Inverse Realized Laplace Transforms for Nonparametric Volatility Density Estimation in Jump-Diffusions

Transcription

1 Iverse Realized Laplace Trasforms for Noparametric Volatility Desity Estimatio i Jump-Diffusios Viktor Todorov ad George Tauche April 212 Abstract We develop a oparametric estimator of the stochastic volatility desity of a discretelyobserved Itô semimartigale i the settig of a icreasig time spa ad fier mesh of the observatio grid. There are two steps. The first is aggregatig the high-frequecy icremets ito the realized Laplace trasform, which is a robust oparametric estimate of the uderlyig volatility Laplace trasform. The secod step is usig a regularized kerel to ivert the realized Laplace trasform. The two steps are relatively quick ad easy to compute, so the oparametric estimator is practicable. We derive bouds for the mea squared error of the estimator. The regularity coditios are sufficietly geeral to cover empirically importat cases such as level jumps ad possible depedecies betwee volatility moves ad either diffusive or jump moves i the semimartigale. Mote Carlo work idicates that the oparametric estimator is reliable ad reasoably accurate i realistic estimatio cotexts. A empirical applicatio to 5-miute data for three large-cap stocks, , reveals the importace of big short-term volatility spikes i geeratig high levels of stock price variability over ad above that iduced by price jumps. The applicatio also shows how to trace out the dyamic respose of the volatility desity to both positive ad egative jumps i the stock price. Keywords: Laplace trasform, stochastic volatility, ill-posed problems, regularizatio, oparametric desity estimatio, high-frequecy data. We wish to thak the Editor, Associate Editor ad two aoymous referees for their detailed commets that led to sigificat improvemets. We also thak Tim Bollerslev, Fracis Diebold, Jia Li, Adrew Patto as well as may semiar participats. Todorov s work was partially supported by NSF grat SES Associate Professor, Departmet of Fiace, Kellogg School of Maagemet, Northwester Uiversity, Evasto, IL 628; v-todorov@orthwester.edu. Professor, Departmet of Ecoomics, Duke Uiversity, Durham, NC 2778; george.tauche@duke.edu. 1

2 1 Itroductio Cotiuous-time models are widely used i empirical fiace to model the evolutio of fiacial asset prices. Absece of arbitrage (uder some techical coditios) implies that traded fiacial assets should be semimartigales ad typically most, if ot all, applicatios restrict attetio to Itô semimartigales, i.e., semimartigales with characteristics absolutely cotiuous i time. Thus the stadard asset pricig model for the log-fiacial price X t is of the form dx t = α t dt + V t dw t + dj t, (1) where α t ad V t > are processes with càdlàg paths, W t is a Browia motio, J t is a jump process. I Sectio 2 further below we give the techical coditios for the compoets comprisig X i (1), while here we briefly describe their mai roles i determiig the dyamics of X. α t captures risk premium (ad possibly risk-free rate) ad is well-kow to be preset but will ot be the object of iterest i this paper. As will be see below, it gets filtered out i our estimatio process. The jump compoet, J t, highlighted by Bardorff-Nielse ad Shephard (24), amog others, accouts accordig to earlier empirical evidece typically for 5 to 15 percet of the total variace of the icremets i X. Jumps reflect the fact that fiacial time series exhibit very sharp shortterm moves icompatible with the cotiuous sample paths implied by diffusive models. Much of the evidece o jumps has bee adduced usig very high frequecy data, see e.g., Bardorff- Nielse ad Shephard (26) ad Ait-Sahalia ad Jacod (29); earlier efforts usig coarsely sampled (daily) data were at best mildly successful i hadlig both jumps ad diffusive price moves. Similar to α t, J t will be filtered out i our aalysis. The volatility term, V t, represets the domiat compoet of the variace of icremets i X ad thus is most widely studied, see e.g., Myklad ad Zhag (29) ad the may refereces therei. The volatility process is well kow to be egatively correlated with the icremets i the drivig diffusio W t. This egative correlatio is the so called leverage effect, a term due to Black (1976), ad the effect 2

3 has bee extesively documeted i a wide variety of studies usig various statistical methods. Of course, price ad volatility ca have depedece beyod the leverage effect as i the symmetric GARCH processes, e.g., Klüppelberg et al. (24). This paper develops a method to estimate oparametrically from high-frequecy data (by way of a Laplace trasform) the margial law of the stochastic volatility process V t as well as its coditioal law for certai iterestig evets. For reasos just described, we develop the desity estimatio method withi a very geeral settig where V t ad W t ca be depedet ad jumps ad a drift term are preset as well. Although the precedig discussio is for X beig a fiacial asset price, the results i this paper obviously apply to ay statistical applicatio where highfrequecy observatios of a Itô semimartigale (which icludes may cotiuous-time models) are available. O ituitive level, our method ca be described as follows. We use a cosie trasformatio of appropriately re-scaled high-frequecy returs data (which is aki to usig a bouded ifluece fuctio) that essetially separates out the jumps ad the drift, thereby leavig (essetially) the diffusive piece scaled by V t. Averagig the trasform over time yields the realized Laplace trasform of volatility studied i Todorov ad Tauche (212). This trasform estimates the real-valued Laplace trasform of the uderlyig spot volatility process, ad it further achieves this without ay eed for staggerig of price icremets, explicit trucatio, or other techiques ivolvig tuig parameters commoly used for jump-robust measures of volatility. Real Laplace trasforms uiquely idetify the distributio of oegative radom variables, so a secod step i the estimatio is to ivert the realized Laplace trasform of volatility ad thereby recover a estimate of the volatility desity. The task of ivertig real Laplace trasforms also arise i aalysis of certai physical pheomea. Historically, iversio of the real Laplace trasform, where trasform values are oly available o the oegative real axis but ot the etire complex plae, was amog the most otorious of all ill-posed problems. However, recet regularizatio algorithms developed by Kryzhiy (23a,b) alog with the availability of high 3

4 speed computig equipmet (for ested umerical itegratios) reder the iversio a quick ad easy task to compute i a matter of a few miutes usig stadard software like Matlab. The role of regularizatio i this cotext is to guaratee statistical cosistecy whe the volatility Laplace trasform is recovered with samplig error, as is case here. Our oparametric volatility desity estimate is a itegral o R + of the realized Laplace trasform multiplied by a (determiistic) regularized kerel. To aalyze the asymptotic behavior of the above itegral, the local uiform asymptotics of the realized Laplace trasform derived i Todorov ad Tauche (212) does ot suffice. Here, therefore, we derive the asymptotic behavior of the realized Laplace trasform cosidered as a process i a weighted L 2 (R + ) space, which requires i particular to boud the discretizatio error for icreasig values of the argumet of the Laplace trasform. We further derive bouds o the magitude of the error i the oparametric desity estimatio due to the regularizatio. Combiig these results, we are able to boud the mea squared error of our oparametric volatility desity estimate ad show that it achieves optimal rate of covergece (for the assumed smoothess of the desity). We ca compare our method with the decovolutio approach of Va Es et al. (23) ad Comte ad Geo-Catalot (26). The methods of these papers are developed for a process without jumps, i.e., without J t i (1), ad with stochastic volatility V t idepedet from the Browia motio W t i (1) (ad further without drift i the case of Comte ad Geo-Catalot (26)). I such a setup, the logarithm of the squared price icremets is (approximately) a sum of sigal (the log volatility) ad oise, ad oe ca apply a decovolutio kerel (Va Es et al. (23)) or a pealized projectio (Comte ad Geo-Catalot (26)) to geerate a oparametric estimate of the distributio of the log volatility. The rate of covergece of this estimatio depeds, of course, o the smoothess of the volatility desity, with a logarithmic rate i the least favorable situatio. I our case, similar rates apply o comparable smoothess coditios for the desity, but uder the much weaker ad empirically plausible assumptios o jumps ad leverage for X i (1). O a more practical level, the method here avoids takig logarithms of 4

5 high-frequecy squared price icremets, which could be problematic i some istaces, because the log trasformatio iflates (despite the ceterig) the o-trivial umber of zero returs due to discretizatio, ad it further weighs more heavily the smaller icremets which are more proe to microstructure oise effects. We test our method o simulated data that mimics a typical data set available i fiace ad fid that the method ca recover reasoably well the volatility desity. We further provide guidace o the choice of the regularizatio parameter. I a empirical applicatio, we ivestigate the distributio of the spot volatility for three largecap stocks. Earlier work by Aderse et al. (21) ivestigated the distributio of daily realized volatility (which is the sum of the squared daily high-frequecy icremets) of fiacial series, exchage rates i their case. Here we go oe step further ad recover the distributio of spot volatility. Spot ad realized volatility differ due to the time-aggregatio as well as the presece of price jumps. The evidece suggests that the desity of spot volatility is less cocetrated aroud the mode with more mass i the extreme tails tha that of a (jump-robust) realized volatility measure. This latter fidig uderscores the presece of short-term volatility spikes. We also ivert the realized Laplace trasform o days followig a sigificat price jump ad provide oparametric evidece that volatility icreases sigificatly after jumps with dimiishig impact over time. Overall, the oparametric aalysis sheds light o the importace of the variability of the stochastic volatility process V t i accoutig for big moves i asset prices i additio to pure price jumps geerated by J t. The rest of the paper is orgaized as follows. Sectio 2 states the problem ad the assumptios ad presets the mai asymptotic results. Sectio 3 reports umerical experimets o the iversio of real-valued Laplace trasform usig our proposed method, both i the case whe the latter is the true trasform ad the case where it is recovered from high-frequecy observatios of a jump-diffusio process. Sectio 4 reports o the empirical applicatio to high-frequecy stock data. Sectio 5 cocludes. Sectio 6 cotais the proof of the theoretical results. 5

6 2 Mai results We start with describig our estimatio method ad derivig its asymptotic behavior. 2.1 Setup ad defiitios We first state the ecessary assumptios that we will eed. The observed process X is give by its dyamics specified i (1) ad is defied o a filtered probability space (Ω, F, (F t ) t, P) which satisfies the usual coditios. Our assumptio for X is give i the followig. Assumptio A. For the process X specified i (1) assume A1. For every t we have E { α t 2 + V t 2 + J t 2} K for some positive costat K. A2. J t is a jump process of the form J t = t R κ(x) µ(ds, dx) + t R κ (x)µ(ds, dx) where µ is iteger-valued measure o R + R with compesator ν(ds, dx); µ(ds, dx) = µ(ds, dx) ν(ds, dx); κ(x) is cotiuous fuctio with κ(x) = x aroud zero, or is idetically zero aroud zero, ad it is zero outside a compact set cotaiig the zero; κ (x) = x κ(x). A3. For every t,s we have E ( V t V s 2 F s t ) Ks t t s for some F s t -measurable radom variable K s t with E K s t 2+ι < K for some positive costat K ad arbitrary small ι >, ad E J t J s p K t s for every p (β, 2], some β [, 2) ad a positive costat K. A4. V t is a statioary ad α-mixig process with α mix t small ι >, where = O(t 1 ι ) whe t for some arbitrary α mix t = sup A F, B F t P(A B) P(A)P(B), F = σ(v s, s ) ad F t = σ(v s, s t). (2) Assumptio A1 imposes some mild itegrability coditios o the differet compoets of X. Some of them ca be potetially relaxed, but evertheless they are very weak ad satisfied i virtually all parametric models used i empirical fiace. Assumptio A2 specifies the jump process i X t. We ote that there is very little structure that is assumed for the jumps ad i particular time-variatio i the jump compesator (both i the form of time-varyig jump 6

7 size ad time-varyig jump itesity) is allowed for. I assumptio A3 we impose restrictio o the variability i the processes V t ad J t. The part of A3 cocerig V t is very miimal ad i particular is satisfied whe V t is a Itô semimartigale (as is the case i the popular affie jumpdiffusio framework) but it also holds for certai log-memory specificatios (A3 also stregthes slightly the itegrability coditio for V t i A1). We also poit out that assumptio A allows for jumps i V t that ca have arbitrary depedece with J t which will be of practical importace as we will see i the empirical applicatio. The restrictio of A3 o the jump compoet J t is that the so-called Blumethal-Getoor idex of the latter (which ca be radom) is bouded by β. We ote that we allow for β > 1 which meas that ifiite variatio jumps are icluded i our aalysis as well. Fially A4 is a (stadard) mixig coditio o the volatility process ad it is satisfied i wide classes of volatility models. As stated already i the itroductio, our goal i this paper is to recover oparametrically the desity of the spot volatility margial law (with respect to Lebesgue measure), which we deote with f(x) (ad assume to exist almost everywhere, ad it further does ot deped o t as we are iterested i the case whe volatility is a statioary process). Our ext assumptio imposes the ecessary coditios o f(x). Assumptio B. The margial law of the statioary process V t has desity f(x) which is piecewise cotiuous ad has piecewise cotiuous derivatives o [, ) with f(+) ad f (+) possibly ifiite. We further have B1. f (x) = O(x q ) as x ad f (x) = O(x 1/2 ι ) ad f(x) = o(1) as x, for some oegative q < 5/2 ad arbitrary small ι >. B2. f(x) ad f (x) are bouded o R + with f(x) = o(x 1 ι ) ad f (x) = o(x 2 ι ) for some arbitrary small ι > whe x ad f(+) =. The degree of smoothess of the desity aturally impacts the precisio of estimatio as i stadard oparametric desity estimatio (based o direct observatios of the process) ad the 7

8 above assumptio provides such coditios. Assumptio B1 is quite weak ad allows for the desity of V t to explode aroud zero. Assumptio B2 stregthes B1 by rulig out explosios aroud zero ad further requirig a rate of decay of the volatility desity (ad its derivative) at ifiity. Our strategy of estimatig oparametrically the volatility desity from high-frequecy observatios of X t is based o first recoverig the Laplace trasform of the volatility desity ad the ivertig it. To this ed, we deote the real-valued Laplace trasform of the margial distributio of the process V t with L(u) = E [exp ( uv t )], u. (3) I Todorov ad Tauche (212) we have proposed the followig oparametric estimate of the uobserved volatility Laplace trasform from high-frequecy observatio of X t o the discrete equidistat grid, 1,..., j,..., T L(u) = 1 T T i=1 ( ) cos 2u i X, i X = X i X i 1, u, (4) which we refer to as realized Laplace trasform. As show i Todorov ad Tauche (212), for T ad, we have locally uiformly i u L(u) P L(u), (5) ad their is a associated Cetral Limit theorem but we will ot make use of it here. We ote, i particular, that L(u) is robust to presece of jumps i X (the compoet J t i (1)) as well as ay depedece betwee the volatility process V t ad the Browia motio W t. The results that follow will cotiue to hold if the observatio times are o-equidistat but still oradom (by coditioig they ca be also further exteded to the case whe the samplig is radom but idepedet from the process X), i.e., if o the iterval [, T ] we observe X o the discrete grid, = τ(, ) < τ(, 1) <... < τ(, i) <... ({τ(, i) : i, 1} is a double sequece with idexig the sequece of discretizatio grids). We deote (, i) = τ(, i) 8

9 τ(, i 1) ad with πt = sup i: τ(,i) T (, i) the mesh of the grid o [, T ]. The L(u) i the o-equidistat case gets geeralized to L(u) ( 2u (, ) = 1 T i: τ(,i) T (, i) cos i) 1 i X ad we eed π T as T ad for the cosistecy i (5). Further, the limit results i Theorem 1 below cotiue to hold with 1 replaced with πt. The (real-valued) Laplace trasform of a oegative radom variable uiquely idetifies its distributio (see e.g., Feller (1971)). However recoverig the distributio from the Laplace trasform is a ill-posed problem (Tikhoov ad Arsei (1977)) ad hece oe eeds a regularizatio to make the iversio problem a cotiuous operator o the space of Laplace trasforms. Here, we adopt a approach proposed i Kryzhiy (23a,b) ad propose the followig regularized iversio of the true Laplace trasform L(u) f R (x) = L(u)Π(R, xu)du, (6) where R > is a regularizatio parameter ad the kerel Π(R, x) is defied as Π(R, x) = 4 [ ( ) πr u cos(r l(u)) sih 2π 2 2 u 2 si(xu)du + 1 ( ) πr ] u si(r l(u)) + cosh 2 u 2 si(xu)du. + 1 (7) As show i Kryzhiy (23a), f R (x) f(x )+f(x+) 2 for every x > (poitwise) as R where we defie f(x+) = lim y x f(y) ad f(x ) = lim y x f(y). We further have (L 1 (u) L 2 (u))π(r, xu)du K sup L 1 (u) L 2 (u), u R + for ay two Laplace trasforms L 1 (u) ad L 2 (u) (Kryzhiy (23a), Theorem 2) ad a positive costat K, which shows that this is ideed a regularizatio of the ill-posed iversio problem (Tikhoov ad Arsei (1977)). It is easy to develop ituitio about the regularizatio by usig the coectio betwee the regularized ad true desity derived i Kryzhiy (23a) f R (x) = f(u)δ R,x (u)du, δ R,x (u) = 2 ux π(u 2 x 2 si (R l(u/x)), x >. (8) ) 9

10 The fuctio δ R,x (u) is a smooth approximatio of the Dirac delta fuctio at the poit x. The regularizatio parameter R determies the degree of smoothig ad correspods to the choice of the badwith i regular oparametric kerel estimators where oe has direct observatios of the variable of iterest (ulike here where we do ot observe directly V t ). Higher values of R meas that δ R,x (u) is closer to the Dirac delta ad hece this implies less smoothig. However, these higher values ca lead to a good result oly if the precisio of the iput (here the realized Laplace trasform) is high, otherwise the oscillatios i δ R,x (u) will cause very oisy desity estimates. Exactly the opposite holds for low values of R. This is further cofirmed from Figure 1 where we plot the fuctio δ R,x (u) for several differet values of x ad a low ad a high value of R that we will actually use i our umerical work later o. 2 x=.5, R=1. 5 x=.5, R= u x=1., R= u x=5., R= u u x=1., R= u x=5., R= u Figure 1: The fuctio δ R,x (u). The solid lies correspod to the theoretical value of δ R,x (u) give i (8). The dots o the plots correspod to our umerical evaluatio of δ R,x (u) via evaluatig the itegral i (6) with L(u) = exp( ux). A alterative represetatio of δ R,x (u) is as the regularized iverse of the fuctio exp( ux), that is δ R,x (u) = exp( tx)π(r, ut)dt. We ca use this coectio to check the impact o the estimatio of the error due to the umerical itegratio ivolved i computig (6) ad (7). We 1

11 plot the resultig estimates of δ R,x (u) with the dotted lies o Figure 1. As see from the figure, the dotted lies plot o the top of the solid lies (which correspod to the theoretical value of δ R,x (u) i (8)) which idicates that the umerical error is egligible for the values of R used i the computatios (which covers the rage of R that we are goig to use i practice). The feasible aalogue of f R (x), based o the realized Laplace trasform, is aturally defied as f R (x) = L(u)Π(R, xu)du, (9) ad this will be our oparametric estimate of the desity of V t from the discrete observatios of the uderlyig process X, X 1,...,X T. The local uiform asymptotics for L(u) developed i Todorov ad Tauche (212) does ot suffice to study f R (x) sice the itegral i (9) is defied o R +. To aalyze the asymptotics of f R (x) (both poitwise ad as a fuctio) we eed: (i) asymptotics of L(u) cosidered as a process o weighted L 2 (R + ) space ad (ii) bouds for the order of magitude of the regularizatio error. We tur to this ext. 2.2 Iversio of Real-Valued Laplace Trasforms We have the followig asymptotic result for the regularized estimated desity f R (x). Theorem 1 For the process X i (1), we assume that assumptio A holds ad deote f(x) = f(x )+f(x+) 2. Let, T ad R. (a) If assumptio B1 holds the ( f R (x) f(x) ) 2 ) 5 2q = O (R 2( 3 1) log 2 (R), (1) ( ) ( E fr (x) f R (x) = O exp (πr/2) R 2 ((1/β) 1 (β 1)/2) 1/4+ι), ( ) 2 ( ( E fr (x) f R (x) = O exp (πr) R 4 T 1 + (1/2) (1 β/2)+ι)), ι >. (b) If assumptio B2 holds the { ( f(x)) } ( 2 log 2 (R) ( E w(x) fr (x) dx = O exp (πr) R 4 R + R 2 T β/2+ι)), ι >, where w(x) is a bouded oegative-valued fuctio with w(x) = o(x 2 ) for x. 11 (11) (12)

12 If we further deote f R (x) = L(u) Π(R, xu)du, Π(R, xu) = χ(r 2 x)π(r, xu), χ(u) = u 1, (13) the (uder assumptio B2) { ( f(x)) } ( 2 log 2 (R) ( E fr (x) dx = O exp (πr) R 8 R + R 2 T β/2+ι)), ι >. (14) The result of Theorem 1 implies that f R (x) is a cosistet estimate of the volatility desity f(x) at the poits of cotiuity of the latter ad estimates the average of the right ad left limits (which exist by assumptio B) at the poits of discotiuity. The theorem goes oe step further ad provides bouds o the bias ad the variace of the estimator. There are two sources of bias i the estimatio. Oe, that is determiistic, arises from the regularizatio of the iversio, ad aturally depeds oly o the regularizatio parameter R. Its boud is give i (1). The secod source of bias is stochastic ad arises from the discretizatio error, i.e., we do ot observe directly the empirical volatility Laplace trasform 1 T T exp ( uv s) ds but we eed to recover it from high-frequecy data. The magitude of this bias is give by the first expressio i (11) ad aturally depeds oly o the mesh of the observatio grid, i.e., 1/. Fially, the boud o the variace of the estimator is give i the secod expressio i (11). It depeds both o the spa of the data ad the mesh of the observatio grid. The leadig compoet of E ( fr (x) f R (x) (provided is icreasig sufficietly fast relative to T ad R is fixed) is give by 1 T Π(R, xu)π(r, xv)σ u,v dudvdx, where Σ u,v is the log-ru variace-covariace kerel of 1 T Σ u,v = T (exp( uv s) L(u)) ds, i.e., {E [(exp ( uv t ) L(u)) (exp ( vv ) L(v))] + E [(exp ( uv ) L(u)) (exp ( vv t ) L(v))]} dt. I the most commo case of assumptio B2 ad provided T α for some α >, we ca set ) 2 R = γ log(t ) π for some positive γ < α(1 β/2) 1, ad get the squared bias ad the variace of the estimator of (almost) the same order of magitude. Such choice of R will result i (oly) 12

13 logarithmic rate of covergece for our volatility desity estimator. This is ot surprisig give our weak assumptio B2 for the desity f(x). Squared logarithmic rate of covergece for the log-volatility desity, i a settig where X t does ot cotai jumps ad V t is idepedet from W t, is obtaied via a decovolutio approach i Va Es et al. (23) where it is coected with the optimal rate of decovolutig a desity i the presece of super-smooth oise derived i Fa (1991) (i the cotext of i.i.d. data). Va Es et al. (23) assume f(x) is twice cotiuously differetiable ad obtai optimal rate of covergece for their desity estimator of squared logarithmic rate while here we assume oly first-order derivatives for f(x) ad hece we ed up with (almost) logarithmic rate of covergece (which is the optimal decovolutio rate uder this smoothess assumptio for f(x) i the presece of super-smooth oise, see Fa (1991)). Similarly, here if we assume more smoothess coditios for the volatility desity f(x) we ca show that the determiistic bias due to the regularizatio f(x) f R (x) is of smaller order of magitude tha the boud give i (1). This i tur will imply faster rate of covergece for our volatility desity estimator (provided R is chose optimally). Thus we have a atural lik betwee the rate of covergece of our desity estimator ad the degree of smoothess of the ukow volatility desity. This is similar to the results i Comte ad Geo-Catalot (26). I the settig of o drift ad o jumps as well as idepedet W t ad V t, Comte ad Geo-Catalot (26) show that a pealized projectio type volatility desity estimator ca provide faster rates of covergece for smoother volatility desities. Uder the miimal smoothess requiremet for the volatility desity i assumptio B, the relative speed coditio betwee T ad is relatively weak, i.e., as poited above it is of the form T α for some α >. Such a coditio, i particular, is much weaker tha the correspodig requiremet i the problem of parametric estimatio of diffusios from discrete observatios, see e.g., Prakasa Rao (1988). This of course is o surprise ad is a mere reflectio of the much smaller role (i relative terms) played by the discretizatio error i our oparametric volatility desity estimatio. We also poit out that, quite aturally, the discretizatio error is bigger, the 13

14 bigger is the boud o the activity idex of jumps i X, β. This is because higher activity jumps become harder to separate from diffusive iovatios i the icremets of X, which i particular implies that ( cos 2u i X ) cos ( 2u( i X i J)) is larger for a higher value of β (its order of magitude is (β 1)/2 (1/β) 1+ι for ι > arbitrary small). We also poit out that the discretizatio error is of smaller magitude i (12) versus (11) (provided β < 1) because of the extra assumptio o the behavior of the volatility desity aroud zero i assumptio B2 (the volatility desity approaches zero aroud the origi). We ote further that the result i part (a) is poitwise, i.e., for fixed x, while that i part (b) is for the mea itegrated squared error (ote that uder B1, f(x) does ot eed to belog to L 2 (R)). For the estimator f R (x), we provide i (12) a boud for its mea itegrated squared error weighted by a fuctio w(x) that is bouded ad o(x 2 ) as x but is otherwise arbitrary. The role of w(x) is to dow-weight the estimatio error i f R (x) aroud zero. I (13) we propose a slight modificatio of f R (x) which we defie as f R (x). We have f R (x) = χ(r 2 x) f R (x) ad the fuctio χ(r 2 x) serves to dampe our origial desity estimate aroud zero. This dampeig i tur allows us to boud i (14) the mea itegrated squared error of f R (x) with w(x) = 1, i.e., without ay dow-weightig of the estimatio error aroud zero. I our applicatios below we will use f R (x) but we will evaluate it startig from sufficietly small value of x that is above zero, guidace for which ca be easily obtaied from the quatiles of ay oparametric daily itegrated volatility estimates. Fially, the aalysis here ca be easily exteded to recoverig the volatility desity coditioed o some set. Oe iterestig example, that we will cosider i our empirical applicatio, is the occurrece of big price jumps. The aalysis i Bollerslev ad Todorov (211) ca be used to boud the discretiatio error i idetifyig the set of big jumps (uder some coditios for J t ). Also, oe ca cosider a settig of fixed spa, i.e., T fixed, i which f R (x) will recover the desity of the empirical volatility distributio over the give iterval of time. Of course, for this we will eed the smoothess assumptio B to hold for the empirical volatility distributio. 14

15 A example where this will be the case is whe V t is a o-gaussia OU (Orstei-Uhlebeck) process (see equatio (2) below) i which the drivig Lévy process is compoud Poisso (ote that assumptio B allows for discotiuities i the desity). Such models for volatility have bee cosidered for example i Bardorff-Nielse ad Shephard (21). More geerally, however, the desity of the empirical distributio will ot be differetiable ad oe ca istead recover the empirical cumulative distributio of volatility. 3 Numerical Experimets We proceed ext with umerical experimets to test our estimatio method developed i the previous sectio. We first ivestigate how well our estimator ca recover the volatility desity i the ifeasible sceario whe the volatility Laplace trasform is measured without error. We the cosider the feasible sceario where the volatility Laplace trasform is recovered from highfrequecy price data via the realized Laplace trasform. 3.1 Ivertig Kow Laplace Trasforms We use two distributios i our umerical aalysis here which will be the margial laws of two popular volatility specificatios that we will use i our Mote Carlo aalysis below. The first is the Gamma distributio. We deote Y G(a, b) for a radom variable with probability desity f G (x) = ba x a 1 Γ(a) exp ( bx) 1 {x>}, a, b >, (15) with correspodig real-valued Laplace trasform give by L G (u) = ( ) 1 a. (16) 1 + u/b The secod distributio we use is the Iverse-Gaussia. We deote Y IG(µ, ν) for a variable with probability desity give by [ ν f IG (x) = 2πx 3 exp ν ] 2µ 2 (x µ)2 1 x {x>}, (17) 15

16 with correspodig real-valued Laplace trasform { [ L IG (u) = exp (ν/µ) 1 ]} 1 + 2µ 2 u/ν. (18) It is easy to check that the two distributios satisfy assumptio B2. I Table 1 we list all the differet cases cosidered i this sectio ad give the correspodig parameters. We look i particular at settigs with small, average, ad big dispersio aroud the mode of the desity. The Gamma ad Iverse Gaussia distributios are the margial laws of two volatility specificatios, widely used i empirical fiace. The first is the square-root diffusio process give by dv t = κ(θ V t )dt + σ V t db t, κ, σ, θ >, σ 2κθ. (19) The margial distributio of the square-root diffusio process is the Gamma distributio with parameters a = 2κθ σ 2 ad b = 2κ σ 2 i the parametrizatio of (15). The secod volatility specificatio is a o-gaussia OU process give by dv t = κv t dt + dl t, κ >, (2) where L t is a Lévy subordiator. Followig Bardorff-Nielse ad Shephard (21), we specify the o-gaussia OU process via its margial distributio which will be the Iverse-Gaussia (which is self-decomposable ad hece this is possible, see e.g., Sato (1999)) with parametrizatio give i the previous subsectio. It ca be show, see e.g., Todorov et al. (211), that the Lévy [ measure of L t is give by κν exp( ν2 x/(2µ 2 )) x 1.5 2Γ(.5) 2 + ν2 x ]..5 Further, both volatility specificatios 2µ 2 i (19) ad (2) satisfy assumptio A. I Table 1 we report the parameter values of the volatility specificatios correspodig to the differet cases cosidered for their margial distributios. I Table 2 we report the Itegrated Squared Error (ISE) i recoverig the volatility desity from the exact Laplace trasform (over the quatile rage Q.5 -Q.995 ). The precisio across all cases is very high. Whe we use the exact Laplace trasform, there is obviously o estimatio error ad all error is due to the regularizatio ad the umerical itegratio. We cosider a rage 16

17 Table 1: Parameter Settig for the Mote Carlo Case Margial Distributio of V t Parameter Values G-L Gamma κ =.2, θ = 1., σ 2 = 2κθ 4. G-M Gamma κ =.2, θ = 1., σ 2 = 2κθ 2.5 G-H Gamma κ =.2, θ = 1., σ 2 = 2κθ 1.5 IG-L Iverse-Gaussia κ =.2, µ = 1., ν = 3. IG-M Iverse-Gaussia κ =.2, µ = 1., ν = 1. IG-H Iverse-Gaussia κ =.2, µ = 1., ν =.5 Note: Cases G-L, G-M ad G-H correspod to the square-root diffusio process i (19) ad the parameters of the Gamma distributio are give by a = 2κθ ad b = 2κ. σ 2 σ 2 Cases IG-L, IG-M ad IG-H correspod to the o-gaussia OU process i (2) with Iverse-Gaussia margial distributio. Table 2: Itegrated Squared Error of Desity Estimate: Kow Laplace Trasform Case (x)dx Regularizatio Paramater R R = 2. R = 3. R = 4. R = 5. G-L G-M G-H IG-L IG-M IG-H Note: The ISE Q.995 Q.5 (f R (x) f(x)) 2 dx is approximated by a Riema sum with legth of the discretizatio iterval of.5. The rage of itegratio is Q.5 -Q.995 for Q α deotig the α-quatile. Each of the cases is explaied i Table 1. of values for the regularizatio parameter R ad we ca see from Table 2 that R plays a big role i the precisio of the estimatio. Small values of R result i bias due to over-smoothig (recall Figure 1) while very big values i R ca result i a bigger error due to the umerical itegratio. We also poit out that the optimal value of R depeds o the volatility desity which is of course ukow. 3.2 Ivertig Estimated Laplace Trasforms We tur ext to the feasible case where the volatility desity is ot kow ad has to be estimated from high-frequecy observatios of X. I the simulatios, the price process is give by (1) with 17

18 volatility followig either (19) or (2), α t =, J t is a compoud Poisso process with itesity 1/3 (i.e., oe jump every three days o average) ad ormally distributed jump size with mea ad variace of.3. For simplicity i the Mote Carlo setup we set the volatility process idepedet from the price process. Simulatio evidece i Todorov et al. (211) idicates that the leverage effect has egligible effect o the realized Laplace trasform i fiite samples (recall from (5) that leverage has o asymptotic effect o L(u)). I all experimets we set E(V t ) = 1 which implies jumps cotribute approximately 1% of price variatio, cosistet with prior empirical evidece. The uit of time i our simulatio desig is a day ad we assume the spa is T = 3, days with = 76 which correspods to samplig the price process every 5-miutes i a 6.5 hour tradig day for approximately 12 years. I Table 3 we report the precisio i recoverig the volatility desity from a sigle simulatio from each of the scearios for a rage of values of the regularizatio parameter R. Comparig Table 2 with Table 3, ot surprisigly, we ca see that the ISE is orders of magitude higher whe Laplace trasform has to be estimated from the data due to the estimatio error. Nevertheless, provided the appropriate R is used, the error i estimatio is reasoably small. The values of R for which the precisio is highest i the case of estimated Laplace trasform are lower tha the case whe Laplace trasform is kow. This is because estimatio error prevets us from usig kerels with high focusig ability, i.e., we eed to smooth more to remove the effect of the estimatio error. I the case of estimated Laplace trasforms we have a U-shaped patter i the ISE: too large ad too small values of R correspod to bigger ISE. This effect ca be most clearly see from Figure 2 where we plot the estimated desities for three differet values of R for the simulatio sceario G-H. Too low R results i over-smoothig ad relatively big estimatio bias. Icreasig R improves the precisio. However, whe R is too big the estimatio oise gets blow up ad this leads to the oscillatios i the estimated desity ( iherited from the more focused kerel) that ca be see from the last plot o Figure 2. 18

19 Table 3: Itegrated Squared Error of Desity Estimate: Estimated Laplace Trasform Case (x)dx Regularizatio Paramater R R = 1. R = 1.5 R = 2. R = 3. R = 4. G-L G-M G-H IG-L IG-M IG-H Note: The computatios are based o a sigle simulatio from the models give i Table 1 ad volatility Laplace trasform estimate usig the realized Laplace trasform L(u) defied i (4). The ISE is approximated the same way as i the calculatios for Table G H: R = G H: R = G H: R = Figure 2: The effect of regularizatio. The figure shows the recovered desity of V t for a simulated series from model G-H whose itegrated squared error is reported i Table 3. The solid lies correspod to the estimated desity ad the dashed lies to the true oe. 19

20 3.3 Mote Carlo We ow tur to a Mote Carlo study usig the above specified setup ad 1, replicatios. Based o the aalysis i the previous subsectios, the crucial questio is how to pick R as the optimal value of R depeds o the ukow volatility desity. From Figure 2 we kow that whe R is too high for the precisio with which we ca recover the Laplace trasform of volatility from the high-frequecy data, the the recovered desity starts to oscillate. Therefore, a reasoable ad very easy rule is to set R as the largest value which results with a miimum umber of violatios of the quasicocavity coditios (see e.g. Koeker ad Mizera (21)) of the recovered volatility desity. I particular, for the case plotted i Figure 2, this will lead us to pickig the middle value of R = 2.. We implemet this rule i the Mote Carlo study. We ote that this ca lead to a differet value of R for the differet realizatios of the simulated processes. The results from the Mote Carlo are reported i Table 4. As see from the results i the table, we have relatively good precisio with which we ca recover the volatility desities across the differet simulatio scearios. I geeral also the mea itegrated squared error (MISE) is comparable with the miimal ISE from the sigle realizatios of the process reported i Table 3. The hardest case of all simulatio scearios is the IG-H which correspods to iverse-gaussia with very high volatility of volatility. The estimatio error ivolved i this case is relatively big, ecessitatig small values of R to atteuate its effect o the iversio, which i tur leads to some bias. 4 Empirical Applicatio We ext illustrate the use of the developed oparametric techique i a short empirical applicatio. We aalyze three large-cap stocks that are part of the S&P 1 idex: oe i the techology sector (IBM), oe i utilities ad services (Johso ad Johso, abbreviated by its ticker JNJ) ad oe i the fiacial sector (Bak of America, abbreviated by its ticker BAC). The sample period is from April 1997 till December 21, ad we sample every 5-miutes durig the tradig 2

21 Table 4: Mote Carlo Results: MISE Case f 2 (x)dx MISE Case (x)dx MISE R R G-L IG-L G-M IG-M G-H IG-H Note: The computatios are based o 1, replica of the models give i Table 1 with T = 3, ad = 76. The MISE is a sample average over the replicatios of the ISE which i tur is approximated the same way as i the calculatios for Table 2. The choice of the regularizatio parameter R is over the discrete grid 1 :.25 : 3.5 ad is the largest umber of this set with least violatios of quasicocavity of the desity estimate. hours o each tradig day (which is our uit of time) ad this results i 76 high-frequecy retur observatios per day (we omit the price at the ope ad at the close to atteuate potetial special effects with start ad ed of tradig). We exclude days i which there was o tradig of the stock for more tha half of the day. This resulted i a total of 3423 days for IBM, 3421 for BAC ad 342 days for JNJ i our sample. The 5-miute samplig frequecy is coarse eough so that the effect of microstructure oise is egligible. Usig the trucated variatio defied later i (22), we estimate that jumps cotribute the otrivial 11.6, 11.1 ad 12.7 percet of the total price variatio for IBM, BAC ad JNJ stocks respectively. Before turig to the actual estimatio we stadardize the high-frequecy returs, whe usig them i the calculatio of the realized Laplace trasform, i order to accout for the wellkow presece of a diural determiistic withi-day patter i volatility, see e.g., Aderse ad Bollerslev (1997). To this ed, V t i (1) is replaced by Ṽt = V t d(t t ) where V t is our origial statioary volatility process satisfyig assumptio A ad d(s) is a positive differetiable determiistic fuctio o [, 1] that captures the diural patter. The we stadardize each high-frequecy icremet i X with 1/ d i for d i = ĝi ĝ, ĝ i = T T i t X 2 1( i t X α ϖ ), ĝ = 1 t=1 ĝ i, i = 1,..., T, α >, ϖ (, 1/2), i=1 where i t = t 1 + i [i/], for i = 1,..., T ad t = 1,..., T. We set ϖ =.49 ad α = 3 BV t 21

22 for BV t deotig the Bipower variatio of Bardorff-Nielse ad Shephard (24, 26) defied as BV t = π 2 t i=(t 1)+1 i 1X i X. (21) Ituitively, di estimates the determiistic compoet of the stochastic variace, ad the we stadardize the high-frequecy icremets with it. Todorov ad Tauche (212) derive the asymptotic effect of this cleaig procedure but sice d i estimates quite precisely the determiistic patter, aturally this effect is rather small. We plot the estimated desities of the spot volatility obtaied from the estimatio method based o the regularized iverse of the realized Laplace trasform i (9) for each of the three stocks o the three top paels of Figure 3. For ease of iterpretatio we preset the desity estimates for V t (ad ot V t ) i percetage terms, as this is the stadard way of quotig volatility o the market. We ca cotrast these spot volatility desity estimates with estimates of the desity of a jump-robust measure of the daily itegrated volatility obtaied by ivertig its direct empirical Laplace trasform. We measure the itegrated volatility usig the trucated variatio, origially proposed i Macii (29), ad applied here as T V t = t i=(t 1)+1 i X 2 1( i t X α ϖ ), (22) for the same choice of ϖ =.49 ad α = 3 BV t as for the estimatio of the diural compoet of volatility d i. Uder our assumptio A, T V t is a cosistet estimate for the uobservable itegrated volatility (ad this is the reaso for usig it as a bechmark for the volatility desity), i.e., we have for each fixed t 1 (see e.g., Jacod (28)) T V t P t t 1 σ 2 s ds, as. (23) The dashed lies i the top three paels of Figure 3 show the implied desity for the daily itegrated volatility from usig our method to ivert the empirical Laplace trasform of the T V t series, while the three lower paels show stadard kerel-desity estimates of T V t obtaied from 22

23 IBM BAC JNJ Figure 3: Noparametric Spot ad Itegrated Volatility Desity Estimates. The solid lies correspod to our oparametric estimate for the desity of V t while the dashed lies are oparametric desity estimates of daily T V t. The dashed lies i the top plots are based o ivertig the empirical Laplace trasform of T V t usig our estimator i (9) while the oes i the bottom plots are stadard kerel estimates with Silverma s automatic badwidth of h =.79 IQR T 1/5 for IQR deotig the iter-quartile rage. 23

24 the same observatios. There are several coclusios to be made from Figure 3. First, there is sigificat volatility of volatility: the spot volatility ca take values as high as five to six times its modal value. Thus volatility dyamics (ad i particular short-lived sharp chages i it) play a importat role i geeratig tail evets i idividual stock returs i additio to the price jumps. We recall that the realized Laplace trasform is robust to presece of price jumps. Therefore our oparametric separatio of volatility from jumps idetifies a rather otrivial role of stochastic volatility for geeratig extreme evets i asset prices. Comparig the spot with itegrated volatility we ca see a commo patter i the three stocks. The mode of the spot volatility desity is slightly to the left of that for the daily itegrated volatility, ad spot volatility has somewhat fatter distributio tha that of the itegrated volatility. The reaso for this is i the presece of short term volatility moves i the form of volatility jumps ad the mea reversio. The daily itegrated volatility averages out the sharp moves i volatility. Overall our oparametric evidece here poits to stochastic volatility with sigificat volatility of volatility possibly geerated by volatility jumps. We ivestigate further the hypothesis of volatility jumps ad their iteractio with the price jumps by computig coditioal desity estimates. I particular, we will use the methodology developed here to gather oparametric evidece regardig the effect of price jumps o stochastic volatility. I stadard volatility models, volatility is a diffusio process ad thus by costructio stochastic volatility does ot jump whe the price jumps (volatility ad price jumps though ca still be depedet i such settig as volatility ca drive the jump itesity). More recet parametric work has allowed for volatility jumps as i the o-gaussia OU model of Bardorff- Nielse ad Shephard (21), although i some specificatios volatility ad price jumps are costraied to be ucorrelated. To ivestigate the effect of price jumps o volatility we do the followig. We idetify the days i the sample where relatively large jumps occurred (we will be precise about what costitutes a large jump below) ad the costruct the realized Laplace trasform of volatility o a give (fixed) 24

25 umber of days after the day with the large jump. We the use our oparametric procedure to ivert the estimated Laplace trasform ad recover the desity of volatility a fixed umber of days after the occurrece of price jumps. More formally, for some big fixed τ > we defie for ay iteger k 1 the set of days with a big positive, respectively egative, jump as { I ±τ (k) = t = k + 1,..., T : {i = (t k 1) + 1,..., (t k)} { i = 1,..., T : i X ±(α ϖ τ) } }, where α ad ϖ are the same as the oes used i the costructio of T V t. I ±τ (k) is the set of days i the sample where k days ago a big positive or egative jump has occurred. We ca the costruct the realized Laplace trasform o the sets I ±τ (k), i.e., L ±τ 1 t ( X) (k) = I ±τ cos 2u i, (24) (k) t I ±τ (k) i=(t 1)+1 where I ±τ (k) deotes the size of the set I ±τ (k). Fially, we ca ivert L ±τ (k) usig (9). Our goal is to produce a oparametric estimate of the desities of V t 1 {t I τ (k)} ad V t 1 {t I +τ (k)} where the set I ±τ (k) is defied via I ±τ (k) = { t : [ t k 1; t k] } {s > : X s ±τ} =. (25) I our actual applicatio we set τ to τ = E(T V t ) 5/ which is five-stadard deviatio move for the cotiuous part of the high-frequecy retur (the mea of the trucated variatio is estimated by the correspodig sample average). This results i approximately 1 jumps of each sig i our sample to estimate the realized Laplace trasforms i (24). We further set k to 1 ad 22 which amouts to lookig at volatility 1 day ad 1 caledar moth after a big jump. The result of the calculatios are preseted o Figure 4. Comparig the estimated volatility desities o Figure 4 with the ucoditioal oes o Figure 3 we ca see a very proouced shift of the mode towards the right, i.e., volatility uambiguously icreases after a big jump. Iterestigly, the desity of the volatility oe moth after the jump starts movig towards the ucoditioal oe (compared with the desity the day followig the jump). The iterpretatio 25

26 is that a big price jump feeds ito higher future volatility with the effect dimiishig over time due to the mea reversio i volatility. Of course, oe should be careful i iterpretig the above evidece, as high volatility might be geeratig the big price jumps which i tur are followed by higher volatility. This will be the case where price jump itesity depeds o volatility. Nevertheless, our aalysis clearly shows that price jumps are very closely related with the stochastic volatility dyamics ad i particular i terms of parametric volatility modelig we eed models that allow for this coectio as for example i the o-gaussia OU model of Bardorff-Nielse ad Shephard (21). Aother iterestig commo patter across the three stocks is that there is a sigificat spread i the estimated volatility desities o Figure 4. This suggests that the size of the price jumps plays a big role i determiig the size of the impact it has o the future stochastic volatility. Thus the coectio betwee volatility ad price jumps is size depedet. Further, comparig the left ad the right side of Figure 4 we ca see that the volatility desities after a positive ad a egative jump are rather similar for these three stocks. Fially, it is iterestig to poit out that amog the three stocks, the oe whose volatility reacts strogest to the occurrece of price jumps is BAC. This is cosistet with the view that stocks i the fiacial sector are most sesitive to fiacial distress i the form of extreme market evets. 5 Coclusio I this paper we propose a oparametric method for estimatio of the spot volatility desity i a jump-diffusio model from high-frequecy data with icreasig time spa. The method cosists of aggregatig the high-frequecy returs data ito a fuctio kow as realized Laplace trasform, which provides a cosistet estimate of the uobservable real-valued volatility Laplace trasform. O a secod stage the estimated volatility Laplace trasform is iverted usig a regularized kerel method to obtai a estimate of the desity of spot volatility. We derive bouds o the MISE i the desity estimatio ad provide guidace o the feasible choice of the regularizatio parameter. A empirical applicatio for three large-cap stocks idicates the importace of short-term high 26

27 IBM: After Negative Jump IBM: After Positive Jump BAC: After Negative Jump BAC: After Positive Jump JNJ: After Negative Jump JNJ: After Positive Jump Figure 4: Noparametric Volatility Desity Estimates After a Price Jump. The solid (dashed) lie is a oparametric desity estimate of V t over the days i the sample which follow a day (respectively 22 days) after a positive (left side) or egative big jump (right side) i the price. The threshold for the big jumps is set to five stadard deviatios of the cotiuous part of the high-frequecy retur based o the sample mea of T V t of each idividual series. frequecy movemets i volatility that get smoothed over i formig estimates of daily realized variatio, ad it shows how to trace out the dyamic respose of the spot volatility desity to large price jumps of either sig as it relaxes back to the steady state ucoditioal desity. 6 Proofs The proof of Theorem 1 cosists of two parts. The first part is aalysis of the determiistic bias f R (x) f(x) (respectively R (f R(x) f(x)) 2 dx) ad the secod part deals with the estimatio error f R (x) f R (x) (respectively R w(x)( f R (x) f R (x)) 2 dx ad R ( f R (x) f R (x)) 2 dx). I what follows we will deote with K a positive costat that does ot deped o R, x, T ad. 27