design of an adaptive observer and its application to an adaptive pole placement controller abstract in this paper an adaptive observer design method based upon the least square approach is presented for an unknown time invariant continuous linear plant this observer is then implemented in the design of an adaptive pole placement problem for arbitrarily positioning all the poles of the unknown plant the generated feedback control signal is shown to converge to a desired pole placing state feedback control signal

high dimensional minimum variance portfolio estimation based on high frequency data abstract this paper studies the estimation of high dimensional minimum variance portfolio mvp based on the high frequency returns which can exhibit heteroscedasticity and possibly be contaminated by microstructure noise under certain sparsity assumptions on the precision matrix we propose estimators of the mvp and prove that our portfolios asymptotically achieve the minimum variance in a sharp sense in addition we introduce consistent estimators of the minimum variance which provide reference targets simulation and empirical studies demonstrate the favorable performance of the proposed portfolios

bootstrapping realized volatility we propose bootstrap methods for a general class of nonlinear transformations of realized volatility which includes the raw version of realized volatility and its logarithmic transformation as special cases we consider the independent and identically distributed i i d bootstrap and the wild bootstrap wb and prove their first order asymptotic validity under general assumptions on the log price process that allow for drift and leverage effects we derive edgeworth expansions in a simpler model that rules out these effects the i i d bootstrap provides a second order asymptotic refinement when volatility is constant but not otherwise the wb yields a second order asymptotic refinement under stochastic volatility provided we choose the external random variable used to construct the wb data appropriately none of these methods provides third order asymptotic refinements both methods improve upon the first order asymptotic theory in finite samples

bootstrapping integrated covariance matrix estimators in noisy jump diffusion models with non synchronous trading abstract we propose a bootstrap method for estimating the distribution and functionals of it such as the variance of various integrated covariance matrix estimators in particular we first adapt the wild blocks of blocks bootstrap method suggested for the pre averaged realized volatility estimator to a general class of estimators of integrated covolatility we then show the first order asymptotic validity of this method in the multivariate context with a potential presence of jumps dependent microstructure noise irregularly spaced and non synchronous data our results justify using the bootstrap to estimate the covariance matrix of a broad class of covolatility estimators the bootstrap variance estimator is positive semi definite by construction an appealing feature that is not always shared by existing variance estimators of the integrated covariance estimator as an application of our results we also consider the bootstrap for regression coefficients we show that the wild blocks of blocks bootstrap appropriately centered is able to mimic both the dependence and heterogeneity of the scores we provide a proof of construction of bootstrap percentile and percentile t intervals as well as variance estimates in this context this contrasts the traditional pairs bootstrap which is not able to mimic the score heterogeneity even in the simple case where no microstructure noise is present our monte carlo simulations show that the wild blocks of blocks bootstrap improve the finite sample properties of the alternative approach based on the gaussian approximation we illustrate its practical use on high frequency equity data

testing if the market microstructure noise is fully explained by the informational content of some variables from the limit order book abstract in this paper we build tests for the presence of residual noise in a model where the market microstructure noise is a known parametric function of some variables from the limit order book the tests compare two distinct quasi maximum likelihood estimators of volatility where the related model includes a residual noise in the market microstructure noise or not the limit theory is investigated in a general nonparametric framework in the presence of residual noise we examine the central limit theory of the related quasi maximum likelihood estimation approach

testing for jumps in noisy high frequency data a b s t r a c t this paper proposes a robustification of the test statistic of ait sahalia and jacod 2009b for the presence of market microstructure noise in high frequency data based on the pre averaging method of jacod et al 2010 we show that the robustified statistic restores the test s discriminating power between jumps and no jumps despite the presence of market microstructure noise in the data

robust score and portmanteau tests of volatility spillover abstract this paper presents a variety of tests of volatility spillover that are robust to heavy tails generated by large errors or garch type feedback the tests are couched in a general conditional heteroskedasticity framework with idiosyncratic shocks that are only required to have a finite variance if they are independent we negligibly trim test equations or components of the equations and construct heavy tail robust score and portmanteau statistics trimming is either simple based on an indicator function or smoothed in particular we develop the tail trimmed sample correlation coefficient for robust inference and prove that its gaussian limit under the null hypothesis of no spillover has the same standardization irrespective of tail thickness further if spillover occurs within a specified horizon our test statistics obtain power of one asymptotically we discuss the choice of trimming portion including a smoothed p value over a window of extreme observations a monte carlo study shows our tests provide significant improvements over extant garch based tests of spillover and we apply the tests to financial returns data finally based on ideas in patton 2011 we construct a heavy tail robust forecast improvement statistic which allows us to demonstrate that our spillover test can be used as a model specification pre test to improve volatility forecasting

modeling returns volatility realized garch incorporating realized risk measure abstract this study applies realized garch models by introducing several risk measures of intraday returns into the measurement equation to model the daily volatility of e mini s p 500 index futures returns besides using the conventional realized measures realized volatility and realized kernel as our benchmarks we also use generalized realized risk measures realized absolute deviation and two realized tail risk measures realized value at risk and realized expected shortfall the empirical results show that realized garch models using the generalized realized risk measures provide better volatility estimation for the in sample and substantial improvement in volatility forecasting for the out of sample in particular the realized expected shortfall performs best for all of the alternative realized measures our empirical results reveal that future volatility may be more attributable to present losses risk measures the results are robust to different sample estimation windows

causality effects in return volatility measures with random times we provide a structural approach to identify instantaneous causality effects between durations and stock price volatility so far in the literature instantaneous causality effects have either been excluded or cannot be identified separately from granger type causality effects by giving explicit moment conditions for observed returns over random duration intervals we are able to identify an instantaneous causality effect the documented causality effect has significant impact on inference for tick by tick data we find that instantaneous volatility forecasts for e g ibm stock returns must be decreased by as much as 40 when not having seen the next quote change before its conditionally median time also instantaneous volatilities are found to be much higher than indicated by standard volatility assessment procedures using tick by tick data for ibm a naive assessment of spot volatility based on observed returns between quote changes would only account for 60 of the actual volatility for less liquidly traded stocks at nyse this effect is even stronger

large dimensional factor modeling based on high frequency observations abstract this paper develops a statistical theory to estimate an unknown factor structure based on financial high frequency data we derive an estimator for the number of factors and consistent and asymptotically mixed normal estimators of the loadings and factors under the assumption of a large number of cross sectional and high frequency observations the estimation approach can separate factors for continuous and rare jump risk the estimators for the loadings and factors are based on the principal component analysis of the quadratic covariation matrix the estimator for the number of factors uses a perturbed eigenvalue ratio statistic in an empirical analysis of the s p 500 firms we estimate four stable continuous systematic factors which can be approximated very well by a market and industry portfolios jump factors are different from the continuous factors

large volatility matrix inference via combining low frequency and high frequency approaches it is increasingly important in financial economics to estimate volatilities of asset returns however most of the available methods are not directly applicable when the number of assets involved is large due to the lack of accuracy in estimating high dimensional matrices therefore it is pertinent to reduce the effective size of volatility matrices in order to produce adequate estimates and forecasts furthermore since high frequency financial data for different assets are typically not recorded at the same time points conventional dimension reduction techniques are not directly applicable to overcome those difficulties we explore a novel approach that combines high frequency volatility matrix estimation together with low frequency dynamic models the proposed methodology consists of three steps i estimate daily realized covolatility matrices directly based on high frequency data ii fit a matrix factor model to the estimated daily covolatility matrices and iii fit a vector autoregressive mo

robust jump regressions abstractwe develop robust inference methods for studying linear dependence between the jumps of discretely observed processes at high frequency unlike classical linear regressions jump regressions are determined by a small number of jumps occurring over a fixed time interval and the rest of the components of the processes around the jump times the latter are the continuous martingale parts of the processes as well as observation noise by sampling more frequently the role of these components which are hidden in the observed price shrinks asymptotically the robustness of our inference procedure is with respect to outliers which are of particular importance in the current setting of relatively small number of jump observations this is achieved by using nonsmooth loss functions like l1 in the estimation unlike classical robust methods the limit of the objective function here remains nonsmooth the proposed method is also robust to measurement error in the observed processes which is achieved by

ultra high frequency volatility estimation with dependent microstructure noise we analyze the impact of time series dependence in market microstructure noise on the properties of estimators of the integrated volatility of an asset price based on data sampled at frequencies high enough for that noise to be a dominant consideration we show that combining two time scales for that purpose will work even when the noise exhibits time series dependence analyze in that context a refinement of this approach is based on multiple time scales and compare empirically our different estimators to the standard realized volatility

a multiple regime smooth transition heterogeneous autoregressive model for long memory and asymmetries in this paper we propose a flexible model to describe nonlinearities and long range dependence in time series dynamics the new model is a multiple regime smooth transition extension of the heterogeneous autoregressive har model which is specifically designed to model the behavior of the volatility inherent in financial time series the model is able to simultaneously approximate long memory behavior as well as describe sign and size asymmetries a sequence of tests is developed to determine the number of regimes and an estimation and testing procedure is presented monte carlo simulations evaluate the finite sample properties of the proposed tests and estimation procedures we apply the model to several dow jones industrial average index stocks using transaction level data from the trades and quotes database that covers ten years of data we find strong support for long memory and both sign and size asymmetries furthermore the new model when combined with the linear har model is viable and flexible for purposes of forecasting volatility

limit theorems for moving averages of discretized processes plus noise this paper presents some limit theorems for certain functionals of moving averages of semimartingales plus noise which are observed at high frequency our method generalizes the pre averaging approach see bernoulli15 2009 634 658 stochastic process appl 119 2009 2249 2276 and provides consistent estimates for various characteristics of general semimartingales furthermore we prove the associated multidimensional stable central limit theorems as expected we find central limit theorems with a convergence rate n 1 4 if n is the number of observations

variance dynamics joint evidence from options and high frequency returns this paper analyzes the s p 500 index return variance dynamics and the variance risk premium by combining information in variance swap rates constructed from options and quadratic variation estimators constructed from tick data on s p 500 index futures estimation shows that the index return variance jumps the jump arrival rate is not constant over time but is proportional to the variance rate level the variance jumps are not rare events but arrive frequently estimation also identifies a strongly negative variance risk premium the absolute magnitude of which is proportional to the variance rate level

market microstructure noise integrated variance estimators and the accuracy of asymptotic approximations a growing literature has been advocating consistent kernel estimation of integrated variance in the presence of financial market microstructure noise we find that for realistic sample sizes encountered in practice the asymptotic results derived for the proposed estimators may provide unsatisfactory representations of their finite sample properties in addition the existing asymptotic results might not offer sufficient guidance for practical implementations we show how to optimize the finite sample properties of kernel based integrated variance estimators empirically we find that their suboptimal implementation can in some cases lead to little or no finite sample gains when compared to the classical realized variance estimator significant statistical and economic gains can however be recovered by using our proposed finite sample methods

testing for self excitation in jumps this paper extends the notion of self excitation in jumps to a rich class of continuous time semimartingale models proposes statistical tests to detect its presence in a discretely observed sample path at high frequency and derives the tests asymptotic properties our statistical setting is semiparametric except for necessary parametric assumptions on the jump size measure the other components of the semimartingale model are left essentially unrestricted we analyze the finite sample performance of our tests in monte carlo simulations

anova for diffusions and ito processes ito processes are the most common form of continuous semimartingales and include diffusion processes this paper is concerned with the nonparametric regression relationship between two such ito processes we are interested in the quadratic variation integrated volatility of the residual in this regression over a unit of time such as a day a main conceptual finding is that this quadratic variation can be estimated almost as if the residual process were observed the difference being that there is also a bias which is of the same asymptotic order as the mixed normal error term the proposed methodology anova for diffusions and ito processes can be used to measure the statistical quality of a parametric model and non parametrically the appropriateness of a one regressor model in general on the other hand it also helps quantify and characterize the trading hedging error in the case of financial applications

vast volatility matrix estimation for high frequency financial data high frequency data observed on the prices of financial assets are commonly modeled by diffusion processes with micro structure noise and realized volatility based methods are often used to estimate integrated volatility for problems involving a large number of assets the estimation objects we face are volatility matrices of large size the existing volatility estimators work well for a small number of assets but perform poorly when the number of assets is very large in fact they are inconsistent when both the number p of the assets and the average sample size n of the price data on the p assets go to infinity this paper proposes a new type of estimators for the integrated volatility matrix and establishes asymptotic theory for the proposed estimators in the framework that allows both n and p to approach to infinity the theory shows that the proposed estimators achieve high convergence rates under a sparsity assumption on the integrated volatility matrix the numerical studies demonstrate that the proposed estimators perform well for large p and complex price and volatility models the proposed method is applied to real high frequency financial data

covariance measurement in the presence of non synchronous trading and market microstructure noise this paper studies the problem of covariance estimation when prices are observed non synchronously and contaminated by i i d microstructure noise we derive closed form expressions for the bias and variance of three popular covariance estimators namely realised covariance realised covariance plus lead and lag adjustments and the hayashi and yoshida estimator and present a comprehensive investigation into their properties and relative efficiency our main finding is that the ordering of the covariance estimators in terms of efficiency crucially depends on the level of microstructure noise as well as the level of correlation in fact for sufficiently high levels of noise the standard realised covariance estimator without any corrections for non synchronous trading can be most efficient we also propose a sparse sampling implementation of the hayashi and yoshida estimator study the robustness of our findings using simulations with stochastic volatility and correlation and highlight some important practical considerations

realized volatility forecasting and market microstructure noise this paper extends the analytical results for reduced form realized volatility based forecasting in andersen bollerslev and meddahi 2004 to allow for market microstructure frictions and deviations from the basic semi martingale assumption in the observed high frequency returns underlying the realized measures our results builds directly on the powerful eigenfunction representation of the general stochastic volatility class of models originally developed by meddahi 2001 in addition to the traditional realized volatility measure and the role of the underlying sampling frequencies we also explore the forecasting performance of several alternative volatility measures explicitly designed to mitigate the impact of the market microstructure noise our analysis of these alternative robust measures is facilitated by a simple unifled quadratic form representation our results suggest that the detrimental impact of the noise in terms of the accuracy of the forecasts can be substantial and that in empirically realistic situations the linear forecasts based on a simple to implement average estimator deflned by averaging across several sparsely sampled traditional realized volatility measures generally performs on par with best alternative robust measures

forecasting multivariate realized stock market volatility we present a new matrix logarithm model of the realized covariance matrix of stock returns the model uses latent factors which are functions of lagged volatility lagged returns and other forecasting variables the model has several advantages it is parsimonious it does not require imposing parameter restrictions and it results in a positive definite estimated covariance matrix we apply the model to the covariance matrix of size sorted stock returns and find that two factors are sufficient to capture most of the dynamics

high frequency returns jumps and the mixture of normals hypothesis abstract previous empirical studies find both evidence of jumps in asset prices and that returns standardized by realized volatility are approximately standard normal these findings appear to be contradictory using a sample of high frequency returns for 20 heavily traded us stocks we show how microstructure noise distorts the standard deviation and kurtosis of returns normalized using realized variance when returns are standardized using a recently developed realized kernel estimator the resulting series is clearly platykurtotic and the standard normal distribution is soundly rejected moreover daily returns standardized using realized bipower variation an estimator for integrated variance that is robust to the presence of jumps are more consistent with the standard normal distribution these results suggest that there is no empirical contradiction jumps should be included in stock price models

a bound on the order of a group having a large character degree abstract let d be the degree of an irreducible character of a finite group g we can write g d d e for some non negative integer e in this document we prove that if e 1 then g e 6 e 4 this improves an upper bound found by isaacs of the form b e 6 where b is an unknown universal constant we also describe conditions sufficient to sharpen this bound to g e 4 e 3 in addition we remove the appeal to the classification of simple groups which is used in the original paper by isaacs

econometrics of co jumps in high frequency data with noise abstract we establish estimation methods to determine co jumps in multivariate high frequency data with non synchronous observations and market microstructure a rate optimal estimator of the entire quadratic covariation of an ito semimartingale is constructed by a locally adaptive spectral approach thresholding allows to disentangle the co jump from the continuous part we derive a feasible limit theorem for a truncated estimator of integrated covolatility which facilitates asymptotically efficient co volatility estimation in the presence of jumps a test for common jumps is presented simulations and an empirical application to intra day tick data from eurex futures demonstrate the practical value of the approach

testing against constant factor loading matrix with large panel high frequency data abstract in this paper we introduce a nonparametric test against the constancy of the factor loading matrix of a high dimensional continuous time factor model using high frequency data the central limit theorems on the test statistics with and without perturbation are established under the null hypothesis that the factor loading matrix is constant as time evolves the tests perform well in size and power interestingly the test statistic without perturbation converges at a rate that depends not only on the sample size but also on the dimension through the cross sectional dependence of the residual process which is a distinctive feature that contrasts with the low dimensional setting extensive numerical studies including monte carlo simulations and real data analysis validate the performance of our test

occupation density estimation for noisy high frequency data abstract this paper studies the nonparametric estimation of occupation densities for semimartingale processes observed with noise as leading examples we consider the stochastic volatility of a latent efficient price process the volatility of the latent noise that separates the efficient price from the actually observed price and nonlinear transformations of these processes our estimation methods are decidedly nonparametric and consist of two steps the estimation of the spot price and noise volatility processes based on pre averaging techniques and in fill asymptotic arguments followed by a kernel type estimation of the occupation densities our spot volatility estimates attain the optimal rate of convergence and are robust to leverage effects price and volatility jumps general forms of serial dependence in the noise and random irregular sampling the convergence rates of our occupation density estimates are directly related to that of the estimated spot volatilities and the smoothness of the true occupation densities an empirical application involving high frequency equity data illustrates the usefulness of the new methods in illuminating time varying risks market liquidity and informational asymmetries across time and assets

assessment of uncertainty in high frequency data the observed asymptotic variance the availability of high frequency financial data has generated a series of estimators based on intra day data improving the quality of large areas of financial econometrics however estimating the standard error of these estimators is often challenging the root of the problem is that traditionally standard errors rely on estimating a theoretically derived asymptotic variance and often this asymptotic variance involves substantially more complex quantities than the original parameter to be estimated standard errors are important they are used to assess the precision of estimators in the form of confidence intervals to create feasible statistics for testing to build forecasting models based on say daily estimates and also to optimize the tuning parameters the contribution of this paper is to provide an alternative and general solution to this problem which we call observed asymptotic variance it is a general nonparametric method for assessing asymptotic variance avar it provides consistent estimators of avar for a broad class of integrated parameters t dt where the spot parameter process can be a general semimartingale with continuous and jump components the observed avar is implemented with the help of a two scales method its construction works well in the presence of microstructure noise and when the observation times are irregular or asynchronous in the multivariate case the methodology is valid for a wide variety of estimators including the standard ones for variance and covariance and also for more complex estimators such as of leverage effects high frequency betas and semivariance

bootstrapping high frequency jump tests the main contribution of this article is to propose a bootstrap test for jumps based on functions of realized volatility and bipower variation bootstrap intraday returns are randomly generated fro

quasi maximum likelihood estimation of volatility with high frequency data this paper investigates the properties of the well known maximum likelihood estimator in the presence of stochastic volatility and market microstructure noise by extending the classic asymptotic results of quasi maximum likelihood estimation when trying to estimate the integrated volatility and the variance of noise this parametric approach remains consistent efficient and robust as a quasi estimator under misspecified assumptions moreover it shares the model free feature with nonparametric alternatives for instance realized kernels while being advantageous over them in terms of finite sample performance in light of quadratic representation this estimator behaves like an iterative exponential realized kernel asymptotically comparisons with a variety of implementations of the tukey hanning2 kernel are provided using monte carlo simulations and an empirical study with the euro us dollar future illustrates its application in practice

designing realised kernels to measure the ex post variation of equity prices in the presence of noise this paper shows how to use realised kernels to carry out efficient feasible inference on the expost variation of underlying equity prices in the presence of simple models of market frictions the issue is subtle with only estimators which have symmetric weights delivering consistent estimators with mixed gaussian limit theorems the weights can be chosen to achieve the best possible rate of convergence and to have an asymptotic variance which is close to that of the maximum likelihood estimator in the parametric version of this problem realised kernels can also be selected to i be analysed using endogenously spaced data such as that in databases on transactions ii allow for market frictions which are endogenous iii allow for temporally dependent noise the finite sample performance of our estimators is studied using simulation while empirical work illustrates their use in practice

efficient asymptotic variance reduction when estimating volatility in high frequency data abstract this paper shows how to carry out efficient asymptotic variance reduction when estimating volatility in the presence of stochastic volatility and microstructure noise with the realized kernels rk from barndorff nielsen et al 2008 and the quasi maximum likelihood estimator qmle studied in xiu 2010 to obtain such a reduction we chop the data into b blocks compute the rk or qmle on each block and aggregate the block estimates the ratio of asymptotic variance over the bound of asymptotic efficiency converges as b increases to the ratio in the parametric version of the problem i e 1 0025 in the case of the fastest rk tukey hanning 16 and 1 for the qmle the impact of stochastic sampling times and jump in the price process is examined carefully the finite sample performance of both estimators is investigated in simulations while empirical work illustrates the gain in practice

statistical properties of microstructure noise we study the estimation of joint moments of microstructure noise based on high frequency data the estimation is conducted under a nonparametric setting which allows the underlying price process to have jumps the observation times to be irregularly spaced and the noise to be dependent on the price process and to have diurnal features estimators of arbitrary orders of joint moments are provided for which we establish consistency as well as central limit theorems in particular we provide estimators of autocovariances and autocorrelations of the noise simulation studies demonstrate excellent performance of our estimators in the presence of jumps irregular observation times and even rounding empirical studies reveal moderate positive autocorrelations of microstructure noise for the stocks tested

generalized ait sahalia type interest rate model with poisson jumps and convergence of the numerical approximation abstract in this paper we consider a generalized ait sahalia interest rate model with poisson jumps in finance the analytical properties including positivity boundedness and pathwise asymptotic estimations of the solution to this model are investigated moreover we prove that the euler maruyama em numerical solution converges to the true solution of the model in probability finally we apply the em solution to compute some financial quantities a numerical example is provided to demonstrate the effectiveness of our results

econometric analysis of realized volatility and its use in estimating stochastic volatility models summary the availability of intraday data on the prices of speculative assets means that we can use quadratic variation like measures of activity in financial markets called realized volatility to study the stochastic properties of returns here under the assumption of a rather general stochastic volatility model we derive the moments and the asymptotic distribution of the realized volatility error the difference between realized volatility and the discretized integrated volatility which we call actual volatility these properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation intensive methods

empirical evidence on the importance of aggregation asymmetry and jumps for volatility prediction abstract many recent modeling advances in finance topics ranging from the pricing of volatility based derivative products to asset management are predicated on the importance of jumps or discontinuous movements in asset returns in light of this a number of recent papers have addressed volatility predictability some from the perspective of the usefulness of jumps in forecasting volatility key papers in this area include andersen et al 2003 corsi 2004 andersen et al 2007 corsi et al 2008 barndorff et al 2010 patton and shephard 2011 and the references cited therein in this paper we review the extant literature and then present new empirical evidence on the predictive content of realized measures of jump power variations including upside and downside risk jump asymmetry and truncated jump variables constructed using instantaneous returns i e r t q 0 q 6 in the spirit of ding et al 1993 and ding and granger 1996 we also present new empirical evidence on the predictive content of realized measures of truncated large jump variations constructed using truncated squared instantaneous return i e r t 2 i r t where is the threshold jump size our prediction experiments use high frequency price returns constructed using s p500 futures data as well as stocks in the dow 30 and our empirical implementation involves estimating linear and nonlinear heterogeneous autoregressive realized volatility har rv type models we find that past large jump power variations help less in the prediction of future realized volatility than past small jump power variations additionally we find evidence that past realized signed jump power variations which have not previously been examined in this literature are strongly correlated with future volatility and that past downside jump variations matter in prediction finally incorporation of downside and upside jump power variations does improve predictability albeit to a limited extent

a tale of two time scales it is a common practice in finance to estimate volatility from the sum of frequently sampled squared returns however market microstructure poses challenges to this estimation approach as evidenced by recent empirical studies in finance the present work attempts to lay out theoretical grounds that reconcile continuous time modeling and discrete time samples we propose an estimation approach that takes advantage of the rich sources in tick by tick data while preserving the continuous time assumption on the underlying returns under our framework it becomes clear why and where the usual volatility estimator fails when the returns are sampled at the highest frequencies if the noise is asymptotically small our work provides a way of finding the optimal sampling frequency a better approach the two scales estimator works for any size of the noise

parametric estimation from approximate data non gaussian diffusions we study the problem of parameters estimation in indirect observability contexts where x t in r r is an unobservable stationary process parametrized by a vector of unknown parameters and all observable data are generated by an approximating process y varepsilon t which is close to x t in l 4 norm we construct consistent parameter estimators which are smooth functions of the sub sampled empirical mean and empirical lagged covariance matrices computed from the observable data we derive explicit optimal sub sampling schemes specifying the best paired choices of sub sampling time step and number of observations we show that these choices ensure that our parameter estimators reach optimized asymptotic l 2 convergence rates which are constant multiples of the l 4 norm y varepsilon t x t

econometric analysis of multivariate realised qml estimation of the covariation of equity prices under asynchronous trading estimating the covariance between assets using high frequency data is challenging due to market microstructure effects and asynchronous trading in this paper we develop a multivariate realised quasi likelihood qml approach carrying out inference as if the observations arise from an asynchronously observed vector scaled brownian model observed with error under stochastic volatility the resulting realised qml estimator is positive semi definite uses all available data is consistent and asymptotically mixed normal the quasi likelihood is computed using a kalman filter and optimised using a relatively simple em algorithm which scales well with the number of assets we derive the theoretical properties of the estimator and prove that it achieves the efficient rate of convergence we show how to make itobtain the non parametric efficiency bound for this problem the estimator is also analysed usingmonte carlo methods and applied to equity data with varying levels of liquidity

asymptotic error distributions for the euler method for stochastic differential equations we are interested in the rate of convergence of the euler scheme approximation of the solution to a stochastic differential equation driven by a general possibly discontinuous semimartingale and by the asymptotic behavior of the associated normalized error it is well known that for ito s equations the rate is 1 sqrt n we provide a necessary and sufficient condition for this rate to be 1 sqrt n when the driving semimartingale is a continuous martingale or a continuous semimartingale under a mild additional assumption we also prove that in these cases the normalized error processes converge in law the rate can also differ from 1 sqrt n this is the case for instance if the driving process is deterministic or if it is a levy process without a brownian component it is again 1 sqrt n when the driving process is levy with a nonvanishing brownian component but then the normalized error processes converge in law in the finite dimensional sense only while the discretized normalized error processes converge in law in the skorohod sense and the limit is given an explicit form

estimating the degree of activity of jumps in high frequency data we define a generalized index of jump activity propose estimators of that index for a discretely sampled process and derive the estimators properties these estimators are applicable despite the presence of brownian volatility in the process which makes it more challenging to infer the characteristics of the small infinite activity jumps when the method is applied to high frequency stock returns we find evidence of infinitely active jumps in the data and estimate their index of activity

robust high dimensional volatility matrix estimation for high frequency factor model high frequency financial data allow us to estimate large volatility matrices with relatively short time horizon many novel statistical methods have been introduced to address large volatility matr

multivariate realised kernels consistent positive semi definite estimators of the covariation of equity prices with noise and non synchronous trading we propose a multivariate realised kernel to estimate the ex post covariation of log prices we show this new consistent estimator is guaranteed to be positive semi definite and is robust to measurement noise of certain types and can also handle non synchronous trading it is the first estimator which has these three properties which are all essential for empirical work in this area we derive the large sample asymptotics of this estimator and assess its accuracy using a monte carlo study we implement the estimator on some us equity data comparing our results to previous work which has used returns measured over 5 or 10 minutes intervals we show the new estimator is substantially more precise

bootstrapping realized multivariate volatility measures we study bootstrap methods for statistics that are a function of multivariate high frequency returns such as realized regression coefficients and realized covariances and correlations for these measures of covariation the monte carlo simulation results of barndorff nielsen and shephard 2004 show that finite sample distortions associated with their feasible asymptotic theory approach may arise if sampling is not too frequent this motivates our use of the bootstrap as an alternative tool of inference for covariation measures we consider an i i d bootstrap applied to the vector of returns we show that the finite sample performance of the bootstrap is superior to the existing first order asymptotic theory nevertheless and contrary to the existing results in the bootstrap literature for regression models subject to heteroskedasticity in the error term the edgeworth expansion for the i i d bootstrap that we develop here shows that this method is not second order accurate we argue that this is due to the fact that the conditional mean parameters of realized regression models are heterogeneous under stochastic volatility

data based ranking of realised volatility estimators abstract this paper presents new methods for comparing the accuracy of estimators of the quadratic variation of a price process i provide conditions under which the relative accuracy of competing estimators can be consistently estimated as t and show that forecast evaluation tests may be adapted to the problem of ranking these estimators the proposed methods avoid making specific assumptions about microstructure noise and facilitate comparisons of estimators that would be difficult using methods from the extant literature such as those based on different sampling schemes an application to high frequency ibm data between 1996 and 2007 illustrates the new methods

bounding the order of a group with a large character degree abstract let d be the degree of some irreducible character of a finite group g we can write g d d e for some nonnegative integer e and we show that if e 1 then g b e 6 for some universal constant b this result which improves a non polynomial bound of snyder relies on recent work of larsen malle and tiep on character degrees of simple groups and so it uses the simple group classification

econometric analysis of realized covariation high frequency based covariance regression and correlation in financial economics this paper analyses multivariate high frequency financial data using realized covariation we provide a new asymptotic distribution theory for standard methods such as regression correlation analysis and covariance it will be based on a fixed interval of time e g a day or week allowing the number of high frequency returns during this period to go to infinity our analysis allows us to study how high frequency correlations regressions and covariances change through time in particular we provide confidence intervals for each of these quantities

realised quantile based estimation of the integrated variance in this paper we propose a new jump robust quantile based realised variance measure of ex post return variation that can be computed using potentially noisy data the estimator is consistent for the integrated variance and we present feasible central limit theorems which show that it converges at the best attainable rate and has excellent efficiency asymptotically the quantile based realised variance is immune to finite activity jumps and outliers in the price series while in modified form the estimator is applicable with market microstructure noise and therefore operational on high frequency data simulations show that it has superior robustness properties in finite sample while an empirical application illustrates its use on equity data

estimating the integrated volatility with tick observations abstract we develop a volatility estimator that can be directly applied to tick by tick data more specifically we consider a model that allows for i irregular observation times that can be endogenous ii dependent noise that can have diurnal features and be dependent on the latent price process and iii jumps in the latent price process we show that our estimator yields consistent estimates and enjoys the optimal rate of convergence simulation as well as empirical studies demonstrate favorable properties of our proposed estimator

estimation of continuous time linear system parameters from periodic data the parameters of a continuous time linear system are identified by use of integral functions satisfying periodicity conditions explicit equations are derived for the evaluation of integral functions in terms of measured periodic output data it is shown that a walsh function representation of the integral functions has modest computational requirements making implementation on a 16 bit microprocessor feasible simulation and experimental results are provided to illustrate truncation error estimation accuracies and output response errors

estimating the quadratic covariation matrix for asynchronously observed high frequency stock returns corrupted by additive measurement error this paper studies the estimation problem of the covariance matrices of asset returns in the presence of microstructure noise and asynchronicity between the observations across different assets motivated by malliavin and mancino 2002 2009 we propose a new fourier domain based estimator of multivariate ex post volatility which we call the fourier realized kernel frk an advantage of this approach is that no explicit time alignment is required unlike the time domain based methods widely adopted in the existing literature we derive the large sample properties and establish asymptotic normality of our estimator under some general conditions that allow for both temporal and cross sectional correlations in the measurement error process our results can be viewed as frequency domain extension of the asymptotic theories for the multivariate realized kernel estimator of barndorff nielsen et al 2011 we show in extensive simulations that our method outperforms the time domain estimators when two assets with different liquidity are traded asynchronously

do high frequency measures of volatility improve forecasts of return distributions many finance questions require the predictive distribution of returns we propose a bivariate model of returns and realized volatility rv and explore which features of that time series model contribute to superior density forecasts over horizons of 1 to 60 days out of sample this term structure of density forecasts is used to investigate the importance of the intraday information embodied in the daily rv estimates the functional form for log rv dynamics the timing of information availability and the assumed distributions of both return and log rv innovations we find that a joint model of returns and volatility that features two components for log rv provides a good fit to s p 500 and ibm data and is a significant improvement over an egarch model estimated from daily returns

pre averaging estimators of the ex post covariance matrix in noisy diffusion models with non synchronous data we show how pre averaging can be applied to the problem of measuring the ex post covariance of financial asset returns under microstructure noise and non synchronous trading a pre averaged realised covariance is proposed and we present an asymptotic theory for this new estimator which can be configured to possess an optimal convergence rate or to ensure positive semi definite covariance matrix estimates we also derive a noise robust hayashi yoshida estimator that can be implemented on the original data without prior alignment of prices we uncover the finite sample properties of our estimators with simulations and illustrate their practical use on high frequency equity data

is the diurnal pattern sufficient to explain intraday variation in volatility a nonparametric assessment abstract in this paper we propose a nonparametric way to test the hypothesis that time variation in intraday volatility is caused solely by a deterministic and recurrent diurnal pattern we assume that noisy high frequency data from a discretely sampled jump diffusion process are available the test is then based on asset returns which are deflated by the seasonal component and therefore homoskedastic under the null to construct our test statistic we extend the concept of pre averaged bipower variation to a general ito semimartingale setting via a truncation device we prove a central limit theorem for this statistic and construct a positive semi definite estimator of the asymptotic covariance matrix the t statistic after pre averaging and jump truncation diverges in the presence of stochastic volatility and has a standard normal distribution otherwise we show that replacing the true diurnal factor with a model free jump and noise robust estimator does not affect the asymptotic theory a monte carlo simulation also shows this substitution has no discernable impact in finite samples the test is however distorted by small infinite activity price jumps to improve inference we propose a new bootstrap approach which leads to almost correctly sized tests of the null hypothesis we apply the developed framework to a large cross section of equity high frequency data and find that the diurnal pattern accounts for a rather significant fraction of intraday variation in volatility but important sources of heteroskedasticity remain present in the data

predictive density estimators for daily volatility based on the use of realized measures abstract the main objective of this paper is to propose a feasible model free estimator of the predictive density of integrated volatility in this sense we extend recent papers by andersen et al andersen t g bollerslev t diebold f x labys p 2003 modelling and forecasting realized volatility econometrica 71 579 626 and by andersen et al andersen t g bollerslev t meddahi n 2004 analytic evaluation of volatility forecasts international economic review 45 1079 1110 andersen t g bollerslev t meddahi n 2005 correcting the errors volatility forecast evaluation using high frequency data and realized volatilities econometrica 73 279 296 who address the issue of pointwise prediction of volatility via arma models based on the use of realized volatility our approach is to use a realized volatility measure to construct a non parametric kernel estimator of the predictive density of daily volatility we show that by choosing an appropriate realized measure one can achieve consistent estimation even in the presence of jumps and microstructure noise in prices more precisely we establish that four well known realized measures i e realized volatility bipower variation and two measures robust to microstructure noise satisfy the conditions required for the uniform consistency of our estimator furthermore we outline an alternative simulation based approach to predictive density construction finally we carry out a simulation experiment in order to assess the accuracy of our estimators and provide an empirical illustration that underscores the importance of using microstructure robust measures when using high frequency data

a unified approach to volatility estimation in the presence of both rounding and random market microstructure noise abstract widely used volatility estimation methods mainly consider one of the following two simple microstructure noise models random additive noise on log prices or pure rounding errors apparently in real data these two types of noise co exist in this paper we discover a common feature of these two types of noise and propose a unified volatility estimation approach in the presence of both rounding and random noise our data driven method enjoys superior properties in terms of bias and convergence rate we establish feasible central limit theorems and show their superior performance via simulations empirical studies show clear advantages of our method when applied to both stocks data and currency exchange data

groups with a character of large degree relative to a normal subgroup abstract let n be a normal subgroup of a finite group g and be an irreducible character of n which is fixed by the conjugation action of g let be an irreducible character of g that restricts to a multiple of on n then d 1 1 is an integer which divides g n and has g n d 2 we can thus write g n d d e for a non negative integer e and ask what can be said about d and g n for a given e this is a generalization of a problem considered by snyder 12 where he takes d to an irreducible character degree of g and writes g d d e berkovich has shown in 1 that if e 1 then g is a sharply 2 transitive group for e 1 snyder shows in 12 that d is bounded by a function of e this bound is later improved by isaacs in 7 and then by durfee and jensen in 2 and lewis in 9 in this more general version of the problem we will work under the assumption that g n is solvable we will show that for e 0 if d e 1 2 then e divides d and d e 1 is a prime power if in addition either d e 5 e d e e 1 or d e 1 e 1 then there exist groups x y with n x y g such that y x is a sharply 2 transitive group of order d e d e 1

a quasi maximum likelihood approach for integrated covariance matrix estimation with high frequency data estimating the integrated covariance matrix icm from high frequency financial trading data is crucial to reflect the volatilities and covariations of the underlying trading instruments such an objective is difficult due to contaminated data with microstructure noises asynchronous trading records and increasing data dimensionality in this paper we study a quasi maximum likelihood qml approach for estimating an icm from high frequency financial data we explore a novel multivariate moving average time series device that is convenient for evaluating the estimator both theoretically for its asymptotic properties and numerically for its practical implementations we demonstrate that the qml estimator is consistent to the icm and is asymptotically normally distributed efficiency gain of the qml approach is theoretically quantified and numerically demonstrated via extensive simulation studies an application of the qml approach is illustrated through analyzing a high frequency financial trading data set

high frequency covariance estimates with noisy and asynchronous financial data this article proposes a consistent and efficient estimator of the high frequency covariance quadratic covariation of two arbitrary assets observed asynchronously with market microstructure noise this estimator is built on the marriage of the quasi maximum likelihood estimator of the quadratic variation and the proposed generalized synchronization scheme and thus is not influenced by the epps effect moreover the estimation procedure is free of tuning parameters or bandwidths and is readily implementable monte carlo simulations show the advantage of this estimator by comparing it with a variety of estimators with specific synchronization methods the empirical studies of six foreign exchange future contracts illustrate the time varying correlations of the currencies during the 2008 global financial crisis demonstrating the similarities and differences in their roles as key currencies in the global market

a discrete time model for daily s p500 returns and realized variations jumps and leverage effects abstract we develop an empirically highly accurate discrete time daily stochastic volatility model that explicitly distinguishes between the jump and continuous time components of price movements using nonparametric realized variation and bipower variation measures constructed from high frequency intraday data the model setup allows us to directly assess the structural inter dependencies among the shocks to returns and the two different volatility components the model estimates suggest that the leverage effect or asymmetry between returns and volatility works primarily through the continuous volatility component the excellent fit of the model makes it an ideal candidate for an easy to implement auxiliary model in the context of indirect estimation of empirically more realistic continuous time jump diffusion and levy driven stochastic volatility models effectively incorporating the interdaily dependencies inherent in the high frequency intraday data

realized range based estimation of integrated variance abstract we provide a set of probabilistic laws for estimating the quadratic variation of continuous semimartingales with the realized range based variance a statistic that replaces every squared return of the realized variance with a normalized squared range if the entire sample path of the process is available and under a set of weak conditions our statistic is consistent and has a mixed gaussian limit whose precision is five times greater than that of the realized variance in practice of course inference is drawn from discrete data and true ranges are unobserved leading to downward bias we solve this problem to get a consistent mixed normal estimator irrespective of non trading effects this estimator has varying degrees of efficiency over realized variance depending on how many observations that are used to construct the high low the methodology is applied to taq data and compared with realized variance our findings suggest that the empirical path of quadratic variation is also estimated better with the realized range based variance

the realized empirical distribution function of stochastic variance with application to goodness of fit testing abstract we propose a nonparametric estimator of the empirical distribution function edf of the latent spot variance of the log price of a financial asset we show that over a fixed time span our realized edf or redf inferred from noisy high frequency data is consistent as the mesh of the observation grid goes to zero in a double asymptotic framework with time also increasing to infinity the redf converges to the cumulative distribution function of volatility if it exists we exploit these results to construct some new goodness of fit tests for stochastic volatility models in a monte carlo study the redf is found to be accurate over the entire support of volatility this leads to goodness of fit tests that are both correctly sized and relatively powerful against common alternatives in an empirical application we recover the redf from stock market high frequency data we inspect the goodness of fit of several two parameter marginal distributions that are inherent in standard stochastic volatility models the inverse gaussian offers the best overall description of random equity variation but the fit is less than perfect this suggests an extra parameter as available in e g the generalized inverse gaussian is required to model stochastic variance

continuous time analysis of fleeting discrete price moves abstractthis article proposes a novel model of financial prices where i prices are discrete ii prices change in continuous time iii a high proportion of price changes are reversed in a fraction of a second our model is analytically tractable and directly formulated in terms of the calendar time and price impact curve the resulting cadlag price process is a piecewise constant semimartingale with finite activity finite variation and no brownian motion component we use moment based estimations to fit four high frequency futures datasets and demonstrate the descriptive power of our proposed model this model is able to describe the observed dynamics of price changes over three different orders of magnitude of time intervals supplementary materials for this article are available online

on the number of common factors with high frequency data summaryin this paper we introduce a local principal component analysis approach to determining the number of common factors of a continuous time factor model with time varying factor loadings using high frequency data the model is approximated locally on shrinking blocks using discrete time factor models the number of common factors is estimated by minimizing the penalized aggregated mean squared residual error over all shrinking blocks while the local mean squared residual error on each block converges at rate min n 1 4 p where n is the sample size and p is the dimension the aggregated mean squared residual error converges at rate min n 1 2 p this achieves the convergence rate of the penalized criterion function of the global principal component analysis method assuming restrictive constant factor loading an estimator of the number of factors based on the local principal component analysis is consistent simulation results justify the performance of our estimator a real financial dataset is analysed

estimating quadratic variation when quoted prices change by a constant increment abstract for financial assets whose best quotes almost always change by jumping by the market s price tick size one cent five cents etc this paper proposes an estimator of quadratic variation which controls for microstructure effects it measures the prevalence of alternations where quotes jump back to their just previous price it defines a simple property called uncorrelated alternation which under conditions implies that the estimator is consistent in an asymptotic limit theory where jumps become very frequent and small feasible limit theory is developed and in simulations works well

increased correlation among asset classes are volatility or jumps to blame or both abstract we develop estimators and asymptotic theory to decompose the quadratic covariation between two assets into its continuous and jump components in a manner that is robust to the presence of market microstructure noise using high frequency data on different assets classes we find that the recent financial crisis led to an increase in both the quadratic variations of the assets and their correlations however we find little evidence to suggest a change between the relative contributions of the brownian and jump components as both comove co jumps stem from surprising news announcements that occur primarily before the opening of the us market and are also accompanied by an increase in brownian driven correlations

a reduced form framework for modeling volatility of speculative prices based on realized variation measures abstract building on realized variance and bipower variation measures constructed from high frequency financial prices we propose a simple reduced form framework for effectively incorporating intraday data into the modeling of daily return volatility we decompose the total daily return variability into the continuous sample path variance the variation arising from discontinuous jumps that occur during the trading day as well as the overnight return variance our empirical results based on long samples of high frequency equity and bond futures returns suggest that the dynamic dependencies in the daily continuous sample path variability are well described by an approximate long memory har garch model while the overnight returns may be modeled by an augmented garch type structure the dynamic dependencies in the non parametrically identified significant jumps appear to be well described by the combination of an ach model for the time varying jump intensities coupled with a relatively simple log linear structure for the jump sizes finally we discuss how the resulting reduced form model structure for each of the three components may be used in the construction of out of sample forecasts for the total return volatility

minimum norm quadratic estimation of spatial variograms abstract the estimation of spatial variograms a measure of spatial correlation is a critical problem in the implementation of kriging a method for interpolating random fields we consider the use of minimum norm quadratic estimators of the variogram when it is specified up to a finite number of linear parameters we investigate the asymptotic behavior of such estimators for gaussian processes as the number of observations within some bounded region increases the basic conclusion is that we can estimate consistently those functions of the parameters and only those functions that have a nonnegligible impact asymptotically on the kriging procedure in general the behavior of the variogram over relatively short distances is the only aspect of the variogram that is asymptotically important as an example consider a gaussian process z on the real line with unknown constant mean and where is known as the semivariogram of the process and is a finite vector of unknown parameters suppose for 0

testing whether the underlying continuous time process follows a diffusion an infinitesimal operator based approach abstract we develop a nonparametric test to check whether a process can be represented by a stochastic differential equation driven only by a brownian motion our testing procedure utilizes the infinitesimal operator based martingale characterization combined with a generalized spectral approach such a testing procedure is feasible and convenient because the infinitesimal operator of the diffusion process has a closed form expression the proposed test is applicable to both univariate and multivariate processes and has an n 0 1 limit distribution under the diffusion hypothesis simulation and empirical studies show that the proposed test has reasonable performance in small samples

factor garch ito models for high frequency data with application to large volatility matrix prediction abstract several novel large volatility matrix estimation methods have been developed based on the high frequency financial data they often employ the approximate factor model that leads to a low rank plus sparse structure for the integrated volatility matrix and facilitates estimation of large volatility matrices however for predicting future volatility matrices these nonparametric estimators do not have a dynamic structure to implement in this paper we introduce a novel ito diffusion process based on the approximate factor models and call it a factor garch ito model we then investigate its properties and propose a quasi maximum likelihood estimation method for the parameter of the factor garch ito model we also apply it to estimating conditional expected large volatility matrices and establish their asymptotic properties simulation studies are conducted to validate the finite sample performance of the proposed estimation methods the proposed method is also illustrated by using data from the constituents of the s p 500 index and an application to constructing the minimum variance portfolio with gross exposure constraints

exponential convergence of adaptive identification and control algorithms output and equation error adaptive identification algorithms are shown to be exponentially convergent under a deterministic or stochastic persistently exciting or spanning condition on the system inputs together with several other standard conditions an adaptive control algorithm is shown to be exponentially convergent under a deterministic or stochastic persistently exciting condition on the reference trajectory together with some standard conditions

predictive inference for integrated volatility numerous volatility based derivative products have been engineered in recent years this has led to interest in constructing conditional predictive densities and confidence intervals for integrated volatility in this article we propose nonparametric estimators of the aforementioned quantities based on model free volatility estimators we establish consistency and asymptotic normality for the feasible estimators and study their finite sample properties through a monte carlo experiment finally using data from the new york stock exchange we provide an empirical application to volatility directional predictability

continuous time constrained least squares algorithms for recursive parameter estimation of stochastic linear systems by a stabilized output error method discrete time least squares algorithms for recursive parameter estimation have continuous time counterparts which minimize a quadratic functional the continuous time algorithms can also include in equality constraints asymptotic convergence is demonstrated by means of lyapunov methods the constrained algorithms are applied in a stabilized output error configuration for parameter estimation in stochastic linear systems

optimal sparse volatility matrix estimation for high dimensional ito processes with measurement errors stochastic processes are often used to model complex scientiflc problems in flelds ranging from biology and flnance to engineering and physical science this paper investigates rate optimal estimation of the volatility matrix of a high dimensional it o process observed with measurement errors at discrete time points the minimax rate of convergence is established for estimating sparse volatility matrices by combining the multi scale and threshold approaches we construct a volatility matrix estimator to achieve the optimal convergence rate the minimax lower bound is derived by considering a subclass of it o processes for which the minimax lower bound is obtained through a novel equivalent model of covariance matrix estimation for independent but non identically distributed observations and through a delicate construction of the least favorable parameters in addition a simulation study was conducted to test the flnite sample performance of the optimal estimator and the simulation results were found to support the established asymptotic theory

estimating covariation epps effect microstructure noise abstract this paper is about how to estimate the integrated covariance x y t of two assets over a fixed time horizon 0 t when the observations of x and y are contaminated and when such noisy observations are at discrete but not synchronized times we show that the usual previous tick covariance estimator is biased and the size of the bias is more pronounced for less liquid assets this is an analytic characterization of the epps effect we also provide the optimal sampling frequency which balances the tradeoff between the bias and various sources of stochastic error terms including nonsynchronous trading microstructure noise and time discretization finally a two scales covariance estimator is provided which simultaneously cancels to first order the epps effect and the effect of microstructure noise the gain is demonstrated in data

testing for jumps in a discretely observed process we propose a new test to determine whether jumps are present in asset returns or other discretely sampled processes as the sampling interval tends to 0 our test statistic converges to 1 if there are jumps and to another deterministic and known value such as 2 if there are no jumps the test is valid for all ito semimartingales depends neither on the law of the process nor on the coefficients of the equation which it solves does not require a preliminary estimation of these coefficients and when there are jumps the test is applicable whether jumps have finite or infinite activity and for an arbitrary blumenthal getoor index we finally implement the test on simulations and asset returns data

on loss functions and ranking forecasting performances of multivariate volatility models the ranking of multivariate volatility models is inherently problematic because when the unobservable volatility is substituted by a proxy the ordering implied by a loss function may be biased with respect to the intended one we point out that the size of the distortion is strictly tied to the level of the accuracy of the volatility proxy we propose a generalized necessary and sufficient functional form for a class of non metric distance measures of the bregman type which ensure consistency of the ordering when the target is observed with noise an application to three foreign exchange rates is provided 2012 elsevier b v all rights reserved

testing for jumps when asset prices are observed with noise a swap variance approach abstract this paper proposes a new test for jumps in asset prices that is motivated by the literature on variance swaps formally the test follows by a direct application of ito s lemma to the semi martingale process of asset prices and derives its power from the impact of jumps on the third and higher order return moments intuitively the test statistic reflects the cumulative gain of a variance swap replication strategy which is known to be minimal in the absence of jumps but substantial in the presence of jumps simulations show that the jump test has nice properties and is generally more powerful than the widely used bi power variation test an important feature of our test is that it can be applied in analytically modified form to noisy high frequency data and still retain power as a by product of our analysis we obtain novel analytical results regarding the impact of noise on bi power variation an empirical illustration using ibm trade data is also included

a fourier transform method for nonparametric estimation of multivariate volatility we provide a nonparametric method for the computation of instantaneous multivariate volatility for continuous semi martingales which is based on fourier analysis the co volatility is reconstructed as a stochastic function of time by establishing a connection between the fourier transform of the prices process and the fourier transform of the co volatility process a non parametric estimator is derived given a discrete unevenly spaced and asynchronously sampled observations of the asset price processes the asymptotic properties of the random estimator are studied namely consistency in probability uniformly in time and convergence in law to a mixture of gaussian distributions