Adapted Autoregressive Model and Volatility Model with Application

Transcription

1 Journal of Daa Science 11(2013), Adaped Auoregressive Model and Volailiy Model wih Applicaion Naisheng Wang 1 and Yan Lu 2 1 China Securiies Index Co., Ld and 2 Universiy of New Mexico Absrac: Price limis are applied o conrol risks in various fuures markes. In his research, we proposed an adaped auoregressive model for he observed fuures reurn by inroducing dummy variables ha represen limi moves. We also proposed a sochasic volailiy model wih dummy variables. These wo models are used o invesigae he exisence of price delayed discovery effec and volailiy spillover effec from price limis. We give an empirical sudy of he impac of price limis on copper and naural rubble fuures in Shanghai Fuures Exchange (SHFE) by using MCMC mehod. I is found ha price limis are efficien in conrolling copper fuures price, bu he rubber fuures price is disored significanly. This implies ha he effecs of price limis are significan for producs wih large flucuaion and frequen limis hi. Key words: Auoregressive model, MCMC sampling, price delayed discovery effec, price limis, sochasic volailiy model. 1. Inroducion More han wo hirds of he organized markes in he world have price limis (Hall and Kofman, 2001). Price limis describe he highes and lowes prices ha a commodiy or opion is permied o reach in a given rading session. I is an imporan regulaion o conrol risks and o inhibi excessive flucuaion of fuures price. The use of price limis in fuures markes has generaed a grea deal of discussion since he global marke crash of Afer a sudy of he U.S Treasury Bond Fuures behavior, Ma and Rao and Sears (1989a, 1989b) found ha price limis could preven he overreacion of fuures markes o he fundamenal markes informaion. Price may deviae from he rue equilibrium price afer reaching he limi. Bu he deviaion is emporary and will disappear soon. Therefore, i will Corresponding auhor.

2 656 Naisheng Wang and Yan Lu no affec he price-discovery process. Brennan (1986) and Moser (1990) poined ou ha price limis could serve o reduce he poenial reneging of he conracs. The references ha consider posiive effecs of price limis can also be found from Anderson (1984), Greewald and Sein (1991), Arak and Cook (1997) and so on. However, mos of he sudies do no suppor he heory of posiive effecs of price limis. They showed ha by prevening price from converging o he equilibrium price effecively, price limis would slow down he price-discovery process and make markes inefficien. Therefore, price limis are useless. Such sudies include Figlewsk (1984), Lehmman (1989), Fama (1989) and Kim and Rhee (1997) ec. Jiang and Konsaninidi and Skiadopoulos (2012) also proposed a novel approach o examine he effec of US and European news announcemens on he spillover of volailiy across US and European Sock markes. The effeciveness of price limis are usually examined according o price delayed discovery effec, volailiy spillover effec and rading disurbance effec. When a limi is reached, rading sops and he equilibrium price is no observed. Under hypohesis of price delayed discovery effec, price will coninue o reach equilibrium price during he following rading days. Under hypohesis of volailiy spillover effec, rading following a limi move will reflec unrealized flucuaions of ha day. Therefore he period of price flucuaion exends o a longer one. Generally, if he hypoheses of price delayed discovery effec and volailiy spillover effec are acceped, i is considered ha price limis are useless. In his paper, we proposed an auoregressive model for he observed fuures reurn by inroducing dummy variables ha represen limi moves. We also proposed a sochasic volailiy model wih dummy variables. Using hese wo models, we invesigae he exisence of price delayed discovery effec and volailiy spillover effec of copper and naural rubber fuures in SHFE. Due o he difficuly of daa collecion, we didn discuss rading disurbance effec. This paper consiss of five Chapers. Chaper 2 discusses he realizaion of price delayed discovery effec and volailiy spillover effec. Chaper 3 proposes adaped ime series models and discusses parameer esimaion and hypohesis esing. Chaper 4 presens an empirical sudy of he effecs of price limis in SHFE. Lasly, we give a conclusion in Chaper The Influence of Price Limis on Fuures Price In his secion, firs, we review fuures reurn under price limis. We decompose he rue equilibrium fuures reurn in hree pars: he observed reurn, he unrealized par of ha rading day and he unrealized equilibrium reurn carried over from he previous rading day. Based on our decomposiion, we discuss price delayed discovery effec. Then we discuss volailiy spillover effec.

3 Adjused AR Model and SV Model wih Applicaion Fuures Reurn under Price Limis In a marke wih a daily price limi, rading is permied only a prices wihin limis deermined by he selemen price of he previous day. In Chinese fuures markes, he absolue variaion is 3% around selemen price of he previous day under normal siuaion. If he bid is ouside he allowed rading range, i is considered invalid. Therefore if equilibrium price moves ouside he limis, price limis are he price we observe. The relaionship beween he observed fuures price and equilibrium price a h rading day can be described as he following: F 1 (1 + l), F = F e, F 1 (1 l), if F e F 1 (1 + l), if F 1 (1 l) < F e < F 1 (1 + l), if F e F 1 (1 l), where F e is he equilibrium price a ime, F is he observed price and l is he maximum daily limi imposed on he absolue change in fuures price wihin a rading day. In Chinese fuures markes l = 3%. The observed fuures price is equal o he rue equilibrium price only if he fuures equilibrium price falls wihin he price limis. If he fuures price we observe is equal o he limis, he rue equilibrium price mus be higher (or lower) han he price observed. The Log reurns log(f /F 1 ) is commonly used during empirical sudy. I can be wrien as he following: r u, r = ln(f e /F 1 ), r d, if F e F 1 (1 + l), if F 1 (1 l) < F e < F 1 (1 + l), if F e F 1 (1 l), where r = F /F 1, r u = ln(1 + l), and r d = ln(1 l). r u and r d are he limi-up and limi-down of fuures log reurn respecively. We can see ha he observed reurn is no necessarily equal o he rue equilibrium reurn. This is deermined by he fac wheher fuures price of he previous rading day reaches he limis or no. 2.2 Price Delayed Discovery Effec Chou and Wu (1998) decomposed he equilibrium reurn as he sum of he observed reurn and he unrealized pars from he curren rading day. They didn consider he par carried over from he previous rading day. Therefore heir conclusion is no suiable for he siuaion of consecuive limi his. In he following, we decompose he rue equilibrium fuures reurn as hree (1) (2)

4 658 Naisheng Wang and Yan Lu pars: r e = ln(f e /F 1) e e = ln(f /F 1 ) + ln(f e /F ) ln(f 1/F e 1 ) (3) Where E = ln( F / F ) denoes he unrealized par of equilibrium reurn a he h rading = r + E E 1, day because of he exisence of price limis. From (3), we see ha equilibrium reurn is equal o he sum of he observed reurn and he unrealized par of ha rading day where E = ln(f e /F ) denoes he unrealized par of equilibrium reurn a he h rading subracing day because he unrealized of he equilibrium exisence reurn of carried price over limis. from he previous From rading (3), day. we see ha equilibrium reurn is equal o he sum of he observed reurn and he unrealized Paricularly, if here is no limi move a (-1) par of ha rading day subracing he h rading day, E -1 = 0; if here is no limi move unrealized equilibrium reurn carried over from he a h previous rading day, rading E = 0. Oherwise, day. he Paricularly, unrealized par of if equilibrium here isreurn no limi will be move a ( 1) h rading day, E 1 = 0; if here is no limi move a h rading day, carried over o he nex rading day. This is called price delayed discovery effec (see E = 0. Oherwise, he unrealized par of equilibrium reurn will be carried over o he nex Figure rading 1). day. This is called price delayed discovery effec (see Figure 1). Equilibrium reurn disribuion wihou limi-up in he previous rading day Equilibrium reurn disribuion wih limiup in he previous rading day Unrealized par of equilibrium reurn from he previous rading day Equilibrium Reurn The disribuion on he righ is he fuures equilibrium reurn of he day following a limi-up. The unrealized par of equilibrium reurn from he previous rading Figure 1 Figure 1 The disribuion on he righ is he fuures equilibrium reurn of he day following a limiup. The unrealized par of equilibrium reurn from he previous rading day is carried over day is carried o he over nex rading o he day. nex rading day. If price reaches If price reaches he upper he upper limi limi a ahe he -rading h -rading day, he esimaed day, he equilibrium esimaed reurn equilibrium reurn wih price limis a ha day is: wih price limis a ha day is: ˆr e = E[r e r e > r u ], (4). rˆ e = Er [ e r e > r u ] (4) (4) is he condiional expeced reurn given he equilibrium reurn greaer han he limi-up reurn (see Figure 2).

5 (4) is he condiional expeced reurn given he equilibrium reurn greaer han he limi-up reurn (see Figure 2). Adjused AR Model and SV Model wih Applicaion 659 Equilibrium disribuion wihou price limis Condiional equilibrium reurn disribuion under limi-up Equilibrium Reurn In Figure In Figure 2, 2, beween he wo wo sraigh sraigh lines, we lines, see he we disribuion see heunder disribuion he price limis. under he price limis. The curve on he righ indicaes he condiional equilibrium reurn The curve on he righ indicaes he condiional equilibrium reurn disribuion under limi- disribuion under limi-up. The equilibrium reurn of he nex rading day is: up. Figure 2 2 ˆr +1 e r e = E[r+1 r e e > r u ]. (5) The equilibrium reurn of he nex rading day is: Suppose equilibrium reurn follows he simple sochasic model: e e e u rˆ + 1 e = Er [ 1 r r ] r + >. (5) r e = µ + ε, (6) Suppose equilibrium reurn follows he simple sochasic model: where ε is a sequence of independen, idenically disribued, random variables r = µ + ε, (6) e wih mean 0 and variance σ 2. By (4) and (5) we obain (7): Where ε is a sequence of independen, idenically disribued, random variables wih mean E[r e +1 r e > r u ] = 2µ + φ(ξ )/[1 Φ(Ξ )] r u 2 0 and variance σ. By (4) and (5) we obain (7): µ = E[r e +1], (7) where Ξ = (r u µ)/σ. (7) indicaes ha fuures price ends o go up following a limi-up day. On average, equilibrium reurn is greaer han he equilibrium reurn wihou he enforcemen of price limis. This is due o he price delayed discovery effec. Similarly, he fuures price following a limi-down day ends o go down. Expeced equilibrium reurn is lower han he expeced equilibrium reurn wihou price limis. The same conclusion can be derived even we replace (6) by a more complicaed model such as auoregressive processes. Trading sops when price reaches he limis. Under hypohesis of price delayed discovery effec, fuures price will coninue o go up or fall o reach equilibrium price in he following rading days, a leas he nex rading day.

6 660 Naisheng Wang and Yan Lu 2.3 Volailiy Spillover Effec Consider model (6) again. When price reaches he upper limi a he h day, variance of condiional equilibrium reurn of he nex rading day is given as he following: Var[r e +1 r e > r u ] = σ σ 2 { 1 φ(ξ [ ]} ) φ(ξ ) 1 Φ(Ξ ) 1 Φ(Ξ ) Ξ σ 2 = Var[r e ]. (8) The resul is similar even wih a more complicaed model. Because of he high volailiy a he limi-hi-day, large flucuaion will coninue in he following days. This indicaes ha fuures reurn sequence is volailiy clusering. Price flucuaion of he nex rading day following a limi-hi-day will be larger han he flucuaion wihou price limis. This is called volailiy spillover effec. Volailiy spillover effec and price delayed discovery effec is highly relaed. The unrealized par of he equilibrium reurn because of price limis will deliver o he nex rading day. This par of reurn is uncerain. Consequenly, he uncerainy of reurn of he following rading day will increase, resuling in flucuaion increscen. Price limis impede price flucuaion ha should be compleed wihin one rading day. This resuls in a longer period of price flucuaion and volailiy spillover. As a resul, price limis could preven price slump and price jump, could reduce risk and could make markes more raional. 3. Tesing Hypohesis and Mehods In his secion, we discuss how o es price delayed discovery effec and volailiy spillover effec. In lieraure, many researchers adop case sudy. They invesigae he effec of price limis by comparing fuures reurn and flucuaion before he limi move o hose afer he limi move. These mehods work well for sock markes, where a lo of limi moves are observed and differen socks can be sudied ogeher. However, i is no efficien in fuures markes, where a single fuures produc is unlikely o have los of limi moves and differen fuures are no supposed o sudy ogeher because of differen feaures. For example, here are only abou 20 limi moves of copper fuures in SHFE wihin four years ( ). A fuures produc has several conracs a he same ime, bu hey may have similar behavior because of arbirage. If we consider hese conracs ogeher, i is acually duplicae compuaions on he same price. Furhermore, case sudies are hard o deal wih he phenomena of consecuivelimi moves. Therefore, we propose a means model for he observed fuures reurn wih inroducing dummy variables ha represen limi moves. We also propose

7 Adjused AR Model and SV Model wih Applicaion 661 a sochasic volailiy model wih dummy variables. MCMC sampling mehod is used o esimae he parameers and es hypohesis. 3.1 Adaped Auoregressive Model Generally we assume ha he equilibrium reurn sequence is an auoregressive process (AR(1)), i.e., r e = µ + ϕr e 1 + ε, where, (µ, ϕ) are parameers, ε is he error erm wih mean zero. MCMC mehod and EM algorihm can be used o analyze he above model (Wei, 2002). There are some oher researchers who consider dummy variables ha reflec price limis in he above model (Chou and Wu, 1998). We believe he rue equilibrium reurn is inheren, which would no be affeced by he exisence of price limis. Wha has been affeced is he observed reurn. As a resul, in order o examine if price limis have impac on price delayed discovery effec, i is no appropriae o inroduce dummy variable in equilibrium reurn model. We need o consruc a model for he observed reurn and inroduce dummy variable. The model we proposed is as follows: r = µ + ϕr 1 + γδ u 1 + κδ d 1 + ε, ε N (0, σ 2 ), (9) where σ is volailiy; δ 1 u and δd 1 are he dummy variables reflecing wheher here is a limi up or a limi down respecively. They are given as follows: δ u 1 = { 1, r 1 = r u, 0, r 1 < r u, δ d 1 = { 1, r 1 = r d, 0, r 1 > r d. Following a limi up day, price reurn will move from µ + ϕr 1 o µ + ϕr 1 + γ; following a limi down day, price reurn will move from µ+ϕr 1 o µ+ϕr 1 +k. The null hypohesis and alernaive hypohesis are given below: H 1 0 : γ = 0, κ = 0 vs H 1 1 : γ 0, κ 0. (10) Wheher γ equals zero or no indicaes wheher limi up has impac on price delayed discovery effec or no. Similarly, wheher k equals zero or no indicaes wheher limi down has impac on price delayed discovery effec or no. 3.2 Volailiy Model When sudying price volailiy, Chou and Wu (1998) assume ha volailiy is a consan when here are no limi moves. Alhough volailiy spillover is one of he reasons of volailiy clusering and oher heeroscedasic characerisics

8 662 Naisheng Wang and Yan Lu indicaed from he financial asse reurn sequence, here exiss volailiy clusering phenomena in some fuures markes wihou price limis. So i is no ok o consider heeroscedasic characerisics as he resul of price limis. Therefore we use heeroscedasic Volailiy Model in our sudy. The popular Volailiy Models are GARCH (Engle, 1982) model and Sochasic Volailiy model (SV model) (Taylor, 1986). These models can accuraely describe he special characerisic of he financial daa ha has condiional flucuaion depending on ime. Comparing o GARCH model, SV model has he following advanages: (1) SV model considers hisorical daa and he newes informaion, which makes he predicion more accepable; (2) SV model can easily reflecs leverage of differen ypes of informaion. Alhough GARCH model can also reach his goal afer revising, i inroduces more parameers; (3) SV model is more flexible comparing o GARCH model wih fewer resrains on parameer esimaion and esing hypohesis. In addiion, SV model can show he fa ail characerisic of asse reurn sequence. Wih he recen developmen of MCMC compuing echniques, parameer esimaion of SV model is no longer a problem. Therefore, we use SV model o be fuures reurn volailiy model. Generally, suppose volailiy follows he SV(1) process: σ = exp(h /2), h = ω + ψ(h 1 ω) + η, η = N (0, τ 2 ), where ψ ( 1, 1), η and ε are muually independen. Sochasic volailiy model considers he logarihm of volailiy as an auoregressive process, which is ime-dependen. Auoregressive coefficien ψ shows susained saus of flucuaion. η denoes he poenial informaion from rading day 1 o rading day. In order o invesigae he impac on fuures price volailiy due o price limis, we inroduce a dummy variable ha represens wheher here is a limi move or no. The above model becomes: h = ω + ψ(h 1 ω) + ζδ u 1 + λδ d 1 + η. (11) The coefficiens of dummy variables show he influence of price limis on volailiy. ζ > 0 (λ > 0) indicaes ha volailiy increases because of limi move, resuling volailiy spillover. ζ < 0 (λ < 0) indicaes ha volailiy decreases afer limi move. Oherwise, ζ = 0 (λ = 0) indicaes no influence on volailiy from price limis. The null hypohesis and alernaive hypohesis are as he following: H 2 0 : ζ = 0, λ = 0 vs H 2 1 : ζ 0, λ 0. (12) If H0 2 is rejeced, we conclude ha ζ and λ differ from 0 significanly and here exiss volailiy spillover effec.

9 Adjused AR Model and SV Model wih Applicaion Parameer Esimaion and Tesing Mehods Parameer esimaion is he major problem ha resrics he applicaion of SV model. Mehods such as leas square esimaion, maximum likelihood esimaion and oher mehods do no work. Curren popular mehods for parameer esimaion of SV model include pseudo likelihood mehod, generalized momen mehod and MCMC sampling mehod. All of hese mehods require a lo of compuaion. MCMC sampling mehod is one of he efficien ways o esimae he parameers of SV model. The basic idea of MCMC is from Bayes mehod. MCMC sampling mehod considers he parameers as random variables, assigns parameers cerain prior disribuions and calculaes heir poserior disribuions. Random numbers are drawn from condiional poserior disribuion o esimae he parameers. Le ξ be a vecor of random variables wih (µ, ϕ, γ, κ, ω, ψ, ζ, λ, τ). We se up he prior disribuion as he following non-informaive improper prior: where π 0 (µ, ϕ, γ, κ, ω, ψ, ζ, λ, τ) 1, µ (, + ), ϕ ( 1, 1), γ (, + ), κ (, + ), ω (, + ), ψ ( 1, 1), ζ (, + ), λ (, + ), τ (0, ). To ensure model sabiliy of (9) and (11), we resric ϕ and ψ beween 1 and 1. Le r = {r, = 1,, T } be he reurn vecor and h = {h, = 1,, T } be he log volailiy vecor. Based on (9), we can ge he likelihood funcion of he reurn sequence {r } as he following: L(ξ, h r) = T p(r r 1, h, ξ)p(h h 1, ξ). =1 Under he condiion ha he prior disribuion is non-informaive, he join poserior disribuion funcion of he parameers and h is π(ξ, h r) = L(ξ, h r)/ L(ξ, h r)dr. (13) Therefore, Bayes esimaor of he parameers are given by: ˆξ = ξπ(ξ, h r)dhdξ. Noice he dimension of he vecor h is he same as he number of he observed daa, calculaing he above inegral direcly is impossible. MCMC sampling

10 664 Naisheng Wang and Yan Lu mehod, Berger (1985) is a recenly developed efficien Bayes compuaional mehod. Is basic idea is o obain a sample π(ξ, h r) by consrucing a Markov Chain wih saionary disribuion π(ξ, h r). Various saisical inferences are hen derived based on hese samples and large number heory. The mos popular MCMC sampling mehod is Gibbs sampling. Le θ = (µ, ϕ, κ, γ, ω, ψ, ζ, λ, τ, h), following is he basic seps: Sep 1 : Calculae he saionary disribuion π(θ i θ i, r), where θ i denoes he vecor wihou θ i. Sep 2 : Choose a saring poin θ (0) such ha θ (0) = (θ (0) 1,, θ(0) k ), where k is he lengh of vecor θ; Sep 3 : Draw θ (j) i from saionary disribuion π(θ i θ (j 1) i, r)(i = 1, 2,, k); Sep 4 : Repea Sep 3, sampling N imes afer convergence; Sep 5 : Use he N samples afer convergence o perform saisical inference. Based on he sample obained by MCMC mehod, we can esimae he parameers of model (9) and (11) and perform hypohesis esing and oher saisical inference. In he laer empirical sudy, besides esimaing he parameers, we also calculae he highes poserior densiy (HPD) inerval of he coefficiens of dummy variables a cerain significance level. In erms of HPD inerval (Berger, 1985), we use observed daa and he resuled poserior disribuion o find a minimum inerval wihin he domain of parameers such ha he probabiliy of he parameers in he inerval reaches he required confidence level. If he HPD inerval includes null hypoheses H 1 0 and H2 0, we have no reason o rejec H1 0 and H Empirical Sudies 4.1 Daa Currenly here are hree fuures producs: copper, aluminum and naural rubber fuures in SHFE. Aluminum fuures price is quie sable wih seldom limis hi. Therefore copper and naural rubber is used in our sudy. Daa we use for sudy consiss daily closing price and selemen price during hree-monh conrac. Table 1 gives he informaion of he daase. During he period ( ), each produc experienced from recession o marke bubble. Therefore, daa is represenaive. Table 1: Descripion of daa

11 Adjused AR Model and SV Model wih Applicaion 665 Beginning and Trading Minimum ending dae days volailiy uni Copper 1999/2/23 fuures 2003/11/20 Naural rubber fuures 1999/5/ /11/20 Days Days Frequency of up of lower of limis limis limis hi hi hi RMB % RMB % In pracice, when judging wheher here is a limi hi or no, we do no sricly follow he described percenage, bu consider he minimum ick size of he conracs. If we use o denoe he minimum ick size and use SP o represen selemen price of h rading day, he price limis of ( + 1) h rading day is: ( ) P+1 u SP (1 + l) = in, P+1 d = ceil ( SP (1 l) ), (14) where funcion in(x) is he larges ineger no greaer han x and ceil(x) is he smalles ineger no less han x. If he price is higher han P+1 u, a limi up happens. If he price is lower han P+1 d, a limi down happens. This rule is slighly differen from he rule of he fuures exchange. Table 1 gives he beginning and ending daes of he daa in our sudy, number of rading days and informaion on limi his of boh fuures producs. From Table 1, we noice ha frequency of limis hi is quie differen beween he wo fuures producs, indicaing he difference of flucuaion and risk beween hese wo producs. The high frequency of limis hi of naural rubber indicaes high price volailiy and high risk of his produc. The relaively lower frequency of limis hi of copper fuures indicaes relaive sabiliy and relaive lower risk of his produc. 4.2 Empirical Resuls and Analysis We generae MCMC samples for he parameers of models (9) and (11) using BUGS and SAS sofware. The resul indicaes ha he sequence converges around 4000 imes sampling (Figure 3). We hen coninue wih anoher sampling. The converged sampling sequence is used o esimae he parameers. Figure 4 gives he disribuions of sampling sequences of dummy variables afer convergence, which are asympoically symmeric. So we can deermine 90% HPD confidence inerval by calculaing is 5% and 95% quariles.

12 666 Naisheng Wang and Yan Lu Table 2 gives he esimaor, sandard deviaion, MCMC sampling error and HPD confidence inerval of each parameer. Table 3 gives a summary of he effecs of price limis on copper and naural rubber. Parameer Table 2: The resuls of MCMC esimaion Mean Sandard Sampling error 90% HPD inerval deviaion Copper µ(10 2 ) ϕ(10 2 ) γ(10 2 ) ( , ) κ(10 2 ) ( , ) ω ψ ζ ( , ) λ ( , ) τ Naural rubber µ(10 2 ) ϕ(10 2 ) γ(10 2 ) (0.5322, ) κ(10 2 ) ( , ) ω ψ ζ ( , ) λ ( , ) τ Table 3: A summary of he effec of price limis on copper and naural rubber fuures Copper Naural rubber Upper Limi Lower Limi Price Discovery Volailiy Price Discovery Volailiy Slighly Slighly pulling Slighly delay Slighly decreasing spillover back Significanly Slighly Slighly pulling Slighly spillover delay spillover back Price Delayed Discovery Effec For copper fuures, alhough boh he esimaed coefficiens of dummy variables γ and k in he means model are no equal o zero, boh of he 90% HPD con-

13 Adjused AR Model and SV Model wih Applicaion 667 fidence inervals include zero, so we don rejec he null hypoheses H0 1 : γ = 0, κ = 0. Price limis didn have significan price delayed discovery effec on copper fuures. Coefficien γ is greaer han zero, indicaing sligh price delayed discovery effec. Coefficien κ is less han zero, indicaing ha price will be pulled back in he following rading days. Overreacion exiss in copper fuures markes, which is correced in he following rading days. For naural rubber fuures, he esimaed coefficien of γ in he means model is no equal o zero and zero is ouside is 90% HPD confidence inerval. The hypohesis γ = 0 is herefore rejeced. There is significan price delayed discovery effec on naural rubber fuures due o limi up. Coefficien k is less han zero, bu zero is inside is 90% HPD confidence inerval, indicaing sligh delayed discovery effec of naural rubber fuures price afer limi down Volailiy Spillover Effec For copper fuures and naural rubber volailiy model, all he 90% HPD confidence inervals of dummy variables include zero. We can rejec hypohesis H0 2 for boh producs. This indicaes ha no significan volailiy spillover effec on copper fuures and naural rubber fuures is due o price limis. Boh of he dummy variables of limi up are no equal o zero, indicaing sligh volailiy spillover effec on copper fuures and naural rubber fuures because of limi up. Coefficiens of limi down of copper fuures and naural rubber fuures are less han zero and greaer han zero respecively, indicaing ha here is sligh decrease of copper fuures reurn volailiy following limi down. This means ha price limis have cooling-effec on he copper invesors. While naural rubber fuures reurn volailiy has sligh spillover phenomena. From he above analysis, price limis have no adverse impac on copper fuures price. I is efficien o conrol risks for copper fuures. For naural rubber, limi down has no adverse impac, while he limi up hinder fuures price-discovery process. The difference beween he effecs of price limis on naural rubber fuures and copper fuures is relaed o he differen markes behavior of hese wo producs. Copper fuures are relaively sable wih few limis hi. The frequency of limis hi is only 2.02%. While he frequency of limis hi of naural rubber fuures is 6.03%. Price limis have more influence on acive fuures producs han he relaively sable ones. When sudying he effec of price limis on sock price, Kim and Limpaphayom (2000) poined ou hose differen effecs from price limis exised because of differen characerisics of socks. Based on our sudy, heir conclusions can be exended o fuures markes. Table 2 gives he esimaor, sandard deviaion, MCMC sampling error of

14 668 Naisheng Wang and Yan Lu each parameer and HPD confidence inerval of he dummy variables. Table 3 gives a summary of he effec of price limis on copper and naural rubber. Price limis have no adverse impac on copper fuures price. I is efficien o conrol risks for copper fuures. For naural rubber, limi down has no adverse impac, while he limi up hinder fuures price-discovery process. 5. Conclusions This paper sudies he effec of price limis on SHFE heoreically and empirically using auoregressive model, sochasic volailiy model and MCMC sampling echniques. The auhors have he following findings: (1) Price limis have 20. Wei, S.X. (2002). A censored GARCH model of asse reurns wih price limis. Journal no adverse impac on copper fuures price. They are efficien o conrol risks. of Empirical Finance, 9, For naural rubber fuures, alhough price limis efficienly conrol risks, i impede fuures price discovery process. (2) The influence on fuures price from price limis relies on characerisics of fuures producs. Influence is significan for producs wih large flucuaion and frequen limis hi. This finding implies ha he effec of price limis on fuures price relies heavily on naure of fuures producs and he price limis on rubber fuures should be adjused. Appendix: Sampling sequences and sampling disribuions of he coefficiens of Appendix: Sampling Sequences and Sampling Disribuions of he Coefficiens dummy variables of Dummy for Naural Variables Rubber forfuures. Naural Rubber Fuures gamma kappa zea ieraion lambda ieraion ieraion Figure 3 Figure 3 gives a descripion of he convergence of he sampling sequences of he coefficiens of dummy variables for Naural Rubber Fuures. The sequences converge around 3100 sampling. Figure 3 gives a descripion of he convergence of he sampling sequences of he coefficiens of dummy variables for Naural Rubber Fuures. The sequences converge around 3100 sampling. Figure ieraion

15 gamma sample: Adjused AR Model and SV Model wih Applicaion 669 kappa sample: gamma sample: kappa sample: zea sample: zea sample: lambda sample: lambda sample: Figure Figure 4 gives he sampling Disribuion of Dummy Variables Coefficiens for Naural Rubber Fuures. All he disribuions are asympoically symmeric. Acknowledgemens Figure 4 Figure 4 Figure The4 auhors gives he hank sampling he Disribuion referee for his of Dummy careful reading Variables ofcoefficiens he manuscrip for Naural and his Figure helpful 4 gives commens. he sampling Disribuion of Dummy Variables Coefficiens for Naural Rubber Fuures. All he disribuions are asympoically symmeric. References Rubber Fuures. All he disribuions are asympoically symmeric. Anderson, R. W. (1984). The indusrial organizaion of fuures markes: a survey. In The Indusrial Organizaion of Fuures Markes (Edied by R. W. Anderson), D. C. Healh, Lexingon, Massachuses. Arak, M. and Cook, R. E. (1997). Do daily price limis ac as magnes? The case of reasury bond fuures. Journal of Financial Services Research 12, Berger, J. O. (1985). Saisical Decision Theory and Bayesian Analysis. Springer, New York. Brennan, M. J. (1986). A heory of price limis in fuures markes. Journal of Financial Economics 16, Chou, P. H. and Wu, S. (1998). A furher invesigaion of daily price limis. Journal of Financial Sudies 6, Engle, R. F. (1982). Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaion. Economerica 50,

16 670 Naisheng Wang and Yan Lu Fama, E. F. (1989). Perspecives on Ocober 1987 or wha did we learn from he crash? In Black Monday and he Fuure of Financial Markes (Edied by R. W. Kamphuis, Jr., R. C. Kormendi and J. W. H. Wason), Irwin, Homewood, Illinois. Figlewski, S. (1984). Margins and marke inegriy: margin seing for sock index fuures and opions. Journal of Fuures Markes 4, Greenwald, B. C. and Sein, J. C. (1991). Transacional risk, marke crashes, and he role of circui breakers. Journal of Business 64, Hall, A. and Kofman, P. (2001). Regulaory ools and price changes in fuures markes. Ausralian Economic Papers 40, Jiang, G. J., Konsaninidi, E. and Skiadopoulos, G. S. (2012). Volailiy spillovers and he effec of news announcemens. Journal of Banking and Finance 36, Kim, K. A. and Limpaphayom, P. (2000). Characerisics of socks ha frequenly hi price limis: empirical evidence from Taiwan and Thailand. Journal of Financial Markes 3, Kim, K. A. and Rhee, G. S. (1997). Price limi performance: evidence from he Tokyo Sock Exchange. Journal of Finance 52, Lehman, B. N. (1989). Commenary: volailiy, price resoluion, and he effeciveness of price limis. Journal of Financial Services Research 3, Ma, C. K., Rao, R. P. and Sears, R. S. (1989a). Limi moves and price resoluion: he case of he reasury bond fuures marke. Journal of Fuures Markes 9, Ma, C. K., Rao, R. P. and Sears, R. S. (1989b). Volailiy, price resoluion, and he effeciveness of price limis. Journal of Financial Services Research 3, Mao, S. S., Wang, J. L. and Pu, X. L. (1998). Advanced Mahemaical Saisics, 1s ediion. Higher Educaion Press, Beijing. Moser, J. T. (1990). Circui breakers, economic perspecives. Federal Reserve Bank of Chicago 14, Taylor, S. J. (1986). Modeling Financial Time Series. Wiley, Chichser. Wei, S. X. (2002). A censored GARCH model of asse reurns wih price limis. Journal of Empirical Finance 9,

17 Adjused AR Model and SV Model wih Applicaion 671 Received Ocober 20, 2012; acceped March 19, Naisheng Wang Financial Analys China Securiies Index Co., Ld 5F, Bldg B, No. 555 Ying Chun Road, Pudong, Shanghai, , China Yan Lu Assisan Professor Deparmen of Mahemaics and Saisics Universiy of New Mexico MSC , Albuquerque, New Mexico 87131, USA