Publications - Kim Christensen

High-dimensional estimation of quadratic variation based on penalized realized variance

Christensen, K., Nielsen, M. S., Podolskij, M. — Sat, 01 Jul 2023 10:45:49 +0200

In this paper, we develop a penalized realized variance (PRV) estimator of the quadratic variation (QV) of a high-dimensional continuous Itô semimartingale. We adapt the principle idea of regularization from linear regression to covariance estimation in a continuous-time high-frequency setting. We show that under a nuclear norm penalization, the PRV is computed by soft-thresholding the eigenvalues of realized variance (RV). It therefore encourages sparsity of singular values or, equivalently, low rank of the solution. We prove our estimator is minimax optimal up to a logarithmic factor. We derive a concentration inequality, which reveals that the rank of PRV is—with a high probability—the number of non-negligible eigenvalues of the QV. Moreover, we also provide the associated non-asymptotic analysis for the spot variance. We suggest an intuitive data-driven subsampling procedure to select the shrinkage parameter. Our theory is supplemented by a simulation study and an empirical application. The PRV detects about three–five factors in the equity market, with a notable rank decrease during times of distress in financial markets. This is consistent with most standard asset pricing models, where a limited amount of systematic factors driving the cross-section of stock returns are perturbed by idiosyncratic errors, rendering the QV—and also RV—of full rank.

A GMM approach to estimate the roughness of stochastic volatility

Bolko, A. E., Christensen, K., Pakkanen, M. S., Veliyev, B. — Tue, 01 Aug 2023 10:45:49 +0200

We develop a GMM approach for estimation of log-normal stochastic volatility models driven by a fractional Brownian motion with unrestricted Hurst exponent. We show that a parameter estimator based on the integrated variance is consistent and, under stronger conditions, asymptotically normally distributed. We inspect the behavior of our procedure when integrated variance is replaced with a noisy measure of volatility calculated from discrete high-frequency data. The realized estimator contains sampling error, which skews the fractal coefficient toward “illusive roughness.” We construct an analytical approach to control the impact of measurement error without introducing nuisance parameters. In a simulation study, we demonstrate convincing small sample properties of our approach based both on integrated and realized variance over the entire memory spectrum. We show the bias correction attenuates any systematic deviance in the parameter estimates. Our procedure is applied to empirical high-frequency data from numerous leading equity indexes. With our robust approach the Hurst index is estimated around 0.05, confirming roughness in stochastic volatility.

The drift burst hypothesis

Christensen, K., Oomen, R., Renò, R. — Fri, 01 Apr 2022 10:45:49 +0200

The drift burst hypothesis postulates the existence of short-lived locally explosive trends in the price paths of financial assets. The recent U.S. equity and treasury flash crashes can be viewed as two high-profile manifestations of such dynamics, but we argue that drift bursts of varying magnitude are an expected and regular occurrence in financial markets that can arise through established mechanisms of liquidity provision. We show how to build drift bursts into the continuous-time Itô semimartingale model, elaborate on the conditions required for the process to remain arbitrage-free, and propose a nonparametric test statistic that identifies drift bursts from noisy high-frequency data. We apply the test and demonstrate that drift bursts are a stylized fact of the price dynamics across equities, fixed income, currencies and commodities. Drift bursts occur once a week on average, and the majority of them are accompanied by subsequent price reversion and can thus be regarded as “flash crashes.” The reversal is found to be stronger for negative drift bursts with large trading volume, which is consistent with endogenous demand for immediacy during market crashes.

A machine learning approach to volatility forecasting

Christensen, K., Siggaard, M. V., Veliyev, B. — Mon, 18 Jan 2021 10:45:49 +0100

We show that machine learning (ML) algorithms improve one-day-ahead forecasts of realized variance from 29 Dow Jones Industrial Average index stocks over the sample period 2001 - 2017. We inspect several ML approaches: Regularization, tree-based algorithms, and neural networks. Off-the-shelf ML implementations beat the Heterogeneous AutoRegressive (HAR) model, even when the only predictors employed are the daily, weekly, and monthly lag of realized variance. Moreover, ML algorithms are capable of extracting substantial more information from additional predictors of volatility, including firm-specific characteristics and macroeconomic indicators, relative to an extended HAR model (HAR-X). ML automatically deciphers the often nonlinear relationship among the variables, allowing to identify key associations driving volatility. With accumulated local effect (ALE) plots we show there is a general agreement about the set of the most dominant predictors, but disagreement on their ranking. We investigate the robustness of ML when a large number of irrelevant variables, exhibiting serial correlation and conditional heteroscedasticity, are added to the information set. We document sustained forecasting improvements also in this setting.

Roughness in spot variance?

Bolko, A. E., Christensen, K., Pakkanen, M., Veliyev, B. — Thu, 01 Oct 2020 10:45:49 +0200

In this paper, we develop a generalized method of moments approach for joint estimation of the parameters of a fractional log-normal stochastic volatility model. We show that with an arbitrary Hurst exponent an estimator based on integrated variance is consistent. Moreover, under stronger conditions we also derive a central limit theorem. These results stand even when integrated variance is replaced with a realized measure of volatility calculated from discrete high-frequency data. However, in practice a realized estimator contains sampling error, the effect of which is to skew the fractal coefficient toward "roughness". We construct an analytical approach to control this error. In a simulation study, we demonstrate convincing small sample properties of our approach based both on integrated and realized variance over the entire memory spectrum. We show that the bias correction attenuates any systematic deviance in the estimated parameters. Our procedure is applied to empirical high-frequency data from numerous leading equity indexes. With our robust approach the Hurst index is estimated around 0.05, confirming roughness in integrated variance.

The economic value of VIX ETPs

Christensen, K., Christiansen, C., Posselt, A. M. — Wed, 01 Jan 2020 10:45:49 +0100

The fairly new VIX ETPs have been promoted for providing effective and easily accessible diversification, while at the same time having large negative returns. We examine the economic value of using VIX ETPs for diversification of stock–bond portfolios. Our analysis begins in 2009, when the first VIX ETPs are introduced, and therefore only considers the period after the recent financial crisis. For investors with a constant allocation strategy, the diversification benefits of the VIX ETPs do not offset their negative returns. This implies negative economic value of a constant allocation. For a dynamic allocation strategy, including short VIX ETPs in the investment opportunity set can have substantial positive economic value.

The realized empirical distribution function of stochastic variance with application to goodness-of-fit testing

Christensen, K., Thyrsgaard, M., Veliyev, B. — Tue, 01 Oct 2019 10:45:49 +0200

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.

The Economic Value of VIX ETPs

Christensen, K., Christiansen, C., Posselt, A. M. — Wed, 18 Sep 2019 10:45:49 +0200

The fairly new VIX ETPs have been promoted for providing effective and easily accessible diversification. We examine the economic value of using VIX ETPs for diversification of stock-bond portfolios. We consider seven different investment strategies based on short-sales constrained and unconstrained investors who use four different investment styles for their optimization strategy. Our analysis begins in 2009, when the first VIX ETPs are introduced, and therefore only considers the period after the recent financial crisis. For investors prohibited from short selling, the diversification benefits of the VIX ETPs do not offset the negative returns on the VIX ETPs. Hence there is a negative economic value of including VIX ETPs in stock-bond portfolios. This applies to all investment styles. It even applies when adjusting for a simulated market crash. For investors who are not constrained from selling assets short, the results are mixed as the economic value of VIX ETPs vary with respect to investment tyle and product.

The drift burst hypothesis

Christensen, K., Oomen, R., Renò, R. — Mon, 20 Aug 2018 10:45:49 +0200

The drift burst hypothesis postulates the existence of short-lived locally explosive trends in the price paths of financial assets. The recent US equity and treasury flash crashes can be viewed as two high profile manifestations of such dynamics, but we argue that drift bursts of varying magnitude are an expected and regular occurrence in financial markets that can arise through established mechanisms of liquidity provision. We show how to build drift bursts into the continuous-time Itô semimartingale model, discuss the conditions required for the process to remain arbitrage-free, and propose a nonparametric test statistic that identifies drift bursts from noisy highfrequency data. We apply the test to demonstrate that drift bursts are a stylized fact of the price dynamics across equities, fixed income, currencies and commodities. Drift bursts occur once a week on average, and the majority of them are accompanied by subsequent price reversion and can thus be regarded as “flash crashes”. The reversal is found to be stronger for negative drift bursts with large trading volume, which is consistent with endogenous demand for immediacy during market crashes.

Is the diurnal pattern sufficient to explain intraday variation in volatility? A nonparametric assessment

Christensen, K., Hounyo, U., Podolskij, M. — Mon, 01 Jan 2018 10:45:49 +0100

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 Itô 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.

Inference from high-frequency data

Christensen, K., Podolskij, M., Thamrongrat, N., Veliyev, B. — Sun, 01 Jan 2017 10:45:49 +0100

In this paper, we show how to estimate the asymptotic (conditional) covariance matrix, which appears in central limit theorems in high-frequency estimation of asset return volatility. We provide a recipe for the estimation of this matrix by subsampling; an approach that computes rescaled copies of the original statistic based on local stretches of high-frequency data, and then it studies the sampling variation of these. We show that our estimator is consistent both in frictionless markets and models with additive microstructure noise. We derive a rate of convergence for it and are also able to determine an optimal rate for its tuning parameters (e.g., the number of subsamples). Subsampling does not require an extra set of estimators to do inference, which renders it trivial to implement. As a variance–covariance matrix estimator, it has the attractive feature that it is positive semi-definite by construction. Moreover, the subsampler is to some extent automatic, as it does not exploit explicit knowledge about the structure of the asymptotic covariance. It therefore tends to adapt to the problem at hand and be robust against misspecification of the noise process. As such, this paper facilitates assessment of the sampling errors inherent in high-frequency estimation of volatility. We highlight the finite sample properties of the subsampler in a Monte Carlo study, while some initial empirical work demonstrates its use to draw feasible inference about volatility in financial markets.

Fact or friction: Jumps at ultra high frequency

Christensen, K., Oomen, R., Podolskij, M. — Wed, 01 Jan 2014 10:45:49 +0100

A cost-effectiveness analysis of identifying Fusobacterium necrophorum in throat swabs followed by antibiotic treatment to reduce the incidence of Lemierre's syndrome and peritonsillar abscesses

Bank, S., Christensen, K., Kristensen, L. H., Prag, J. — Tue, 01 Jan 2013 10:45:49 +0100

The main purpose of this paper was to estimate the cost per quality-adjusted life year (QALY) saved by identifying Fusobacterium necrophorum in throat swabs followed by proper antibiotic treatment, to reduce the incidence of Lemierre's syndrome and peritonsillar abscesses (PTA) originating from a pharyngitis. The second purpose was to estimate the population size required to indicate that antibiotic treatment has an effect. Data from publications and our laboratory were collected. Monte Carlo simulation and one-way sensitivity analysis were used to analyse cost-effectiveness. The cost-effectiveness analysis shows that examining throat swabs from 15- to 24-year-olds for F. necrophorum followed by antibiotic treatment will probably be less costly than most other life-saving medical interventions, with a median cost of US$8,795 per QALY saved. To indicate a reduced incidence of Lemierre's syndrome and PTA in Denmark, the intervention probably has to be followed for up to 5 years. Identifying F. necrophorum in throat swabs from 15- to 24-year-olds followed by proper antibiotic treatment only requires a reduction of 20-25 % in the incidence of Lemierre's syndrome and PTA to be cost-effective. This study warrants further examination of the effect of antibiotic treatment on the outcome of F. necrophorum acute and recurrent pharyngitis, as well as the effect on Lemierre's syndrome and PTA.

On covariation estimation for multivariate continuous Itô semimartingales with noise in non-synchronous observation schemes

Christensen, K., Podolskij, M., Vetter, M. — Tue, 01 Jan 2013 10:45:49 +0100

Asymptotic theory of range-based multipower variation

Christensen, K., Podolskij, M. — Sun, 01 Jan 2012 10:45:49 +0100

Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data

Christensen, K., Kinnebrock, S., Podolskij, M. — Fri, 01 Jan 2010 10:45:49 +0100

Realised quantile-based estimation of the integrated variance

Christensen, K., Oomen, R., Podolskij, M. — Fri, 01 Jan 2010 10:45:49 +0100

Bias-correcting the realised range-based variance in the presence of market microstructure noise

Christensen, K., Podolskij, M., Vetter, M. — Thu, 01 Jan 2009 10:45:49 +0100

Realized range-based estimation of integrated variance

Christensen, K., Podolskij, M. — Mon, 01 Jan 2007 10:45:49 +0100

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.

Topics on high-frequency financial econometrics

Christensen, K. — Mon, 01 Jan 2007 10:45:49 +0100