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Warp Speed Price Moves: Jumps after Earnings Announcements
... Additionally, the findings of Christensen et al. (2023) based on high-frequency data indicate limited price discovery in the days following earnings announcements. However, their findings are confined to 50 liquid firms due to the extensive size of high-frequency data, while we look at all US stocks and aim to capture the speed of price discovery using daily data. ...
... Jumps in stock prices do relate to news releases, as evidenced by earnings announcements, when over 90% of stocks exhibit at least one jump on their announcement days. This observation aligns with existing literature: the jump's probability is significantly higher during news arrival days (see, e.g., Lee, 2012;Bajgrowicz et al., 2016;Jeon et al., 2022;Christensen et al., 2023). Furthermore, this brief overview illustrates jump clustering behavior, with numerous stocks experiencing multiple jumps following earnings releases. ...
We study the clustering behaviour of stock return jumps modelled by a self/cross-exciting process embedded in a stochastic volatility model. Based on the model estimates, we propose a novel measurement of stock price efficiency characterised by the extent of jump clustering that stock returns exhibit. This measurement demonstrates the capability of capturing the speed at which stock prices assimilate new information, especially at the firm-specific level. Furthermore, we assess the predictability of self-exciting (clustered) jumps in stock returns. We employ a particle filter to sample latent states in the out-of-sample period and perform one-step-ahead probabilistic forecasting on upcoming jumps. We introduce a new statistic derived from predicted probabilities of positive and negative jumps, which has been shown to be effective in return predictions.
Material news events can be potentially important sources of jumps in stock returns. We collect 21 million news articles associated with more than 9,000 publicly-traded companies and use textual analyses to derive measures to summarize the news. We find that stock return jumps (including time-variation in jump-size distributions and jump intensity) are significantly related to news flow frequency and content and those effects increase substantially over the last few decades. The sensitivity of jump probability to news is stronger for firms with higher media visibility, analyst coverage, and institutional ownership. This sensitivity also varies across different news categories.
We examine the speed and mechanism of the price discovery process following earnings announcements in the after‐hours market, a very illiquid trading environment. Prices reflect earnings surprises mostly through changes in quotes rather than through trades. Following positive announcement surprises, ask prices adjust quickly while bid prices are slower to adjust, and vice versa for negative surprises. Returns computed from trade prices underestimate the speed and magnitude of price reactions following announcements relative to returns computed from quotes. These findings emphasize the importance of using quotes and not trade prices when examining intraday price discovery. Because firm announcements such as earnings generally occur in the after‐hours market, using quotes is crucial as trading is sparse. We further illustrate the importance of quotes when examining the price discovery process around analyst recommendation revisions.
If competitive equilibrium is defined as a situation in which prices are such that all arbitrage profits are eliminated, is it possible that a competitive economy always be in equilibrium? Clearly not, for then those who arbitrage make no (private) return from their (privately) costly activity. Hence the assumptions that all markets, including that for information, are always in equilibrium and always perfectly arbitraged are inconsistent when arbitrage is costly. We propose here a model in which there is an equilibrium degree of disequilibrium: prices reflect the information of informed individuals (arbitrageurs) but only partially, so that those who expend resources to obtain information do receive compensation. How informative the price system is depends on the number of individuals who are informed; but the number of individuals who are informed is itself an endogenous variable in the model. The model is the simplest one in which prices perform a well-articulated role in conveying information from the informed to the uninformed. When informed individuals observe information that the return to a security is going to be high, they bid its price up, and conversely when they observe information that the return is going to be low. Thus the price system makes publicly available the information obtained by informed individuals to the uninformed. In general, however, it does this imperfectly; this is perhaps lucky, for were it to do it perfectly , an equilibrium would not exist. In the introduction, we shall discuss the general methodology and present some conjectures concerning certain properties of the equilibrium. The remaining analytic sections of the paper are devoted to analyzing in detail an important example of our general model, in which our conjectures concerning the nature of the equilibrium can be shown to be correct. We conclude with a discussion of the implications of our approach and results, with particular emphasis on the relationship of our results to the literature on "efficient capital markets."
Investors' option-implied fear measures – implied volatility (ATMIV) and put-call implied volatility ratios (P/C) – lead key macroeconomic variables such as industrial capacity utilization and short term interest rates by up to eight quarters. We show that this interaction between fear indices, real activity, and policy variables arises in an equilibrium model where investors learn about the trend-growth regimes of economic data, and the central bank uses a learning-based Taylor rule. The model endogenously generates several time series properties of option prices including the counter (pro) cyclicality of ATMIV (P/C), the V-shape (inverse V-shape) relation between ATMIV (P/C) and monetary policy variables, the positive relation between the level and absolute changes in ATMIV, and an economically significant amount of time variation in the volatility premium.Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.
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 Aït-Sahalia et al. (2011) in the presence of autocorrelated and heteroscedastic market microstructure noise.
A basic result in mathematical finance, sometimes called the fundamental theorem of asset pricing (see [DR 87]), is that for a stochastic process
(St)t Î \mathbb R+(S_{t})_{t\in \mathbb {R}_{+}}
, the existence of an equivalent martingale measure is essentially equivalent to the absence of arbitrage opportunities. In finance the process
(St)t Î \mathbb R+(S_{t})_{t\in \mathbb {R}_{+}}
describes the random evolution of the discounted price of one or several financial assets. The equivalence of no-arbitrage
with the existence of an equivalent probability martingale measure is at the basis of the entire theory of “pricing by arbitrage”.
Starting from the economically meaningful assumption that S does not allow arbitrage profits (different variants of this concept will be defined below), the theorem allows the probability
P on the underlying probability space (Ω,ℱ,P) to be replaced by an equivalent measure Q such that the process S becomes a martingale under the new measure. This makes it possible to use the rich machinery of martingale theory. In particular
the problem of fair pricing of contingent claims is reduced to taking expected values with respect to the measure Q. This method of pricing contingent claims is known to actuaries since the introduction of actuarial skills, centuries ago
and known by the name of “equivalence principle”.
Consider a semimartingale of the form Y-t = Y-0 + integral(t)(0) a(s)ds + integral(t)(0) sigma(s-) dW(s), where a is a locally bounded predictable process and or (the "volatility") is an adapted right-continuous process with left limits and W is a Brownian motion. We consider the realised bipower variation process [GRAPHICS] where r and s are nonnegative reals with r + s > 0. We prove that V(Y;r,s)(n)(t) converges locally uniformly in time, in probability, to a limiting process V(Y;r,s)(t) (the "bipower variation process"). If further or is a possibly discontinuous semimartingale driven by a Brownian motion which may be correlated with W and by a Poisson random measure, we prove that root n (V(Y;r,s)(n) - V(Y;r,s)) converges in law to a process which is the stochastic integral with respect to some other Brownian motion W', which is independent of the driving terms of Y and a. We also provide a [SIC].
Theory suggests that dark pools may facilitate or discourage information acquisition. We find that more dark pool trading leads to greater information acquisition. We measure information acquisition using stock price dynamics around earnings announcements. To overcome endogeneity concerns, we exploit a large exogenous decrease to dark pool trading that results from the implementation of the Security and Exchange Commission’s (SEC’s) Tick Size Pilot Program. The results cannot be explained by lit venue liquidity, algorithmic trading, or informational efficiency. A battery of additional tests, such as documenting a shift in SEC EDGAR searches, supports the information acquisition interpretation.
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.
This paper uses option prices to learn about the equity price uncertainty surrounding information released on earnings announcement dates. To do this, we introduce reduced-form models and estimators to separate price uncertainty about earnings announcements from normal day-to-day volatility. Empirically, we find strong support for the importance of earnings announcements. We find that the anticipated price uncertainty is quantitatively large, varies across time, and is informative about the future return volatility. Finally, we quantify the impact of earnings announcements on formal option pricing models.
Received April 13, 2017; editorial decision February 5, 2018 by Editor Stijn Van Nieuwerburgh. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.
A contrast effect occurs when the value of a previously observed signal inversely biases perception of the next signal. We present the first evidence that contrast effects can distort prices in sophisticated and liquid markets. Investors mistakenly perceive earnings news today as more impressive if yesterday's earnings surprise was bad and less impressive if yesterday's surprise was good. A unique advantage of our financial setting is that we can identify contrast effects as an error in perceptions rather than expectations. Finally, we show that our results cannot be explained by an alternative explanation involving information transmission from previous earnings announcements.
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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.
This paper shows that jumps in financial asset prices are often erroneously identified and are, in fact, rare events accounting for a very small proportion of the total price variation. We apply new econometric techniques to a comprehensive set of ultra high-frequency equity and foreign exchange tick data recorded at millisecond precision, allowing us to examine the price evolution at the individual order level. We show that in both theory and practice, traditional measures of jump variation based on lower-frequency data tend to spuriously assign a burst of volatility to the jump component. As a result, the true price variation coming from jumps is overstated. Our estimates based on tick data suggest that the jump variation is an order of magnitude smaller than typical estimates found in the existing literature.
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.
The simultaneous occurrence of jumps in several stocks can be associated with major financial news, triggers short-term predictability in stock returns, is correlated with sudden spikes of the variance risk premium, and determines a persistent increase (decrease) of stock variances and correlations when they come along with bad (good) news. These systemic events and their implications can be easily overlooked by traditional univariate jump statistics applied to stock indices. They are instead revealed in a clearly cut way by using a novel test procedure applied to individual assets, which is particularly effective on high-volume stocks.
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 Θ = ∫ θt dt, where the spot parameter process θ 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.
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.
Applying tests for jumps to financial data sets can lead to an important number of spurious detections. Bursts of volatility are often incorrectly identified as jumps when the sampling is too sparse. At a higher frequency, methods robust to microstructure noise are required. We argue that whatever the jump detection test and the sampling frequency, a large number of spurious detections remain because of multiple testing issues. We propose a formal treatment based on an explicit thresholding on available test statistics. We prove that our method eliminates asymptotically all remaining spurious detections. In Dow Jones stocks between 2006 and 2008, spurious detections can represent up to 90% of the jumps detected initially. For the stocks considered, jumps are rare events, they do not cluster in time, and no cojump affects all stocks simultaneously, suggesting jump risk is diversifiable. We relate the remaining jumps to macroeconomic news, prescheduled company-specific announcements, and stories from news agencies that include a variety of unscheduled and uncategorized events. The vast majority of news does not cause jumps but may generate a market reaction in the form of bursts of volatility.
We investigate how individual equity prices respond to continuous and jumpy market price moves and how these different market price risks, or betas, are priced in the cross section of expected stock returns. Based on a novel high-frequency data set of almost 1,000 stocks over two decades, we find that the two rough betas associated with intraday discontinuous and overnight returns entail significant risk premiums, while the intraday continuous beta does not. These higher risk premiums for the discontinuous and overnight market betas remain significant after controlling for a long list of other firm characteristics and explanatory variables.
We 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 non-smooth loss functions (like L1) in the estimation. Unlike classical robust methods, the limit of the objective function here remains non-smooth. The proposed method is also robust to measurement error in the observed processes which is achieved by locally smoothing the high-frequency increments. In an empirical application to financial data we illustrate the usefulness of the robust techniques by contrasting the behavior of robust and OLS-type jump regressions in periods including disruptions of the financial markets such as so called “flash crashes.”
A five-factor model directed at capturing the size, value, profitability, and investment patterns in average stock returns performs better than the three-factor model of Fama and French (FF, 1993). The five-factor model's main problem is its failure to capture the low average returns on small stocks whose returns behave like those of firms that invest a lot despite low profitability. The model's performance is not sensitive to the way its factors are defined. With the addition of profitability and investment factors, the value factor of the FF three-factor model becomes redundant for describing average returns in the sample we examine.
Part I Introduction and Preliminary Material.- 1.Introduction .- 2.Some Prerequisites.- Part II The Basic Results.- 3.Laws of Large Numbers: the Basic Results.- 4.Central Limit Theorems: Technical Tools.- 5.Central Limit Theorems: the Basic Results.- 6.Integrated Discretization Error.- Part III More Laws of Large Numbers.- 7.First Extension: Random Weights.- 8.Second Extension: Functions of Several Increments.- 9.Third Extension: Truncated Functionals.- Part IV Extensions of the Central Limit Theorems.- 10.The Central Limit Theorem for Random Weights.- 11.The Central Limit Theorem for Functions of a Finite Number of Increments.- 12.The Central Limit Theorem for Functions of an Increasing Number of Increments.- 13.The Central Limit Theorem for Truncated Functionals.- Part V Various Extensions.- 14.Irregular Discretization Schemes. 15.Higher Order Limit Theorems.- 16.Semimartingales Contaminated by Noise.- Appendix.- References.- Assumptions.- Index of Functionals.- Index.
We study after-hours trading (AHT), price contributions, and price discovery following quarterly earnings announcements released outside of the normal trading hours. For Standard & Poor’s (S&P) 500 index stocks from 2004–2008, AHT is heightened on announcement days. A significant portion of the price change and price discovery occurs immediately after the earnings releases. Prices in AHT show a large degree of informational efficiency, further demonstrating the importance of price discovery in AHT. We also provide evidence suggesting that firms prefer after-hours earnings announcements, as trades are mainly from informed traders, and those trades are relied upon to convey information to the general public.
Does limited attention among investors affect stock returns? We compare the response to earnings announcements on Friday, when investor inattention is more likely, to the response on other weekdays. If inattention influences stock prices, we should observe less immediate response and more drift for Friday announcements. Indeed, Friday announcements have a 15% lower immediate response and a 70% higher delayed response. A portfolio investing in differential Friday drift earns substantial abnormal returns. In addition, trading volume is 8% lower around Friday announcements. These findings support explanations of post-earnings announcement drift based on underreaction to information caused by limited attention. Copyright (c) 2009 the American Finance Association.
Using a new data set consisting of six years of real-time exchange-rate quotations, macroeconomic expectations, and macroeconomic realizations, we characterize the conditional means of U.S. dollar spot exchange rates. In particular, we find that announcement surprises produce conditional mean jumps; hence high-frequency exchange-rate dynamics are linked to fundamentals. The details of the linkage are intriguing and include announcement timing and sign effects. The sign effect refers to the fact that the market reacts to news in an asymmetric fashion: bad news has greater impact than good news, which we relate to recent theoretical work on information processing and price discovery.
This paper proposes a robustification of the test statistic of Aït-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.
For firms seeking to strategically combine their resources with those of other firms, two popular alternative governance structures emerge: alliance or acquisition. In this paper, we propose a dyadic perspective to examine how and why configurations of two firms' resources and capabilities affect the costs and benefits associated with each governance structure. More specifically, we posit that factors such as (1) the resource similarity and complementarity between a pair of firms, (2) the combined relational capabilities of a pair of firms, and (3) the partner-specific knowledge between a pair of firms will affect the likelihood of observing that pair of firms forming an alliance vs. engaging in an acquisition. We test and find support for our hypotheses using extensive longitudinal data from a sample of the largest firms in the United States from 1991 to 2000. Copyright © 2007 John Wiley & Sons, Ltd.
Summary Realized kernels use high-frequency data to estimate daily volatility of individual stock prices. They can be applied to either trade or quote data. Here we provide the details of how we suggest implementing them in practice. We compare the estimates based on trade and quote data for the same stock and find a remarkable level of agreement.
We identify some features of the high-frequency data, which are challenging for realized kernels. They are when there are local trends in the data, over periods of around 10 minutes, where the prices and quotes are driven up or down. These can be associated with high volumes. One explanation for this is that they are due to non-trivial liquidity effects.
The paper examines volatility activity and its asymmetry and undertakes further specification analysis of volatility models based on it. We develop new nonparametric statistics using high frequency option-based VIX data to test for asymmetry in volatility jumps. We also develop methods to estimate and evaluate, using price data alone, a general encompassing model for volatility dynamics where volatility activity is unrestricted. The nonparametric application to VIX data, along with model estimation for S&P Index returns, suggests that volatility moves are best captured by infinite variation pure-jump martingale with symmetric jump distribution. The latter provides a parsimonious generalization of the jump-diffusions commonly used for volatility modeling.
We consider a stochastic process driven by diffusions and jumps. Given a discrete record of observations, we devise a technique for identifying the times when jumps larger than a suitably defined threshold occurred. This allows us to determine a consistent non-parametric estimator of the integrated volatility when the infinite activity jump component is Lévy. Jump size estimation and central limit results are proved in the case of finite activity jumps. Some simulations illustrate the applicability of the methodology in finite samples and its superiority on the multipower variations especially when it is not possible to use high frequency data.
This paper presents a generalized pre-averaging approach for estimating the integrated volatility, in the presence of noise. This approach also provides consistent estimators of other powers of volatility -- in particular, it gives feasible ways to consistently estimate the asymptotic variance of the estimator of the integrated volatility. We show that our approach, which possesses an intuitive transparency, can generate rate optimal estimators (with convergence rate n-1/4).
We propose an econometric model that captures the effects of market microstructure on a latent price process. In particular, we allow for correlation between the measurement error and the return process and we allow the measurement error process to have a diurnal heteroskedasticity. We propose a modification of the TSRV estimator of quadratic variation. We show that this estimator is consistent, with a rate of convergence that depends on the size of the measurement error, but is no worse than n−1/6. We investigate in simulation experiments the finite sample performance of various proposed implementations.
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.
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.
The space-fractional diffusion equation models anomalous super-diffusion. Its solutions are transition densities of a stable Lévy motion, representing the accumulation of power-law jumps. The tempered stable Lévy motion uses exponential tempering to cool these jumps. A tempered fractional diffusion equation governs the transition densities, which progress from super-diffusive early-time to diffusive late-time behavior. This article provides finite difference and particle tracking methods for solving the tempered fractional diffusion equation with drift. A temporal and spatial second-order Crank–Nicolson method is developed, based on a finite difference formula for tempered fractional derivatives. A new exponential rejection method for simulating tempered Lévy stables is presented to facilitate particle tracking codes.
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 Itô’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.
We develop and implement a method for maximum likelihood estimation in closed-form of stochastic volatility models. Using Monte Carlo simulations, we compare a full likelihood procedure, where an option price is inverted into the unobservable volatility state, to an approximate likelihood procedure where the volatility state is replaced by proxies based on the implied volatility of a short-dated at-the-money option. The approximation results in a small loss of accuracy relative to the standard errors due to sampling noise. We apply this method to market prices of index options for several stochastic volatility models, and compare the characteristics of the estimated models. The evidence for a general CEV model, which nests both the affine Heston model and a GARCH model, suggests that the elasticity of variance of volatility lies between that assumed by the two nested models.
We test for price discontinuities, or jumps, in a panel of high-frequency intraday stock returns and an equiweighted index constructed from the same stocks. Using a new test for common jumps that explicitly utilizes the cross-covariance structure in the returns to identify non-diversifiable jumps, we find strong evidence for many modest-sized, yet highly significant, cojumps that simply pass through standard jump detection statistics when applied on a stock-by-stock basis. Our results are further corroborated by a striking within-day pattern in the significant cojumps, with a sharp peak at the time of regularly scheduled macroeconomic news announcements.
Recent asset-pricing models incorporate jump risk through Lévy processes in addition to diffusive risk. This paper studies how to detect stochastic arrivals of small and big Lévy jumps with new nonparametric tests. The tests allow for robust analysis of their separate characteristics and facilitate better estimation of return dynamics. Empirical evidence of both small and big jumps based on these tests suggests that models for individual equities and overall market indices require incorporating Lévy-type jumps. The evidence of small jumps also helps explain why jumps in the market index are uncorrelated with jumps in its component equities.
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. Comment: Published in at http://dx.doi.org/10.1214/08-AOS640 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)