Financial Econometrics Introduction to Realized Variance

Transcription

1 Financial Economerics Inroducion o Realized Variance Eric Zivo May 16, 2011 Ouline Inroducion Realized Variance Defined Quadraic Variaion and Realized Variance Asympoic Disribuion Theory for Realized Variance

2 Reading APDVP, chaper 12. Andersen, Bollerslev, Diebold, Labys (ABDL): The Disribuion of Realized Exchange Rae Volailiy JASA, 2001 Andersen, Bollerslev, Diebold, Labys: Modeling and Forecasing Realized Volailiy ECTA, 2003 Barndorff-Nielsen and Shephard (BNS): Esimaing Quadraic Variaion Using Realized Variance JAE 2002 Barndorff-Nielsen and Shephard: Economeric Analysis of Realized Volailiy and Is Use in Esimaing Sochasic Volailiy Models JRSSB, 2002.

3 Inroducion Key problem in financial economerics: modeling, esimaion and forecasing of condiional reurn volailiy and correlaion. Derivaives pricing, risk managemen, asse allocaion Condiional volailiy is highly persisen Inheren problem: condiional volailiy is unobservable Tradiional laen variable models: ARCH-GARCH, Sochasic volailiy (SV)basedonsquaredreurns difficul esimaion high frequency daa no uilized sandardized reurns no Gaussian Imprecise forecass mulivariae exensions are difficul

4 New approach uses esimaes of laen volailiy based on high frequency daa (realized variance measures) Volailiy is observable Tradiional ime series models are applicable High dimensional mulivariae modeling is feasible Consrucion of Realized Variance Measures p i, = log-price of asse i a ime (aligned o common clock) p =(p 1,,...,p n, ) 0 = n 1 vecor of log prices = fracion of a rading session associaed wih he implied sampling frequency, m =1/ = number of sampled observaions per rading session T = number of days in he sample mt oal observaions

5 Example (FX marke): Prices are sampled every 30 minues and rading akes place24hoursperday m =4830-minue inervals per rading day =1/ Example (Equiy marke): Prices are sampled every 5 minues and rading akes place 6.5 hours per day m =785-minues inervals per rading day =1/ Inra-day coninuously compounded (cc) reurns from ime 1+(j 1) o 1+j r i, 1+j = p i, 1+j p i, 1+(j 1), j =1,...,m r 1+j = p 1+j p 1+(j 1), j =1,...,m Reurns for day r i, = r i, 1+ + r i, r i, 1+m r = r 1+ + r r 1+m Realized variance (RV) for asse i on day i, = mx ri, 1+j 2, =1,...,T j=1

6 Realized volailiy (RVOL) for asse i on day : RV OL (m) i, = r i, Realized log-volailiy (RLVOL) : RLV OL (m) i, =ln(rv OL (m) i, ) The n n realized covariance (RCOV) marix on day RCOV (m) = The n n marix RCOV (m) mx r 1+j r 0 1+j, =1,...,T j=1 will be posiive definie provided n<m The realized correlaion beween asse i and asse j RCOR (m) i,j, = = RCOV (m) s RCOV (m) i,i RCOV (m) RV OL (m) i, i,j i,j RV OL (m) j, RCOV (m) j,j

7 RV measures over h days: i, (h) = hx j=1 i,+j RCOV (m) X h i,j, (h) = RCOV (m) +k k=1 QuadraicReurnVariaionandRealizedVariance Two fundamenal quesions abou RV are: Q1 Wha does RV esimae? Q2 Are RV esimaes economically imporan?

8 Answers are provided in a number of imporan papers: Andersen, Bollerslev, Diebold, Labys (ABDL): The Disribuion of Realized Exchange Rae Volailiy JASA, 2001 Andersen, Bollerslev, Diebold, Labys: Modeling and Forecasing Realized Volailiy ECTA, 2003 Barndorff-Nielsen and Shephard (BNS): Esimaing Quadraic Variaion Using Realized Variance JAE 2002 Barndorff-Nielsen and Shephard: Economeric Analysis of Realized Volailiy and Is Use in Esimaing Sochasic Volailiy Models JRSSB, Coninuous ime arbirage-free log-price process le p() denoe he univariae log-price process for a represenaive asse defined on a complee probabiliy space (Ω, F, P), evolving in coninuous ime over he inerval [0,T]. Le F be he σ field reflecing informaion a ime such ha F s F for 0 s T. Resul: If p() is in he class of special semi-maringales hen i has he represenaion p() =p(0) + A()+M(), A(0) = M(0) = 0 where A() is a predicable drif componen of finie variaion, and M() is a local maringale. Noe:jumps are allowed in boh A() and M().

9 Example: Arihemaic Brownian Moion dp() = μd + σdw() W () = Sandard Brownian Moion Z Z p() = p(0) + μ d + σ dw () 0 0 = p(0) + μ + σ W () Hence Z A() = μ d = μ 0 Z M() = σ dw () =σ W () 0 Le mt be a posiive ineger indicaing he number of reurn observaion obained by sampling m =1/ imes per day for T days Theccreurnonassei over he period [,] is r(, ) =p() p( ) The daily cc and cumulaive reurns are r(, 1) = p() p( 1) r() = p() p(0)

10 Le Π M = {0 = 0 < 1 < < M = } be any pariion of he inerval [0,] ino M inervals and define kπ M k = max ( j+1 j ) j=0,...,m 1 Definiion: The quadraic variaion (QV) of he reurn process from ime 0 o is [r]() =p lim kπ M k 0 M 1 X j=0 {p( j+1 ) p( j )} 2 as M The QV process measures he realized sample pah variaion of he squared reurn process. QV is a unique and invarian ex-pos realized volailiy measure ha is essenially model free. Resul: QV for an Io Diffusion Process Le p() be described by he sochasic differenial equaion dp() = μ()d + σ()dw (),W() = Wiener process, where μ() and σ () may be random funcions, wih daily reurn process Z Z r() = μ(s)ds + σ(s)dw (s) 0 0 Z Z r(, 1) = μ(s)ds + σ(s)dw (s) 1 1 Then Z [r]() = σ(s)ds 0 Z QV [r]() [r]( 1) = σ(s)ds = IV 1 where IV denoes inegraed variance for day.

11 Example: QV for Wiener process p() = W () dp() = dw (), σ() =1 Then [r]() = QV = Z Z σ(s)ds = ds = 0 0 Z Z σ(s)ds = ds =1 1 1 The definiion of QV implies he following convergence resul for semi-maringales: p [r]() [r]( 1) QV, as m Tha is, daily RV converges in probabiliy o he daily incremen in QV. This answers he firs quesion Q1. Remark: As noed by ABDL, QV is relaed o, bu disinc from, he daily condiional reurn variance. Tha is, in general QV 6=var(r(, 1) F 1 )

12 Resul (ABDL 2001): If (i) he price process p() is square inegrable; (ii) he mean process A() is coninuous; (iii) he daily mean process, {A(s) A( 1)} s [ 1,], condiional on informaion a ime is independen of he reurn innovaion process, {M(u)} u [ 1,], (iv) he daily mean process, {A(s) A( 1)} s [ 1,], is a predeermined funcion over [ 1,], hen for 0 1 T var(r(, 1) F 1 )=E[QV F 1 ] Tha is, he condiional reurn variance equals he condiional expecaion of he daily QV process. Noe: he ex pos value of is an unbiased esimaor for he condiional reurn variance var(r(, 1) F 1 ): E[ F 1 ]=E[QV F 1 ]=var(r(, 1) F 1 ) Therefore, is economically imporan which answers he second quesion Q2.

13 Remark: The resricions on he condiional mean process allow for realisic price processes. price process is allowed o exhibi deerminisic inra-day seasonal variaion. mean process can be sochasic as long as i remains a funcion, over he inerval [ 1,], of variables in F 1. jumps are allowed in he reurn innovaion process M() leverage effecs caused by conemporaneous correlaion beween reurn innovaions and innovaions o he volailiy process are allowed. Resuls for Iô processes Le p() be described by he sochasic differenial equaion dp() = μ()d + σ()dw (),W() = Wiener process wih daily reurn process Z Z r(, 1) = μ(s)ds σ(s)dw (s) Noe: There may be leverage effecs. Tha is, σ() may be correlaed wih W (). For example, dσ() = μ()d + σ()d W () cov(dw (),d W ()) 6= 0

14 Recall, he daily incremen o QV is given by Z QV = 1 σ2 (s)ds = IV where IV denoes daily inegraed variance (IV). Resul: Since p QV, i follows ha p IV Remark: IV plays a cenral in opion pricing wih sochasic volailiy (e.g. Hull and Whie, 1987) Resul (ABDL (2003)): If he mean process, μ(s), and volailiy process, σ(s), are independen of he Wiener process W (s) over [ 1,] hen µz r(, 1) σ{μ(s),σ(s)} s [ 1,] N μ(s)ds, IV 1 where σ{μ(s),σ(s)} s [ 1,] denoes he σ field generaed by (μ(s),σ(s)) s [ 1,]. Since R 1 μ(s)ds is generally very small for daily reurns and is a consisen esimaor of IV,forIô processes daily reurns should follow a normal mixure disribuion wih as he mixing variable. As a resul, reurns sandardized by realized volailiy should be sandard normal r /RV OL (m) N(0, 1) If here are jumps in dp (), hen p IV bu reurns are no longer condiionally normally disribued.

15 Asympoic Disribuion Theory for Realized Variance For a diffusion process, he consisency of sampling frequency per day,, going o zero. for IV relies on he Convergence resul is no aainable in pracice as i is no possible o sample coninuously ( is bounded from below by highes observable sampling frequency) Theory suggess sampling as ofen as possible o ge he mos accurae esimae of IV. Marke microsrucure fricions evenually dominae he behavior of RV as 0, which implies a pracical lower bound on for observed daa. For > 0, will be a noisy esimae of IV. Defineheerrorin u ( ) = for a given as IV or = IV + u ( ) Resul (BNS (2001)): For he Io diffusion model under he assumpion ha mean and volailiy processes are joinly independen of W (), u ( ) m = (m) (RV m IV ) 2 IQ 2 IQ d N(0, 1) as m where Z IQ = is he inegraed quariciy (IQ). Hence, µ A N 1 σ4 (s)ds IV, 2 IQ m

16 Remarks: converges o IV a rae m, The asympoic disribuion of is mixed-normal since IQ is random. IQ may be consisenly esimaed using he following scaled version of realized quariciy (RQ) RQ (m) = m 3 RQ(m) mx r 1+j 4 j=1 p IQ as m Thefeasibleasympoicdisribuionfor is µ A N IV, 2 3 RQ(m) which implies s v dse( 2 )= 3 RQ(m) = u 2 3 mx r 1+j 4 j=1

17 r Q: Wha is asympoic disribuion of RV OL (m) =? A: Use dela-mehod Recall, if θ A N(θ, V ) and if g(θ) is coninuous and differeniable, hen g(θ) A N(g(θ),g 0 (θ) 2 V ) Le θ = µ and V = 2 3 RQ(m), and g RV OL (m). = r = Then by dela-mehod which suggess RV OL (m) A N dse(rv OL (m) )= p IV, 2 RQ (m) 12 v u 2 12 RQ(m)

18 BNS find ha he finie sample disribuion of and RV OL (m) can be quie far from heir respecive asympoic disribuions for moderaely sized m. Using he dela mehod BNS show ha he asympoic disribuion of µ RLV OL (m) 2, µ RLV OL (m) 2 ln(iv ) v u 2 3 RQ(m) A N (0, 1) is closer o is finie sample asympoic disribuion han he asympoic disribuions of and RV OL (m). BNS (2004) exend he above asympoic resuls o cover he mulivariae case, providing asympoic disribuions for RCOV (m) and RCOR (m) i,j,, as well as realized regression esimaes. These limiing disribuions are much more complicaed han he ones presened above, and he reader is referred o BNS (2004) for full deails and examples.

19 Pracical Problems in he Consrucion of RV Inra-day prices/quoes are no discree observaions from idealized coninuousime process p() =p()+error() =observed price process p() =rue price process error() represens marke microsrucure noise (bid/ask bounce, rounding, price alignmen, invenory effecs) Exisence of error() causes serious problems - bias, inconsisency of as m Empirical Analysis of RV See Powerpoin Summary of Some Famous Published Papers by ABDL and BNS