Torben Gustav Andersen

• PhD Economics; Yale University
• Northwestern University

The VIX index is computed as a weighted average of two model-free option-implied variance measures for maturities around 30 days. This induces an interpolation error which, prior to 2014 (old VIX), could be sizable and especially pertinent under stressed market conditions. We quantify the VIX interpolation errors and identify underlying sources. We...

This paper focuses on the task of detecting local episodes involving violation of the standard It\^o semimartingale assumption for financial asset prices in real time that might induce arbitrage opportunities. Our proposed detectors, defined as stopping rules, are applied sequentially to continually incoming high-frequency data. We show that they a...

The volatility of financial asset returns displays pronounced variation over the trading day. Our goal is nonparametric inference for the average intraday volatility pattern, viewed as a function of time-of-day. The functional inference is based on a long span of high-frequency return data. Our setup allows for general forms of volatility dynamics,...

This paper studies the properties of predictive regressions for asset returns in economic systems governed by persistent vector autoregressive dynamics. In particular, we allow for the state variables to be fractionally integrated, potentially of different orders, and for the returns to have a latent persistent conditional mean, whose memory is dif...

This paper develops parameter instability and structural change tests within predictive regressions for economic systems governed by persistent vector autoregressive dynamics. Specifically, in a setting where all – or a subset – of the variables may be fractionally integrated and the predictive relation may feature cointegration, we provide sup-Wal...

We extend the classic ”martingale-plus-noise” model for high-frequency returns to accommodate an error correction mechanism and endogenous pricing errors. It is motivated by (i) novel empirical evidence documenting that microstructure noise exhibits frequently changing patterns of serial dependence which are interwoven with innovations to the effic...

We study the temporal behavior of the cross‐sectional distribution of assets' market exposure, or betas, using a large panel of high‐frequency returns. The asymptotic setup has the sampling frequency of returns increasing to infinity, while the time span of the data remains fixed, and the cross‐sectional dimension of the panel is either fixed or in...

Increasing evidence points towards the episodic emergence of pockets with extreme return persistence. This notion refers to intraday periods of non-trivial duration, for which stock returns are highly positively autocorrelated. Such episodes include, but are not limited to, gradual jumps and prolonged bursts in the drift component. In this paper, w...

This paper provides a guide to high-frequency option trade and quote data disseminated by the Options Price Reporting Authority (OPRA). We present a comprehensive overview of the U.S. option market, including details on market regulation and the trading processes for all 16 constituent option exchanges. We review the existing literature that utiliz...

This paper studies standard predictive regressions in economic systems governed by persistent vector autoregressive dynamics for the state variables. In particular, all – or a subset – of the variables may be fractionally integrated, which induces a spurious regression problem. We propose a new inference and testing procedure – the Local speCtruM (...

This paper studies the predictability of the Japanese equity market, focusing on the forecasting power of nonparametric volatility and tail risk measures obtained from options data on the S&P 500 and Nikkei 225 market indices. The Japanese market is notoriously difficult to forecast using standard predictive indicators. We confirm that country-spec...

In this paper, we develop the first formal nonparametric test for whether the observation errors in option panels display spatial dependence. The panel consists of options with different strikes and tenors written on a given underlying asset. The asymptotic design is of the infill type—the mesh of the strike grid for the observed options shrinks as...

We provide unifying inference theory for parametric nonlinear factor models based on a panel of noisy observations. The panel has a large cross-section and a time span that may be either small or large. Moreover, we incorporate an additional source of information, provided by noisy observations on some known functions of the factor realizations. Th...

We explore the pricing of tail risk as manifest in index options across international equity markets. The risk premium associated with negative tail events displays persistent shifts, unrelated to volatility. This tail risk premium is a potent predictor of future returns for all the indices, while the option-implied volatility only forecasts the fu...

We develop parametric inference procedures for large panels of noisy option data in a setting, where the underlying process is of pure-jump type, i.e., evolves only through a sequence of jumps. The panel consists of options written on the underlying asset with a (different) set of strikes and maturities available across the observation times. We co...

We develop a nonparametric test for whether return volatility exhibits time-varying intraday periodicity using a long time series of high-frequency data. Our null hypothesis, commonly adopted in work on volatility modeling, is that volatility follows a stationary process combined with a constant time-of-day periodic component. We construct time-of-...

We study short-maturity ("weekly") S&P 500 index options, which provide a direct way to analyze volatility and jump risks. Unlike longer-dated options, they are largely insensitive to the risk of intertemporal shifts in the economic environment. Adopting a novel seminonparametric approach, we uncover variation in the negative jump tail risk, which...

We study the dynamic relation between market risks and risk premia using time series of index option surfaces. We find that priced left tail risk cannot be spanned by market volatility (and its components) and introduce a new tail factor. This tail factor has no incremental predictive power for future volatility and jump risks, beyond current and p...

We analyze the high-frequency dynamics of S&P 500 equity-index option prices by constructing an assortment of implied volatility measures. This allows us to infer the underlying fine structure behind the innovations in the latent state variables driving the evolution of the volatility surface. In particular, we focus attention on implied volatiliti...

Following the “flash crash” on May 6, 2010, warning signals for impending market stress have been in high demand, yet only the VPIN metric of Easley, López de Prado, and O’Hara (ELO) has claimed success. In addition, ELO find the metric useful in predicting short-term volatility. VPIN involves decomposing volume into active buys and sells. We utili...

Some fundamental questions regarding equity-index return dynamics are difficult to address due to the latent character of spot volatility. We exploit tick-by-tick option quotes to compute a novel “Corridor Volatility” index which may serve as an observable proxy for short-term volatility. Exploiting this index, we find that equity-index volatility...

The intraday trading patterns in the E-mini S&P 500 futures contract between January 2008 and November 2011 are consistent with the following invariance relationship: The return variation per transaction is log-linearly related to trade size, with a slope coefficient of -2. This association applies both across the pronounced intraday diurnal patter...

First version: March 10, 2008; This version: July 2, 2008; Preliminary Draft: Comments Welcome We develop a novel approach to estimating the integrated variance of a general jump-diffusion with stochastic volatility. Our approach exploits the relationship between the speed (distance traveled per fixed time unit) and passage time (time taken to trav...

We provide a first in-depth look at robust estimation of integrated quarticity (IQ) based on high-frequency data. IQ is the key ingredient enabling inference about volatility and the presence of jumps in financial time series and is thus of considerable interest in applications. We document the significant empirical challenges for IQ estimation pos...

The Volume-Synchronized Probability of Informed trading (VPIN) metric is introduced by Easley, López de Prado, and O'Hara (2011a) as a real-time indicator of order flow toxicity. They find the measure useful in monitoring order flow imbalances and conclude it may help signal impending market turmoil, exemplified by historical high readings of the m...

In Andersen and Bondarenko (2014), using tick data for S&P 500 futures, we establish that the VPIN metric of Easley, López de Prado, and O'Hara (ELO), by construction, will be correlated with trading volume and return volatility (innovations). Whether VPIN is more strongly correlated with volume or volatility depends on the exact implementation. He...

The Volume-Synchronized Probability of Informed trading (VPIN) metric is introduced by Easley, Lopez de Prado, and O'Hara (2011a) as a real-time indicator of order flow toxicity. They find the measure useful in monitoring order flow imbalances and conclude it may help signal impending market turmoil, exemplified by historical high readings of the m...

Following the much publicized "flash crash" in the U.S. financial markets on May 6, 2010, much work has been done in terms of developing reliable warning signals for impending market stress. However, this has met with limited success, except for one measure. The VPIN, or Volume-synchronized Probability of INformed trading, metric is introduced by E...

Wu, Bethel, Gu, Leinweber, and Ruebel (2013b) provide a response to our commentary, "Reflecting on the VPIN Dispute.'' They interpret a few comments in our piece as questioning their VPIN implementation procedures. This is not the case. In fact, the results of WBGLR (2013a) are fully consistent with our general VPIN critique. On the other hand, it...

In Andersen and Bondarenko (2014), using tick data for S&P 500 futures, we establish that the VPIN metric of Easley, Lopez de Prado, and O'Hara (ELO), by construction, will be correlated with trading volume and return volatility (innovations). Whether VPIN is more strongly correlated with volume or volatility depends on the exact implementation. He...

We provide a first in-depth look at robust estimation of integrated quarticity (IQ) based on high frequency data. IQ is the key ingredient enabling inference about volatility and the presence of jumps in financial time series and is thus of considerable interest in applications. We document the significant empirical challenges for IQ estimation pos...

A number of fundamental questions regarding the equity-index return dynamics are difficult to address due to the latent character of spot volatility. For example, it is hard to discern much about the nature of volatility jumps or the features of the negative (spot) return-volatility correlation. We show that, in theory, the VIX index can be useful...

We develop a new parametric estimation procedure for option panels observed with error which relies on asymptotic approximations assuming an ever increasing set of observed option prices in the moneyness- maturity (cross-sectional) dimension, but with a fixed time span. We develop consistent estimators of the parameter vector and the dynamic realiz...

Current practice largely follows restrictive approaches to market risk measurement, such as historical simulation or RiskMetrics. In contrast, we propose flexible methods that exploit recent developments in financial econometrics and are likely to produce more accurate risk assessments, treating both portfolio-level and asset-level analysis. Asset-...

We provide a first in-depth look at robust estimation of integrated quarticity (IQ) based on high frequency data. IQ is the key ingredient enabling inference about volatility and the presence of jumps in financial time series and is thus of considerable interest in applications. We document the significant empirical challenges for IQ estimation pos...

We extend the analytical results for reduced form realized volatility based forecasting in ABM (2004) to allow for market microstructure frictions in the observed high-frequency returns. Our results build on the eigenfunction representation of the general stochastic volatility class of models developed byMeddahi (2001). In addition to traditional r...

Glossary Definition of the Subject Introduction Model Specification Realized Volatility Applications Estimation Methods Future Directions Acknowledgments Bibliography

The VIX index is computed as a weighted average of SPX option prices over a range of strikes according to specific rules regarding market liquidity. Using tick-by-tick observations on the underlying options, we document that this strike range varies sub-stantially in how much coverage it provides of the distribution of future S&P 500 index prices,...

The VIX index is computed as a weighted average of SPX option prices over a range of strikes according to specific rules regarding market liquidity. It is explicitly designed to provide a model-free option-implied volatility measure. Using tick-by-tick observations on the underlying options, we document a substantial time variation in the coverage...

This article reviews basic notions of return variation in the context of a continuous-time arbitrage-free asset pricing model and discusses some of their applications. We first define return variation in the infeasible continuous-sampling case. Then we introduce realized measures obtained from high-frequency observations which provide consistent an...

The model-free implied volatility (MFVOL) has gained notoriety in recent years, partially due to the Chicago Board Option Exchange (CBOE) publication of the VIX index which is based on the MFVOL. This volatility measure extracts the expected future return volatil- ity from the full cross-section of options with dierent strikes for a given maturity...

This paper provides an introduction to alternative models of uncertain commodity prices. A model of commodity price movements is the engine around which any valuation methodology for commodity production projects is built, whether discounted cash flow (DCF) models or the recently developed modern asset pricing (MAP) methods. The accuracy of the val...

This paper provides an introduction to alternative models of uncertain commodity prices. A model of commodity price movements is the engine around which any valuation methodology for commodity production projects is built, whether discounted cash flow (DCF) models or the recently developed modern asset pricing (MAP) methods. The accuracy of the val...

Recent Developments in GARCH Modeling.- An Introduction to Univariate GARCH Models.- Stationarity, Mixing, Distributional Properties and Moments of GARCH(p, q)#x2013 Processes.- ARCH(#x221E ) Models and Long Memory Properties.- A Tour in the Asymptotic Theory of GARCH Estimation.- Practical Issues in the Analysis of Univariate GARCH Models.- Semipa...

Given the importance of return volatility on a number of practical financial management decisions, the efforts to provide good real- time estimates and forecasts of current and future volatility have been extensive. The main framework used in this context involves stochastic volatility models. In a broad sense, this model class includes GARCH, but...

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 ar...

Realized volatility is a nonparametric ex-post estimate of the return variation. The most obvious realized volatility measure is the sum of finely-sampled squared return realizations over a fixed time interval. In a frictionless market the estimate achieves consistency for the underlying quadratic return variation when returns are sampled at increa...

Stochastic volatility is the main way time-varying volatility is modelled in financial markets. The development of stochastic volatility is reviewed, placing it in a modeling and historical context. Some recent trends in the literature are highlighted.

This paper provides an introduction to alternative models of uncertain commodity prices. A model of commodity price movements is the engine around which any valuation methodology for commodity production projects is built, whether discounted cash flow (DCF) models or the recently developed modern asset pricing (MAP) methods. The accuracy of the val...

The notion of model-free implied volatility (MFIV), constituting the basis for the highly publicized VIX volatility index, can be hard to measure with accuracy due to the lack of precise prices for options with strikes in the tails of the return distribution. This is reflected in practice as the VIX index is computed through a tail-truncation which...

We provide an empirical framework for assessing the distributional properties of daily specu- lative returns within the context of the continuous-time modeling paradigm traditionally used in asset pricing finance. Our approach builds directly on recently developed realized variation measures and non-parametric jump detection statistics constructed...

This chapter demonstrates how important it is to recognize time-varying volatility and correlation in value at risk estimation. It also illustrates how new techniques in multivariate time-series estimation could be usefully brought to bear to address some of the problems that pushed banks toward historical simulation and stress testing. Then, it co...

A growing literature documents important gains in asset return volatility forecasting via use of realized variation measures constructed from high-frequency returns. We progress by using newly developed bipower variation measures and corresponding nonparametric tests for jumps. Our empirical analyses of exchange rates, equity index returns, and bon...

We develop a sequential procedure to test the adequacy of jump-diffusion models for return distributions. We rely on intraday data and nonparametric volatility measures, along with a new jump detection technique and appropriate conditional moment tests, for assessing the import of jumps and leverage effects. A novel robust-to-jumps approach is util...

Using a unique high-frequency futures dataset, we characterize the response of U.S., German and British stock, bond and foreign exchange markets to real-time U.S. macroeconomic news. We find that news produces conditional mean jumps; hence high-frequency stock, bond and exchange rate dynamics are linked to fundamentals. Equity markets, moreover, re...

Volatility has been one of the most active and successful areas of research in time series econometrics and economic forecasting in recent decades. This chapter provides a selective survey of the most important theoretical developments and empirical insights to emerge from this burgeoning literature, with a distinct focus on forecasting application...

Using a unique high-frequency futures dataset, we characterize the response of U.S., German and British stock, bond and foreign exchange markets to real-time U.S. macroeconomic news. We find that news produces conditional mean jumps; hence high-frequency stock, bond and exchange rate dynamics are linked to fundamentals. Equity markets, moreover, re...

We investigate whether bonds can hedge volatility risk in the U.S.\ Treasury market, as predicted by most 'affine' term structure models. To this end, we construct powerful and model-free empirical measures of the quadratic yield variation for a cross-section of fixed-maturity zero-coupon bonds ('realized yield volatility') through the use of high-...

We investigate whether bonds span the volatility risk in the U.S. Treasury market, as predicted by most 'affine' term structure models. To this end, we construct powerful and model-free empirical measures of the quadratic yield variation for a cross-section of fixed- maturity zero-coupon bonds ('realized yield volatility') through the use of high-f...

Using high-frequency data on deutschmark 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...

What do academics have to offer market risk management practitioners in financial institutions? Current industry practice largely follows one of two extremely restrictive approaches: historical simulation or RiskMetrics. In contrast, we favor flexible methods based on recent developments in financial econometrics, which are likely to produce more a...

We selectively survey, unify and extend the literature on realized volatility of financial asset returns. Rather than focusing exclusively on characterizing the properties of realized volatility, we progress by examining economically interesting functions of realized volatility, namely realized betas for equity portfolios, relating them both to the...

Publisher Summary This chapter provides a unified continuous-time, frictionless, no-arbitrage framework for systematically categorizing the various volatility concepts, measurement procedures, and modeling procedures. Volatility has been one of the most active areas of research in empirical finance and time series econometrics during the past decad...

We develop general model-free adjustment procedures for the calculation of unbiased volatility loss functions based on practically feasible realized volatility benchmarks. The procedures, which exploit recent nonparametric asymptotic distributional results, are both easy-to-implement and highly accurate in empirically realistic situations. We also...

This paper presents a stock-flow consistent macroeconomic model in which financial fragility in firm and household sectors evolves endogenously through the interaction between real and financial sectors. Changes in firms' and households' financial practices produce long waves. The Hopf bifurcation theorem is applied to clarify the conditions for th...

This paper provides an introduction to alternative models of uncertain commodity prices. A model of commodity price movements is the engine around which any valuation methodology for commodity production projects is built, whether discounted cash flow (DCF) models or the recently developed modern asset pricing (MAP) methods. The accuracy of the val...

Estimation and forecasting for realistic continuous-time stochastic volatility models is hampered by the lack of closed-form expressions for the likelihood. In response, Andersen, Bollerslev, Diebold, and Labys ("Econometrica", 71 (2003), 579-625) advocate forecasting integrated volatility via reduced-form models for the realized volatility, constr...

We characterize the response of U.S., German and British stock, bond and foreign exchange markets to real-time U.S. macroeconomic news. Our analysis is based on a unique data set of highfrequency futures returns for each of the markets. We find that news surprises produce conditional mean jumps; hence high-frequency stock, bond and exchange rate dy...

A large literature over several decades reveals both extensive concern with the question of time-varying betas and an emerging consensus that betas are in fact time-varying, leading to the prominence of the conditional CAPM. Set against that background, we assess the dynamics in realized betas, vis-a-vis the dynamics in the underlying realized mark...

We characterize the response of U.S., German and British stock, bond and foreign exchange markets to real-time U.S. macroeconomic news. Our analysis is based on a unique data set of high-frequency futures returns for each of the markets. We find that news surprises produce conditional mean jumps; hence high-frequency stock, bond and exchange rate d...

We find that an intuitively appealing and fairly manageable continuous-time model provides an excellent characterization of the U.S. short-term interest rate over the post Second World War period. Our three-factor jump-diffusion model consists of elements embodied in existing specifications, but our approach appears to be the first to successfully...