alizadeh, Brand, and Diebold (00), and Brand and Diebold (006), he range-based volailiy esimaor appears robus o microsrucure noise such as bid-ask bounce. By adding microsrucure noise o he Mone Carlo simulaion, Shu and Zhang (006) also suppor ha he finding of Alizadeh, Brand, and Diebold (00), ha range esimaors are fairly robus oward microsrucure effecs. 1 See Garman and Klass (1980), Beckers(1983), Ball and Torous (1984), Wiggins (1991), Rogers and Sachell (1991), Kuniomo (199), Yang and Zhang (000), Alizadeh, Brand and Diebold (00), Brand and Diebold (006), Brand and Jones (006), Chou (005, 006), Cheung (007), Marens and van Dijk (007), Chou and Wang (007), Chou, Liu and Wu (007), and Chou and Liu (008a,b). 3
Cox and Rubinsein (1985) saed he puzzle ha despie he elegan heory and he suppor of simulaion resuls, he range-based volailiy esimaor has performed poorly in empirical sudies. Chou (005) argued ha he failure of all he range-based models in he lieraure is caused by heir ignorance of he emporal movemens of price range. Using a proper dynamic srucure for he condiional expecaion of range, he condiional auoregressive range (CARR) model, proposed by Chou (005), successfully resolves his puzzle and reains is superioriy in empirical forecasing abiliies. The in-sample and ou-of-sample volailiy forecasing using S&P 500 index daa shows ha he CARR model does provide more accurae volailiy esimaor compared wih he GARCH model. Similarly, Brand and Jones (006) formulae a model ha is analogous o Nelson s (1991) EGARCH model, bu uses he square roo of he inra-day price range in place of he absolue reurn. Boh sudies find ha he range-based volailiy esimaors offer a significan improvemen over heir reurn-based counerpars. Moreover, Chou, Liu, and Wu (007) exend CARR o a mulivariae conex using he dynamic condiional correlaion (DCC) model proposed by Engle (00a). They find ha his range-based DCC model performs beer han oher reurn-based volailiy models in forecasing covariances. This paper will also review alernaive range-based mulivariae volailiy models. Recenly, many sudies use high frequency daa o ge an unbiased and highly efficien esimaor for measuring volailiy, see Andersen e al. (003) and McAleer and Medeiors (008) for a review. The volailiy buil by non-parameric mehods is called realized volailiy, which is calculaed by he sum of non-overlapping squared reurns wihin a fixed ime inerval. Marens and van Dijk (007) replace he squared 4
reurn by he price range o ge a more efficien esimaor, namely he realized range. In heir empirical sudy, he realized range significanly improves over realized reurn volailiy. In addiion, Chrisensen and Podolskij (007) independenly develop he realized range and show ha his esimaor is consisen and relaive efficien under some specific assumpions. The reminders are laid ou as follows. Secion II inroduces he price range esimaors. Secion III describes he range-based volailiy models, including univariae and mulivariae ones. Secion IV presens he realized range. The financial applicaions of range volailiy are provided in Secion V. Finally, he conclusion is showed in Secion VI. II. The Price Range Esimaors A few price range esimaors and heir esimaion efficiency are briefly inroduced and discussed in his secion. A significan pracical advanage of he price range is ha for many asses, daily opening, highes, lowes, and closing prices are readily available. Mos daa suppliers provide daily highes/lowes as summaries of inra-day aciviy. For example, Daasream records he inraday price range for mos securiies, including equiies, currencies and commodiies, going back o 1955. Thus, range-based volailiy proxies are herefore easily calculaed. When using his record, he addiional informaion yields a grea improvemen when used in financial applicaions. Roughly speaking, knowing hese records allows us o ge closer o he real underlying process, even if we do no know he whole pah of asse prices. For an asse, le s define he following variables: O = he opening price of he h rading day, 5
C H L = he closing price of he h rading day, = he highes price of he h rading day, = he lowes price of he h rading day. The Parkinson (1980) esimaor efficiency inuiively comes from he fac ha he price range of inraday gives more informaion regarding he fuure volailiy han wo arbirary poins in his series (he closing prices). Assuming ha he asse price follows a simple diffusion model wihou a drif erm, his esimaor ˆ σ P can be wrien: 1 ˆ σ P = (ln H ln L ). (1) 4ln Bu insead of using wo daa poins, he highes and lowes prices, four daa poins, he opening, closing, highes and lowes prices, migh also give exra informaion. Garman and Klass (1980) propose several volailiy esimaors based on he knowledge of he opening, closing, highes and lowes prices. Like Parkinson (1980), hey assume he same diffusion process and propose heir esimaor ˆGS σ as: ˆGK 0.511[ln( H / L )] 0.19{ln( C / O )[ln( H ) ln( L ) ln( O )] σ = + [ln( H / O ) ln( L / O )]} 0.383[ln( C / O )]. () As menioned in Garman and Klass (1980), heir esimaor can be presened pracically as ˆ σ = 0.5[ln( H / L )] [ln 1][ln( C / O )]. GK Since he price pah canno be moniored when markes are closed, however, Wiggins (1991) finds ha he boh Parkinson esimaor and Garman-Klass esimaor 6
are sill biased downward compared o he radiional esimaor, because he observed highs and lows are smaller han he acual highs and lows. Garman and Klass (1980) and Grammaikos and Saunders (1986), neverheless, esimae he poenial bias using simulaion analysis and show ha he bias decreases wih an increasing number of ransacion. Therefore, i is relaively easy o adjus he esimaes of daily variances o eliminae he source of bias. Because Parkinson (1980) and Garman and Klass (1980) esimaors implicily assume ha log-price follows a geomeric Brownian moion wih no drif erm, furher refinemens are given by Rogers and Sachell (1991) and Kuniomo (199). Rogers and Sachell (1991) add a drif erm in he sochasic process ha can be incorporaed ino a volailiy esimaor using only daily opening, highes, lowes, and closing prices. Their esimaor ˆ σ RS can be wrien: 1 RS = n n n n n n N n= N ( H O ) ( H O ) ( C O ) ˆ σ ln ln ln ( L O ) ( L O ) ( C O ) + ln ln ln n n n n n n. (3) Rogers, Sachell, and Yoon (1994) repor ha he Rogers-Sachell esimaor yields heoreical efficiency gains compared o he Garman-Klass esimaor. They also repor ha he Rogers-Sachell esimaor appears o perform well wih changing drif and as few as 30 daily observaions. Kuniomo (199) uses he opening and closing prices o esimae a modified range corresponding o a hypohesis of a Brownian bridge of he ransformed log-price. This basically also ries o correc he highes and lowes prices for he drif erm: 7
( H L ) ˆ σ 1 ln ˆ ˆ K = n n, (4) β N p= N { } where wo esimaors Hˆ = Arg Max P [ O + ( C O ) ] + ( C O ) [ n 1, n] n i and L = Arg Min P [ O + ( C O ) ] + ( C O ) [ n 1, n] Pi i { } ˆ n n n n i n n i are denoed as P i i i he end-of-he-day drif correcion highes and lowes prices. β 6 /( Nπ ) correcion parameer. n n n i n n i N = is a Finally, Yang and Zhang (000) make furher refinemens by deriving a price range esimaor ha is unbiased, independen of any drif, and consisen in he presence of opening price jumps. Their esimaor ˆYZ σ hus can be wrien 1 ˆYZ = ln(o n /C n-1)-ln(o n /C n-1) (N-1) n= -N σ k + ln(o ˆ n /C n-1)-ln(o n /C n-1) + (1-k) σ (N-1) n= -N RS, (5) where k 0.34 = 1.34 + ( N + 1) ( N 1). The symbol X is he uncondiional mean of X, and σ RS is he Rogers-Sachell esimaor. The Yang-Zhang esimaor is simply he sum of he esimaed overnigh variance, he esimaed opening marke variance, and he Rogers and Sachell (1991) drif independen esimaor. The resuling esimaor herefore explicily incorporaes a erm for he closed marke variance. Shu and Zhang (006) invesigaes he relaive performance of he four range-based volailiy esimaors including Parkinson, Garman-Klass, Rogers-Sachell, and Yang-Zhang esimaors for S&P 500 index daa, and finds ha he price range esimaors all perform very well when an asse price follows a coninuous geomeric 8
Brownian moion. However, significan differences among various range esimaors are deeced if he asse reurn disribuion involves an opening jump or a large drif. In erm of efficiency, all previous esimaors exhibi very subsanial improvemens. Defining he efficiency measure of a volailiy esimaor ˆi σ as he esimaion variance compared wih he close-close esimaor ˆ σ, ha is: Eff ˆ ( σ i ) Var ˆ =. (6) Var ( σ ) ( ˆ σ i ) Parkinson (1980) repors a heoreical relaive efficiency gain ranging from.5 o 5, which means ha he esimaion variance is.5 o 5 imes lower. Garman and Klass (1980) repor ha heir esimaor has an efficiency of 7.4; while he Yang and Zhang (000) and Kuniomo (199) variance esimaors resul in a heoreical efficiency gain of, respecively, 7.3 and 10. III. The Range-based Volailiy Models This secion provides a brief overview of he models used o forecas range-based volailiy. In wha follows, he models are presened in increasing order of complexiy. For an asse, he range of he log-prices is defined as he difference beween he daily highes and lowes prices in a logarihm ype. I can be denoed by: R = ln( H ) ln( L ). (7) According o Chrisoffersen (00), for he S&P500 daa he auocorrelaions of he range-based volailiy, R, show more persisence han he squared-reurn auocorrelaions. Thus, range-based volailiy esimaor of course could be used 9
insead of he squared reurn for evaluaing he forecass from volailiy models, and wih he ime series of R, one can easily consrucs a volailiy model under he radiional auoregressive framework. Insead of using he daa of range, neverheless, Alizadeh, Brand, and Diebold (00) focus on he variable of he log range, ln( R ), since hey find ha in many applied siuaions, he log range approximaely follows a normal disribuion. Therefore, all he models inroduced in he secion excep for Chou s CARR model are esimaed and forecased using he log range. The following range-based volailiy models are firs inroduced in some simpe specificaions, including random walk, moving average (MA), exponenially weighing moving average (EWMA), and auoregressive (AR) models. Hanke and Wichern (005) hink hese models are fairly basic echniques in he applied forecasing lieraure. Addiionally, we also provide some models a a much higher degree of complexiy, such as he sochasic volailiy (SV), CARR and range-based mulivariae volailiy models. The Random Walk Model The log range ln( R ) can be viewed as a random walk. I means ha he bes forecas of he nex period s log range is his period s esimae of log range. As in mos papers, he random walk model is used as he benchmark for he purpose of comparison. E[ln( R + 1) I ] = ln( R ), (8) where I is he informaion se a ime. The esimaor 1 E[ln( R + ) I ] is obained 10
condiional on I. The MA Model MA mehods are widely used in ime series forecasing. In mos cases, a moving average of lengh N where N= 0, 60, 10 days is used o generae log range forecass. Choosing hese lenghs is fairly sandard because hese values of N correspond o one monh, hree monhs and six monhs of rading days respecively. The expression for he N day moving average is shown below: N 1 1 E[ln( R ) I ] = ln(r ). (9) + 1 j N j= 0 The EWMA Model EWMA models are also very widely used in applied forecasing. In EWMA models, he curren forecas of log range is calculaed as he weighed average of he one period pas value of log range and he one period pas forecas of log range. This specificaion is appropriae provided he underlying log range series has no rend. E[ln(R ) I ] = λe[ln(r ) I ] + (1 λ)ln(r ). (10) + 1 1 The smoohing parameer, λ, lies beween zero and uniy. If λ is zero hen he EWMA model is he same as a random walk. If λ is one hen he EWMA model places all of he weigh on he pas forecas. In he esimaion process he opimal value of λ was chosen based on he roo mean squared error crieria. The opimal λ is he one ha records he lowes MSE. The AR Model 11
This model uses an auoregressive process o model log range. There are n lagged values of pas log range o be used as drivers o make a one period ahead forecas. n E[ln(R )] β β ln(r ) = +. (11) + 1 0 i + 1 i i= 1 The Discree-ime Range-based SV Model Alizadeh, Brand, and Diebold (00) presen a formal derivaion of he discree ime SV model from he coninuous ime SV model. The condiional disribuion of log range is approximaely Gaussian: + ρ β, (1) ln R + 1 ln R ~ N[ln R (ln R 1 ln R), ] where = T N, T is he sample period and N is he number of inervals. The parameer β models he volailiy of he laen volailiy. Following Harvey, Ruiz, and Shephard (1994), a linear sae space sysem including he sae equaion and he signal equaion can be wrien: ( ) ln R = ln R + ρ ln R ln R + β υ. (13) ( i+ 1) i ( i+ 1) ln f ( s ) = γ ln R + E ln f ( s ) + ε * i,( i+ 1) i i,( i+ 1) ( i+ 1). (14) Equaion (13) is he sae equaion and Equaion (14) is he signal equaion. In Equaion (14), E is he mahemaical expecaion operaor. The sae equaion errors are i.i.d. N(0,1) and he signal equaion errors have zero mean. A wo-facor model can be represened by he following sae equaion. ln R = ln R + ln R + ln R. ( i+ 1) 1,( i+ 1),( i+ 1) ln R = ρ ln R + β υ. (15) 1,( i+ 1) 1, 1, i 1 1,( i+ 1) 1
ln R = ρ ln R + β υ.,( i+ 1),, i,( i+ 1) The error erms υ 1 and υ are conemporaneously and serially independen N(0,1) random variables. They esimae and compare boh one-facor and wo-facor laen volailiy models for currency fuures prices and find ha he wo-facor model shows more desirable regression diagnosics. The Range-based EGARCH model Brand and Jones (006) incorporae he range informaion ino he EGARCH model, named by he range-based EGARCH model. The model significanly improves boh in-sample and ou-of-sample volailiy forecass. The daily log range and log reurns are defined as he followings: ln( R ) I 1 ~ N(0.43 ln h,0.9 ) +, r I 1 ~ N(0, h ), (16) where h is he condiional volailiy of he daily log reurn r. Then, he range-based EGARCH for he daily volailiy can be expressed by: ln h ln h = κ( θ ln h ) + φ X + δ r / h, (17) R 1 1 1 1 1 where θ is denoed as he long-run mean of he volailiy process, and κ is denoed as he speed of mean revering. The coefficien δ decides he asymmeric effec of lagged reurns. The innovaion, ln( R ) 0.43 ln h =, (18) 0.9 R 1 1 X 1 is defined as he sandardized deviaion of he log range from is expeced value. I means φ is used o measure he sensiiviy o he lagged log ranges. In shor, he range-based EGARCH model is jus replaced he innovaion erm of he modified EGARCH by he sandardized log range. 13
The CARR Model This secion provides a brief overview of he CARR model used o forecas range-based volailiy. The CARR model is also a special case of he muliplicaive error model (MEM) of Engle (00b). Insead of modeling he log range, Chou (005) focuses he process of he price range direcly. Wih he ime series daa of price range R, Chou (005) presens he CARR model of order (p,q), or CARR (p,q) is shown as R = λ ε, ε ~ f (.), p λ = ω + α R + β λ i i j j i= 1 j= 1 q, (19) where λ is he condiional mean of he range based on all informaion up o ime, and he disribuion of he disurbance erm ε, or he normalized range, is assumed o have a densiy funcion f (.) wih a uni mean. Since ε is posiively valued given ha boh he price range R and is expeced value λ are posiively valued, a naural choice for he disribuion is he exponenial disribuion. The equaion of he condiional expecaion of range can be easily exended o incorporae oher explanaory variables, such as rading volume, ime-o-mauriy, lagged reurn. p i= 1 i i q j = 1 j j L λ = ω + α R + β λ + l X. (0) k k = 1 k This model is called he CARR model wih exogenous variables, or CARRX model. The CARR mode essenially belongs o a symmeric model. In order o describe he leverage effec of financial ime series, Chou (006) divides he whole 14
price range ino wo single-side price ranges, upward range and downward range. Furher, he defines UPR, he upward range, and DNR, he downward range as he differences beween he daily highs, daily lows and he opening price respecively, a ime. In oher words, UPR DNR = ln( H ) ln( O ), (1) = ln( O ) ln( L ). () Similariy, wih he ime series of single-side price range, UPR or DNR, Chou (006) exends he CARR model o Asymmeric CARR (ACARR) model. In volailiy forecasing, he asymmeric model also performs beer han he symmeric model. The Range-based DCC model The mulivariae volailiy models have been exensively researched in recen sudies. They provide relevan financial applicaions in various areas, such as asse allocaion, hedging and risk managemen. Lauren and Rombous (006) offer a review of he mulivariae volailiy models. As o he exension of he univariae range models, Fernandes, Moa and Rocha (005) propose one kind of mulivariae CARR model using he formula Cov(X,Y)=[V(X+Y)-V(X)-V(Y)] /. Analogous o he Fernandes, Moa and Rocha s (005) work, Brand and Diebold (006) use no-arbirage condiions o build he covariances in erms of variances. However, his kind of mehod can subsanially apply o a bivariae case. Chou, Liu, and Wu (007) combine he CARR model wih he DCC model of Engle (00a) o propose a range-based volailiy model, which uses he ranges o 15
replace he GARCH volailiies in he firs sep of DCC. They conclude ha he range-based DCC model performs beer han oher reurn-based models (MA100, EWMA, CCC, reurn-based DCC, and diagonal BEKK) hrough he saisical measures, RMSE and MAE based on four benchmarks of implied and realized covariance. The DCC model is a wo-sep forecasing model which esimaes univariae GARCH models for each asse and hen calculaes is ime-varying correlaion by using he ransformed sandardized residuals from he firs sep. The relaed discussions abou he DCC model can be found in Engle and Sheppard,001, Engle (00a), and Cappiello, Engle and Sheppard (006). I can be viewed as a generalizaion of he consan condiional correlaion (CCC) model proposed by Bollerslev (1990). The condiional covariance marix H of a k 1 reurn vecor r in CCC ( r Ω 1 ~ N(0, H ) ) can be expressed as H = D RD, (3) where D a k k diagonal marix wih ime-varying sandard deviaions h i, of he i h reurn series from GARCH on he i h diagonal. R is a sample correlaion marix of r. The DCC is formulaed as he following specificaion: H = D R D, 1 = diag{ Q} Qdiag{ Q 1 R }, (4) Q, Z = D 1 r, = S ( ιι A B) + A Z 1Z 1 + B Q 1 where ι is a vecor of ones and is he Hadamard produc of wo idenically sized 16
marices which is compued simply by elemen by elemen muliplicaion. Q and S are he condiional and uncondiional covariance marices of he sandardized residual vecor Z came from GARCH, respecively. A and B are esimaed parameer marices. Mos cases, however, se hem as scalars. In a word, DCC differs from CCC only by allowing R o be ime varying. IV. The realized range volailiy There has been much research widely invesigaed for measuring volailiy due o he use of high frequency daa. In paricular, he realized volailiy, calculaed by he sum of squared inra-day reurns, provides a more efficien esimae for volailiy. The review o realized volailiy are discussed in Andersen e al. (001), Andersen e al. (003), Barndorff-Nielsen and Shephard (003), Andersen e al. (006a,b), and McAleer and Mederos (008). Marens and van Dijk (007) and Chrisensen and Podolskij (007) replace he squared inra-day reurn by he high-low range o ge a new esimaor called realized range. Iniially, we assume ha he asse price P follows he geomeric Brownian moion: dp = µ Pd + σ Pdz, (5) where µ is he drif erm, σ is he consan volailiy, and z is a Brownian moion. There are τ equal-lengh inervals divided in a rading day. The daily realized volailiy RV a ime can be expressed by: τ = (ln, i ln, i 1), (6) i= 1 RV P P 17
where P, i is he price for he ime i on he rading day, and is he ime inerval. Then, τ is he rading ime lengh in a rading day. Moreover, he realized range RR is: 1 RR H L τ = (ln, i ln, i 1), (38) 4ln i= 1 where H, i and L, i are he highes price and he lowes price of he i h inerval on he h rading day, respecively. As menioned before, several sudies sugges improving efficiency by using he open and close prices, like Garman and Klass (1980). Furhermore, assuming ha P follows a coninuous sample pah maringale, Chrisensen and Podolskij (007) propose inegraed volailiy and show his range esimaor remains consisen in he presence of sochasic volailiy. ln P = ln P + µ ds + σ dz 0 0 s 0 s, for 0 <. (7) The obvious and imporan quesion is ha he realized range should be seriously affeced by microsrucure noise. Marens and van Dijk (007) consider a bias-adjusmen procedure, which scales he realized range by using he raio of he average level of he daily range and he average level of he realized range. They find ha he scaled realized range is more efficien han he (scaled) realized volailiy. V. The Financial Applicaions and Limiaions of he Range Volailiy The range menioned in his paper is a measure of volailiy. From he heoreical poins of view, i indeed provides a more efficien esimaor of volailiy han he 18
reurn. I is inuiively reasonable due o more informaion provided by he range daa. In addiion, he reurn volailiy neglecs he price flucuaion, especially as exising a near disance beween he closing prices of wo rading days. We can herefore conclude ha he high-low volailiy should conain some addiional informaion compared wih he close-o-close volailiy. Moreover, he range is readily available, which has low cos. Hence, mos researches relaed o volailiy may be applied on he range. Poon and Granger (003) provide exensive discussions of he applicaions of volailiies in he financial markes. The range esimaor undoubedly has some inheris shorcomings. I is well known ha he financial asse price is very volaile and is easy o be influences by insananeous informaion. In saisics, he range is very sensiive o he ouliers. Chou (005) provides an answer by using he quanile range. For example, he new range esimaor can be calculaed by he difference beween he op and he boom 5% observaions on average. In heory, many range esimaors in previous secions depend on he assumpion of coninuous-ime geomeric Brownian moion. The range esimaors derived from Parkinson (1980) and Garman and Klass (1980) require a geomeric Brownian moion wih zero drif. Rogers and Sachell (1991) allow a nonzero drif, and Yang and Zhang (000) furher allow overnigh price jumps. Moreover, only finie observaions can be used o build he range. I means he range will appear some unexpeced bias, especially for he asses wih lower liquidiy and finie ransacion volume. Garman and Klass (1980) poined ha his will produce he laer opening and early closing. They also said he difference beween he observed highs and lows will be less han 19
ha beween he acual highs and lows. I means ha he calculaed high-low esimaor should be biased downward. In addiion, Becker (1983) poined ha he highes and lowes prices may be raded by disadvanaged buyers and sellers. The range values migh herefore be less represenaive for measuring volailiy. Before he range was adaped by he dynamic srucures, however, is applicaion is very limied. Based on he SV framework, Gallan, Hsu, and Tauchen (1999) and Alizadeh, Brand, and Diebold incorporae he range ino he equilibrium asse pricing models. Chou (005) and Brand and Jones (006), on he oher hand, fill he gap beween a discree-ime dynamic model and range. Their works give a large exension for he applicaions of range volailiy. In he early sudies, Bollerslev, Chou, and Kroner (199) give good illusraions of he condiional volailiy applicaions. Based on he condiional mean-variance framework, Chou and Liu (008a) show ha he economic value of volailiy iming for he range is significan in comparison o he reurn. I means ha we can apply he range volailiy on some pracical cases. In addiion, Corrado and Truong (007) repor ha he range esimaor has similar forecasing abiliy of volailiy compared wih he implied volailiy. However, he implied volailiies are no available for many asses and he opion markes are no sufficien in many developed counries. In such cases, he range is more pracical. More recenly, Kalev and Duong (008) uilize Marens and van Dijk s (007) realized range o es he Samuelson Hypohesis for he fuures conrac. VI. Conclusion Volailiy plays a cenral role in many areas of finance. In view of he heoreical and pracical sudies, he price range provides an inuiive and efficien esimaor of 0
volailiy. In his paper, we begin our discussion by reviewing he range esimaors. There has been a dramaic increase in he number of publicaions on his work since Parkinson (1980) inroduced he high/low range. From hen on, some new ranges are considered wih opening and closing price. The new range esimaors are disribued feasible weighs o he differences among he highes, lowes, opening, and closing. Through he analysis, we can gain a beer undersanding of he naure of range. Some dynamic volailiy models combined wih range are also inroduced in his sudy. They are led ino broad applicaions in finance. Especially, he CARR model incorporaes boh he superioriy of range in forecasing volailiy and he elasiciy of he GARCH model. Moreover, he range-based DCC model, which combines CARR wih DCC, conribues o he mulivariae applicaions. This research may provide an alernaive o risk managemen and asse allocaion. A las, realized range replace he squared inra-day reurn of realized volailiy by he high-low range o ge a more efficien esimaor. Undoubedly, he range is sensiive o ouliers in saisics, and however only few researches menion his problem. I s useful and meaningful o uilize he quanile range o replace he sandard range o ge a robus measure of range. Moreover, he mulivariae works for range are sill in is infancy. Fuure research is obviously required for his opic. 1
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