Bayesian range-based estimation of stochastic volatility models

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1 Finance Researc Letters ( Bayesian range-based estimation of stocastic volatility models Micael W. Brandt a,b,, Cristoper S. Jones c a Fuqua Scool of Business, Duke University, Duram, NC 7708, USA b National Bureau of Economic Researc, Cambridge, MA 038, USA c Marsall Scool of Business, University of Soutern California, Los Angeles, CA 90089, USA Received 6 July 005; accepted 5 September 005 Available online 9 September 005 Abstract Alizade, Brandt, and Diebold [00. Journal of Finance 57, ] propose estimating stocastic volatility models by quasi-maximum likeliood using data on te daily range of te log asset price process. We suggest a related Bayesian procedure tat delivers exact likeliood based inferences. Our approac also incorporates data on te daily return and accommodates a nonzero drift. We illustrate troug a Monte Carlo experiment tat quasi-maximum likeliood using range data alone is remarkably close to exact likeliood based inferences using bot range and return data. 005 Elsevier Inc. All rigts reserved. JEL classification: G0; G; C; C5 Keywords: Stocastic volatility models; Price range; Bayesian estimation. Introduction Alizade, Brandt and Diebold (ABD, 00 propose estimating stocastic volatility models by quasi-maximum likeliood using data on te daily range of te log asset price process. Teir procedure is simple yet igly efficient, compared to standard daily return-based inferences, for two reasons. First, te daily range is a muc less noisy measure of daily volatility tan absolute or squared daily returns (see Parkinson, 980, and Andersen and Bollerslev, 998. Second, ABD * Corresponding autor. Fax: address: mbrandt@duke.edu (M.W. Brandt /$ see front matter 005 Elsevier Inc. All rigts reserved. doi:0.06/j.frl

2 0 M.W. Brandt, C.S. Jones / Finance Researc Letters ( demonstrate tat te distribution of te log range is close to being Gaussian, wic results in fairly accurate Kalman filter based quasi-maximum likeliood estimates. We address tree open issues in ABD s range-based estimation approac: ( te use of an approximate as opposed to exact likeliood function, ( te use of only daily range data wen daily return data are also readily available, and (3 te assumption of a zero drift of te log asset price process to derive te distribution of te log range. We accomplis tis by developing a closely related Bayesian procedure based on te exact joint distribution of te daily log range and log return, relying eavily on te simulation-based tecniques introduced by Jacquier, Polson and Rossi (JPR, 994. We implement our procedure in te context of a Monte Carlo experiment to examine te extent to wic exact likeliood inferences based on bot range and return data dominates ABD s approximate likeliood approac based on range data alone. We find tat te gains in efficiency are minimal. Our results confirm ABD s conjecture tat te distribution of te log range is sufficiently close to being Gaussian to deliver virtually exact maximum likeliood estimates. Tey also suggest tat, at least in te absence of a drift, absolute or squared daily returns contain little information not already revealed troug te daily range.. Model Consider te following stylized stocastic volatility model: ds τ = µdτ + τ dw Sτ, S τ d ln τ = κ(ln ln τ dτ + βdw τ. ( In tis model, te log volatility ln τ evolves as a mean-reverting Ornstein Ulenbeck process wit mean ln and mean reversion parameter κ>0. Te two Brownian motions are assumed independent. Since we observe stock prices at discrete times t {0,,,...,T}, we discretize te model by assuming tat te return volatility is constant at t over te interval t <τ t. Witin eac interval, te stock price ten follows a standard geometric Brownian motion: ds τ = µdτ + t dw Sτ for t <τ t, ( S τ and te conditional distribution of log volatility from one discrete interval to te next is approximately: ln t ln t N [ ln t + κ(ln ln t, β ]. (3 Tis discretization of te continuous time stocastic volatility model is not original. It is sared by ABD (00, Duffie and Singleton (993, Gallant et al. (997, Harvey et al. (994, Jacquier et al. (994, Kim et al. (998, and Taylor (994, among oters. Like us, tese autors assume tat for discrete time periods te volatility is constant and tat from one period to te next te log volatility is conditionally normal and mean reverting.

3 M.W. Brandt, C.S. Jones / Finance Researc Letters ( For t τ t, letx τ = ln S τ ln S t, so tat x t is te log return for te period ending at time t. Define te range D t as D t = max x τ min x τ. t τ t t τ t Under te assumption tat x τ is a martingale, ABD derive te density function of D t and illustrate tat it is approximately lognormal. Our approac differs from tat of ABD in tat we focus on te joint distribution of x τ and D τ and in tat we allow for a nonzero drift in x τ. Te drift is assumed to be constant and is denoted by a. We decompose te joint distribution of te range and return as p(d t,x t a, t = p(x t a, t p(d t x t,a, t. Te first term on te rigt is Gaussian and is terefore trivial to evaluate. Te second term, te density of te range conditional on te contemporaneous return, is nonstandard and is derived in Appendix A. Te appendix furter sows tat p(d t x t,a, t is in fact independent of a. 3. Econometric metod We rewrite te discretized volatility model in more familiar notation as ln t = α + δ ln t + σν t. We will assume tat we ave diffuse prior information about te parameters of te stocastic volatility process, meaning (4 (5 (6 p(α,δ,σ σ σ σ. (7 Te lower bound on σ is set to a small positive number and is required to insure proper posteriors (see Joannes and Polson, 005. In practice, as long as te algoritm is initialized wit a reasonable value of σ, te bound is never it. We analyze tis model following Tanner and Wong s (987 principle of data augmentation. In tis framework, we calculate te posterior distribution of te model parameters numerically using Markov cain Monte Carlo metods, in wic a Markov cain is constructed to generate te posterior as its invariant distribution. By generating a sufficiently long cain, te posterior distribution may be caracterized wit arbitrary precision. Te cain is constructed by alternatively drawing latent data (in tis case, te T return volatilities given te model parameters and drawing model parameters given latent data. Given latent volatilities, drawing te model parameters in tis case is simple, since te setup conforms to te standard Gaussian linear regression framework. Parameter draws are based on standard OLS estimates, ˆα, ˆδ, and ˆσ. Specifically, σ is drawn from te inverted gamma distribution IG ( ˆσ (T 3, T.Giventedrawofσ, te distribution of [α, δ] is bivariate normal wit mean [ˆα, ˆδ] and covariance matrix σ(h H, were H is a (T matrix wose tt row is equal to [, ln t ]. Oter studies, suc as JPR, equivalently coose to model te log variance process.

4 04 M.W. Brandt, C.S. Jones / Finance Researc Letters ( Te draw of te latent volatility pat is more complicated because it is from a nonstandard distribution. In addition, te ig dimensionality of te latent data makes te application of standard random number generation metods difficult. We terefore adapt te approac of JPR and decompose te multivariate draw into a sequence of univariate draws. Rater tan drawing te vector all at once, we instead draw eac element t conditional on te rest of te vector (denoted t. Since tis draw is also conditional on te current draw of te parameter vector (denoted θ for sortand, we can write our target density as f t ( t = p( t t,x,d,θ, (8 were x and D denote te time series of returns and log ranges. Te Markovian nature of te process greatly reduces te relevant conditioning set, so tat te density may be written as f t ( t = p( t t, t+,x t,d t,θ. (9 As in JPR, we cycle troug te vector drawing a new value to replace te old for eac t. At te end of eac full cycle, a new draw from te parameter vector s conditional distribution replaces te previous draw. Using Bayes rule, we can decompose te target distribution into te product of some simpler kernels: f t ( t = p( t t, t+,x t,d t,θ p( t t, θp( t+ t, θp(x t t, θp(d t t,x t,θ { exp ( ln t m } { exp ( } xt p(d t t,x t, t s t t were m = α( δ + δ(ln t+ + ln t + δ and s = σ + δ result from completing te square of te first two lognormal kernels. Because tis is te density of a nonstandard distribution, we perform draws from it using te Metropolis Hastings algoritm. We terefore form wat JPR term a cyclic Metropolis cain. Since te Metropolis algoritm is an accept/reject procedure, we must select a proposal density to generate candidate values of t. Ideally, tis proposal density sould approximate te target density closely. To te extent tat te two differ, te proposal density sould blanket te target, meaning tat it sufficiently often generates candidate values tat are in te tails of te target distribution. If te blanketing is poor or if it is excessive (too many draws in te tail of te target, ten te convergence of te cain may be slow. As a proposal density we use an approximation of p( t t, t+,d t,θ. Because we do not condition on x t, te proposal density sould on average ave a iger variance tan te target density. Tis could potentially result in a slow-mixing cain, but it is oterwise unproblematic. In any case, since x t contains relatively little information about t wen compared wit D t,tis effect will prove to be minor and of no real concern. We approximate te density p( t t, t+,d t,θ by first decomposing it, as we did wit te target density, into te product of simpler kernels, or (0 ( p( t t, t+,d t,θ p( t t, θp( t+ t, θp(d t t,θ, (

5 M.W. Brandt, C.S. Jones / Finance Researc Letters ( were p( t t,θ and p( t+ t,θ are lognormal densities for t and t+, respectively. Te tird kernel, p(d t t,θ, is of a nonstandard distribution. ABD sow, owever, tat D t is approximately lognormal wit mean ln t and standard deviation 0.9. Letting p (D t t,θ denote tis approximate lognormal distribution for D t, our proposal density is f p ( t p( t t p( t+ t p (D t t { exp ( ln t m } { exp ( ln Dt 0.43 ln } t t s D t 0.9 { exp ( ln t m } { exp ( ln t m } t s s { exp ( ln t m }, t s were m and s are te same as before, m = ln D t , s = 0.9, m = m s + m s s + s and s =. s + s In summary, te proposal density for t is lognormal wit mean parameter m and variance parameter s. Furtermore, te density does not depend on te previous draw of t. Te Metropolis Hastings algoritm dictates tat te probability of replacing te old draw t wit te candidate draw t is equal to [ ft ( t min f ] p( t (5 f t ( t f p ( t,. If te draw is not accepted, te new value of t is set equal to te previous draw. Draws performed in tis way, wen combined wit draws from te conditional distribution of θ =[α, δ, σ ], produce a Markov cain wose invariant distribution is te joint distribution of θ and. Exceptions to tis procedure occur for te draws of te first and last values of te volatility process, and T.For T, te draw is te same, except tat te conditioning variable T + no longer exists. For, te conditioning variable 0 does not exist. We will assume, owever, tat te process as been running for a sufficient period prior to te beginning of our sample so tat te value is distributed according to te stationary distribution of te process, if suc a distribution exists. Appendix B details tese variations of te procedure. 4. Monte Carlo results Here we present simulation evidence to compare our approac wit tose of bot ABD and JPR. Parameter values are from JPR, were we consider tree sets of parameters corresponding to teir intermediate case for te coefficient of variation of t. Under tese parameters, we simulate volatility pats t of lengt 500 according to (6. Witin eac period, we simulate (3 (4 In fact, te skewness and kurtosis of te true density are sligtly lower tan tose of te lognormal. Te approximate density sould terefore be expected to perform well in blanketing te target density, wic sould elp avoid te slow-mixing problem identified by Mengersen and Tweedie (996.

6 06 M.W. Brandt, C.S. Jones / Finance Researc Letters ( Brownian motion wit mean zero and volatility t by discretizing te interval into 000 subperiods. Te range and return are computed for eac period from tese simulated values. We generate 5000 Monte Carlo samples for bot te metod outlined in Section 3 and a Bayesian version of te metod presented in ABD. Te two approaces are identical except tat te ABD metod replaces te true likeliood for bot D t and x t wit te approximate lognormal likeliood for D t only. Tus, te ABD approac differs bot because it uses an approximation and because it does not use returns data. For eac sample, we run te Gibbs sampler for,000 iterations and discard te first 000. Te remaining draws are averaged to compute posterior means, and tese estimates are used to compute te means and root mean squared errors reported in Table. Results for te return-based estimates are taken directly from JPR, were we ave alved teir reported numbers for α and σ since we are working wit a log volatility rater tan log variance model. Te results illustrate te benefits of using te range, particularly for estimating te volatility of volatility parameter σ. For tis parameter, te RMSEs of te two range-based estimators is approximately alf as large as tose for te JPR approac in all tree cases. For te oter two parameters, te improvement is limited to te low and medium persistence cases. For te ig persistence case, te range-based estimates of α and δ are actually less precise (bot more biased and variable tan te return based estimates. Intuitively, and as ABD illustrate, te improvement for estimating te volatility of volatility parameter σ comes from te fact tat te iger signal to noise ratio of te range allows te range-based estimators to better differentiate noise in te volatility proxy from fluctuations in volatility. Te more volatility fluctuates, or te lower te persistence of volatility, te greater is te potential benefit from range-based volatility estimation. Te improved precision for α Table Monte Carlo simulation results Low persistence Medium persistence Hig persistence α δ σ α δ σ α δ σ True parameter values Return-based estimates (from JPR (0.70 (0.046 (0.034 (0.70 (0.046 (0.033 (0.070 (0.00 (0.040 Range-based estimates using ABD metod (0. (0.033 (0.08 (0.097 (0.06 (0.05 (0.079 (0.0 (0.03 Range and return-based estimates using exact likeliood (0.5 (0.033 (0.07 (0.099 (0.06 (0.05 (0.080 (0.0 (0.0 Notes. Te table reports means and root mean squared errors (in parenteses for tree separate Monte Carlo exercises. Return-based estimates are values taken from JPR. Since teir model is written in terms of log variance, we divide teir α and σ parameters by two for comparison wit our results, wic are in terms of log volatility. Te final two sets of results are eac based on our own calculations, were eac statistic is computed from 5000 Monte Carlo simulations. In every case, volatility time series are generated using te parameters reported in te top row, and intraday price pats are ten simulated conditional on tose volatilities using te Euler discretization wit 000 time steps per day. Range-based estimates use te approac of ABD, wic relies on a Gaussian approximation of te density of te log range. Range and return-based estimates use te Bayesian metod developed in tis paper, wic differs from te ABD approac bot because it incorporates returns data and because it uses te exact likeliood function.

7 M.W. Brandt, C.S. Jones / Finance Researc Letters ( and δ are a by-product of estimating σ more precisely (since δ and σ are linked troug te unconditional volatility of volatility. It appears from our results tat for te ig persistence case tis indirect effect on te estimates of α and δ is swamped by oter differences between te estimators. For example, JPR use loose but proper conjugate priors wile we use improper diffuse priors. Altoug te effects of tis difference sould be very minor, it could explain te results for α and δ in te ig-persistence case. Finally, te results suggest tat little is to be gained from using te exact likeliood of te range and return data relative to te approximate likeliood of te range data alone employed by ABD. One caveat ere is tat we simulated log prices by assuming a zero drift in order to be consistent wit JPR and ABD. In a situation were te drift is likely to be large, suc as wit alternative asset classes, our new metod may prove more useful since it is able to account for nonzero drift witout difficulty. Acknowledgments We tank Dmitry Livdan for excellent researc support and Entropia for providing computational resources. Appendix A. Conditional distributions of te range Consider a driftless Brownian motion x τ wit volatility over te interval τ [0, ]. We want Prob[D dd, x d x ], were D = M m and M = max x τ, m= min x τ. t τ t t τ t Assume x 0 = 0, so tat x can be interpreted as a log return. Decompose te joint density: Prob [ (M m dd, x d x ] [ = Prob[x d x ] Prob (M m dd x = x, ]. }{{}}{{} P P Te density P is just Gaussian. Te joint density of M and m is Prob[M dv, m du x = x,] 4k [ (k(u + v x ] ( φ k(u+v x = φ ( x [ 4k(k (k(u + v + x v ] ( φ k(u+v+ x v φ (. x A cange of variable yields te density P : Prob [ (M m dd x = x, ] = 4k [ (kd x ] ( φ kd x φ ( ( x D x (A. (A. (A.3 (A.4

8 08 M.W. Brandt, C.S. Jones / Finance Researc Letters ( k(k ( kd x φ ( kd x φ ( x k(k ( (k D + x φ ( (k D+ x φ ( x (A.5 for D x, were φ(z= exp( z // π. Finally, we note tat te zero drift assumption was unnecessary, as te conditional density of te range given te contemporaneous return does not depend on te drift of te process. Only te Gaussian component P depends on te drift of x, and tat is in te obvious way. To see te drift independence of D, assume instead tat x τ ad drift a. Ten if 0 <τ<: [ xτ x ] N ( [ ] τ a,σ [ τ τ τ ]. Te conditional distribution of x τ given x as mean aτ + τ(x a = τx and variance σ (τ τ. Tus, te marginal distribution of x τ is independent of a. For 0 <τ <τ <, te covariance between x τ and x τ is also independent of a simply because te conditional covariance of a multivariate normal is never a function of its mean. Tus, te entire pat of x τ between 0 and is independent of a after conditioning on x, wic means tat D is independent of a as well. Appendix B. Drawing latent volatility for t = and t = T Te proposal and target densities presented in Section must be modified to draw te first and last values of te latent volatility process. For te final period volatility, T, te target density is somewat simpler due to te lack of a subsequent period. For tis value, te target density becomes were now f T ( T = p( T T,x T,D T p( T T p(x T T p(d T T,x T { exp ( ln T m } { exp T T s ( xt T } p(d T T,x T, (B. m = α + δ ln T and s = σ. (B. As before, te proposal density used is a lognormal wit mean and standard deviation parameters m = m s + m s s + s and s =, s + s (B.3 were m and s are te same as before. Similarly, te target density for initial volatility,, reflects te fact tat tere is no preceding volatility. If te volatility process is stationary, owever, ten te draw of sould be consistent wit te stationary distribution, wic is lognormal. We terefore write te target density as te

9 M.W. Brandt, C.S. Jones / Finance Researc Letters ( product of te unconditional density and some more familiar terms: were f ( = p(,x,d p( p( p(x p(d,x { exp ( ln α/( δ σ/ δ { exp ( } x p(d,x { exp ( ln m s } exp } { exp { ( ln α δ ln } σ ( x } p(d,x, (B.4 m = α( δ + δ(ln t+ + ln t (B.5 + δ and s = σ + δ. Again, given te previous definitions of m and s, te proposal density is lognormal wit mean and standard deviation parameters References m = α( δ δ + δ(ln α and s = σ. (B.6 Alizade, S., Brandt, M.W., Diebold, F.X., 00. Range-based estimation of stocastic volatility models. Journal of Finance 57, Andersen, T.G., Bollerslev, T., 998. Answering te skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review 39, Duffie, D., Singleton, K., 993. Simulated moments estimation of Markov models of asset prices. Econometrica 6, Gallant, A.R., Hsie, D.A., Taucen, G.E., 997. Estimation of stocastic volatility models wit diagnostics. Journal of Econometrics 8, Harvey, A., Ruiz, E., Separd, N., 994. Multivariate stocastic variance models. Review of Economic Studies 6, Jacquier, E., Polson, N.G., Rossi, P.E., 994. Bayesian analysis of stocastic volatility models. Journal of Business and Economic Statistics, Joannes, M., Polson, N.G., 005. MCMC metods for financial econometrics. In: Aït-Saalia, Y., Hansen, L. (Eds., Handbook of Financial Econometrics. Kim, S., Separd, N., Cib, S., 998. Stocastic volatility: Likeliood inference and comparison wit ARCH models. Review of Economic Studies 65, Mengersen, K.L., Tweedie, R.L., 996. Rates of convergence of te Hastings and Metropolis algoritms. Annals of Statistics 4, 0. Parkinson, M., 980. Te extreme value metod for estimating te variance of te rate of return. Journal of Business 53, Tanner, M., Wong, W., 987. Te calculation of posterior distributions by data augmentation. Journal of te American Statistical Association 8, Taylor, S.J., 994. Modelling stocastic volatility. Matematical Finance 4,