nonlinear shrinkage estimation of large integrated covariance matrices integrated covariance matrices arise in intra day models of asset returns which allow volatility to change across the trading day when the number of assets is large the natural estimator of such a matrix suffers from bias contributed from extreme eigenvalues we introduce a novel nonlinear shrinkage estimator for the integrated covariance matrix which shrinks the extreme eigenvalues of a realized covariance matrix back to an acceptable level and enjoys a certain asymptotic efficiency when the number of assets is of the same order as the number of data points novel maximum exposure and actual risk bounds are derived when our estimator is used in constructing the minimum variance portfolio compared to other methods our estimator performs favorably in both simulations and a real data analysis

threshold bipower variation and the impact of jumps on volatility forecasting this study reconsiders the role of jumps for volatility forecasting by showing that jumps have a positive and mostly significant impact on future volatility this result becomes apparent once volatility is separated into its continuous and discontinuous components using estimators which are not only consistent but also scarcely plagued by small sample bias with the aim of achieving this we introduce the concept of threshold bipower variation which is based on the joint use of bipower variation and threshold estimation we show that its generalization threshold multipower variation admits a feasible central limit theorem in the presence of jumps and provides less biased estimates with respect to the standard multipower variation of the continuous quadratic variation in finite samples we further provide a new test for jump detection which has substantially more power than tests based on multipower variation empirical analysis on the s p500 index individual stocks and us bond yields shows that the proposed techniques improve significantly the accuracy of volatility forecasts especially in periods following the occurrence of a jump

required relationship between objective function and pareto frontier orders practical implications it is well known that the weighted sum aggregate objective function fails to capture pareto points that are located on a concave region of the pareto frontier conditions are established for a general aggregate objective function to capture points on a concave pareto frontier region specifically these conditions are developed in terms of the relative orders of the pareto frontier and aggregate objective function these critical conditions can be used in practice to allow an aggregate objective function to capture any potentially desirable pareto point conversely failure to satisfy these critical conditions can represent a pivotal failure point of the general application of design optimization because we may fail to obtain the design we seek

normal boundary intersection a new method for generating the pareto surface in nonlinear multicriteria optimization problems this paper proposes an alternate method for finding several pareto optimal points for a general nonlinear multicriteria optimization problem such points collectively capture the trade off among the various conflicting objectives it is proved that this method is independent of the relative scales of the functions and is successful in producing an evenly distributed set of points in the pareto set given an evenly distributed set of parameters a property which the popular method of minimizing weighted combinations of objective functions lacks further this method can handle more than two objectives while retaining the computational efficiency of continuation type algorithms this is an improvement over continuation techniques for tracing the trade off curve since continuation strategies cannot easily be extended to handle more than two objectives

subsampling high frequency data the main contribution of this paper is to propose a novel way of conducting inference for an important general class of estimators that includes many estimators of integrated volatility a subsampling scheme is introduced that consistently estimates the asymptotic variance for an estimator thereby facilitating inference and the construction of valid confidence intervals the new method does not rely on the exact form of the asymptotic variance which is useful when the latter is of complicated form the method is applied to the volatility estimator of ait sahalia et al 2011 in the presence of autocorrelated and heteroscedastic market microstructure noise

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

on the willmore energy of shells under infinitesimal deformations in this paper we apply tensor calculus and differential geometry to consider shell structure using tensor calculus we examine the stationarity of the willmore energy under infinitesimal deformations of the surfaces in 3 we obtain the class of the surfaces which does not change its willmore energy under infinitesimal deformations in particular a special kind of deformation is considered infinitesimal bending which preserves the arc length the change of the willmore energy under such deformations is determined also we give a new proof of a well known theorem that reads that the total mean curvature of a surface is stationary under an infinitesimal bending applying tensor calculus

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

on the pareto optimum sensitivity analysis in multicriteria optimization to analyse the trade off relations among the set of criteria in multicriteria optimization pareto optimum sensitivity analysis is systematically studied in this paper original contributions cover two parts theoretical demonstrations are firstly made to validate the gradient projection method in pareto optimum sensitivity analysis it is shown that the projected gradient direction evaluated at a given pareto optimum in the design variable space rigorously corresponds to the tangent direction of the pareto curve surface at that point in the objective space this statement holds even for the change of the set of active constraints in the perturbed problem secondly a new active constraint updating strategy is proposed which permits the identification of the active constraint set change to determine the influence of this change upon the differentiability of the pareto curve and finally to compute directional derivatives in non differentiable cases this work will highlight some basic issues in multicriteria optimization some numerical problems are solved to illustrate these novelties

infinitesimal bending influence on the volume change applying tensor calculus we discuss change of the volume under infinitesimal bending of the surface we obtain that the variation of the volume bounded by the surface s and the cone joining the origin to the boundary of the surface under infinitesimal bending of s with the vector field of translation s equals one third of the flux of the field s through the given surface s an example is analysed and graphically presented the paper points to the application of the obtained result in the calculation of the volume of a solid model

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

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

on the estimation of integrated covariance matrices of high dimensional diffusion processes we consider the estimation of integrated covariance icv matrices of high dimensional diffusion processes based on high frequency observations we start by studying the most commonly used estimator the realized covariance rcv matrix we show that in the high dimensional case when the dimension p and the observation frequency n grow in the same rate the limiting spectral distribution lsd of rcv depends on the covolatility process not only through the targeting icv but also on how the covolatility process varies in time we establish a marcenko pastur type theorem for weighted sample covariance matrices based on which we obtain a marcenko pastur type theorem for rcv for a class mathcal c of diffusion processes the results explicitly demonstrate how the time variability of the covolatility process affects the lsd of rcv we further propose an alternative estimator the time variation adjusted realized covariance tvarcv matrix we show that for processes in class mathcal c the tvarcv possesses the desirable property that its lsd depends solely on that of the targeting icv through the marcenko pastur equation and hence in particular the tvarcv can be used to recover the empirical spectral distribution of the icv by using existing algorithms

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

the distribution of realized exchange rate volatility using high frequency data on deutschemark and yen returns against the dollar we construct model free estimates of daily exchange rate volatility and correlation that cover an entire decade our estimates termed realized volatilities and correlations are not only model free but also approximately free of measurement error under general conditions which we discuss in detail hence for practical purposes we may treat the exchange rate volatilities and correlations as observed rather than latent we do so and we characterize their joint distribution both unconditionally and conditionally noteworthy results include a simple normality inducing volatility transformation high contemporaneous correlation across volatilities high correlation between correlation and volatilities pronounced and persistent dynamics in volatilities and correlations evidence of long memory dynamics in volatilities and correlations and remarkably precise scaling laws under temporal aggregation

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

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

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

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

nondomination level coordinate system for parameterizing and exploring pareto frontiers this paper presents a method for defining a barycentric coordinate system on a k objective pareto frontier that is constructed from nondomination levels calculated over subsets of the objectives taken k 1 at a time unlike past approaches these nondomination level coordinates are not limited by the dimensionality convexity or curvature of the frontier and they have an inherent meaning as relative preferences for the competing objectives this intuitive behavior makes the coordinates particularly useful as the basis for parametric models of the frontier and for conducting sensitivity studies and interactive trade space exploration the method is demonstrated on three mathematical example problems exhibiting different geometric properties and on a conceptual wing sizing problem

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

ability of objective functions to generate points on nonconvex pareto frontiers new ground is broken in our understanding of objective functions ability to capture pareto solutions for multi objective design optimization problems it is explained why widely used objective functions fail to capture pareto solutions when the pareto frontier is not convex in objective space and the means to avoid this limitation when possible is provided these conditions are developed and presented in the general context ofn dimensional objective space and numerical examples are provided an important point is that most objective function structures can be made to generate nonconvex pareto frontier solutions if the curvature of the objective function can be varied by setting one or more parameters because the occurrence of nonconvex efficient frontiers is common in practice the results are of direct practical usefulness

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

alternative models for stock price dynamics abstract this paper evaluates the role of various volatility specifications such as multiple stochastic volatility sv factors and jump components in appropriate modeling of equity return distributions we use estimation technology that facilitates nonnested model comparisons and use a long data set which provides rich information about the conditional and unconditional distribution of returns we consider two broad families of models 1 the multifactor loglinear family and 2 the affine jump family both classes of models have attracted much attention in the derivatives and econometrics literatures there are various tradeoffs in considering such diverse specifications if pure diffusion sv models are chosen over jump diffusions it has important implications for hedging strategies if logarithmic models are chosen over affine ones it may seriously complicate option pricing comparing many different specifications of pure diffusion multifactor models and jump diffusion models we find that 1 log linear models have to be extended to two factors with feedback in the mean reverting factor 2 affine models have to have a jump in returns stochastic volatility or probably both models 1 and 2 are observationally equivalent on the data set in hand in either 1 or 2 the key is that the volatility can move violently as we obtain models with comparable empirical fit one must make a choice based on arguments other than statistical goodness of fit criteria the considerations include facility to price options to hedge and parsimony the affine specification with jumps in volatility might therefore be preferred because of the closed form derivatives prices

jump robust volatility estimation using nearest neighbor truncation abstract we propose two new jump robust estimators of integrated variance that allow for an asymptotic limit theory in the presence of jumps specifically our medrv estimator has better efficiency properties than the tripower variation measure and displays better finite sample robustness to jumps and small zero returns we stress the benefits of local volatility measures using short return blocks as this greatly alleviates the downward biases stemming from rapid fluctuations in volatility including diurnal intraday u shape patterns an empirical investigation of the dow jones 30 stocks and extensive simulations corroborate the robustness and efficiency properties of our nearest neighbor truncation estimators

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

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

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

mathematical and pragmatic perspectives of physical programming physical programming pp is an emerging multiobjective and design optimization method that has been applied successfully in diverse areas of engineering and operations research the application of pp calls for the designer to express preferences by defining ranges of differing degrees of desirability for each design metric although this approach works well in practice it has never been shown that the resulting optimal solution is not unduly sensitive to these numerical range definitions pp is shown to be numerically well conditioned and its sensitivity to designer input with respect to optimal solution is compared with that of other popular methods the important proof is provided that all solutions obtained through pp are pareto optimal and the notion of pareto optimality is extended to one of pragmatic implication the important notion of p dominance that extends the concept of pareto optimality beyond the cases minimize and maximize is introduced p dominance is shown to lead to the important concept of generalized pareto optimality numerical results are provided that illustrate the favorable numerical properties of physical programming

change of the willmore energy under infinitesimal bending of membranes change of the willmore energy as the special case of elastic bending energy under infinitesimal bending of the vesicle membrane is discussed a membrane is thought of as a smooth surface in r 3 because its thickness is much smaller than its lateral dimension variation of the willmore energy at the surface point under infinitesimal bending of that surface as well as the condition for the stationarity of the willmore energy are given some examples are visualized variation of the willmore energy of a compact path connected smooth surface in the euclidean 3 space is reduced to a line integral of a special vector field and it is found that the willmore energy of a boundary free surface is stationary under an infinitesimal bending also the willmore energy of a minimal surface is stationary under its infinitesimal bending