Infinite Variation Jumps Research Articles

Drift burst test statistic in the presence of infinite variation jumps

We consider the test statistic devised by Christensen, Oomen and Renò in 2020 to obtain insight into the causes of flash crashes occurring at particular moments in time in the price of a financial asset. Under an Ito semimartingale model containing a drift component, a Brownian component and finite variation jumps, it is possible to identify when the cause is a drift burst (the statistic explodes) or otherwise (the statistic is asymptotically Gaussian). We complete the investigation showing how infinite variation jumps contribute asymptotically. The result is that the jumps never cause the explosion of the statistic. Specifically, when there are no bursts, the statistic diverges only if the Brownian component is absent, the jumps have finite variation and the drift is non-zero. In this case the triggering is precisely the drift. We also find that the statistic could be adopted for a variety of tests useful for investigating the nature of the data generating process, given discrete observations.

Estimation of Tempered Stable Lévy Models of Infinite Variation

Truncated realized quadratic variations (TRQV) are among the most widely used high-frequency-based nonparametric methods to estimate the volatility of a process in the presence of jumps. Nevertheless, the truncation level is known to critically affect its performance, especially in the presence of infinite variation jumps. In this paper, we study the optimal truncation level, in the mean-square error sense, for a semiparametric tempered stable Lévy model. We obtain a novel closed-form 2nd-order approximation of the optimal threshold in a high-frequency setting. As an application, we propose a new estimation method, which combines iteratively an approximate semiparametric method of moment estimator and TRQVs with the newly found small-time approximation for the optimal threshold. The method is tested via simulations to estimate the volatility and the Blumenthal-Getoor index of a generalized CGMY model and, via a localization technique, to estimate the integrated volatility of a Heston type model with CGMY jumps. Our method is found to outperform other alternatives proposed in the literature when working with a Lévy process (i.e., the volatility is constant), or when the index of jump intensity Y is larger than 3/2 in the presence of stochastic volatility.

Statistical Inference of Spot Correlation and Spot Market Beta under Infinite Variation Jumps

Abstract Empirical evidence has revealed that the jumps in financial markets appear to be very frequent. This study considers the statistical inference of the spot correlation and the spot market beta between two different assets using high-frequency data, in a setting where both the cojumps and the individual jumps in the underlying driving processes could be of infinite variation. Starting from the estimation of the spot covariance, we propose consistent estimators of the spot correlation and the spot market beta when the jump processes involved are general semimartingales. The second-order approximation for the estimators, namely, the central limit theorems, is established under the assumption that the jumps around zero are of stable Lévy type. Our estimation procedure is based on the empirical characteristic function of the increments of the processes and the application of the polarization identity; the bias terms stemming from the jumps are removed iteratively. The finite sample performances of the proposed estimators and other existing estimators are assessed and compared by using datasets simulated from various models. Our estimators are also applied to some real high-frequency financial datasets.

Rate-optimal estimation of the Blumenthal–Getoor index of a Lévy process

The Blumenthal–Getoor (BG) index characterizes the jump measure of an infinitely active Lévy process. It determines sample path properties and affects the behavior of various econometric procedures. If the process contains a diffusion term, existing estimators of the BG index based on high-frequency observations achieve rates of convergence which are suboptimal by a polynomial factor. In this paper, a novel estimator for the BG index and the successive BG indices is presented, attaining the optimal rate of convergence. If an additional proportionality factor needs to be inferred, the proposed estimator is rate-optimal up to logarithmic factors. Furthermore, our method yields a new efficient volatility estimator which accounts for jumps of infinite variation. All parameters are estimated jointly by the generalized method of moments. A simulation study compares the finite sample behavior of the proposed estimators with competing methods from the financial econometrics literature.

On finite difference schemes for partial integro-differential equations of Lévy type

In this article we introduce a finite difference approximation for integro-differential operators of Lévy type. We approximate solutions of possibly degenerate integro-differential equations by treating the nonlocal operator as a second-order operator on the whole unit ball, eliminating the need for truncation of the Lévy measure which is present in the existing literature. This yields an approximation scheme with significantly reduced computational cost, especially for Lévy measures corresponding to processes with jumps of infinite variation.

Lack of Fit Test for Infinite Variation Jumps at High Frequencies

In this paper, we are concerned with testing for infinite variation jumps in addition to a continuous local martingale component driven by Brownian motion using high-frequency data. We developed a lack of fit type test based on the empirical distribution of the devolatized increments. Under the null that the jump component is of finite variation, the empirical process associated with the devolatized increments converges to a Gaussian process in the Skorohod topology. Under the alternative hypothesis that the jumps are of infinite variation, that empirical process explodes instead. Theoretical results show good performance of the size and power. Simulation studies justify the theory. A real financial data set is analyzed.

Estimating spot volatility in the presence of infinite variation jumps

We propose a kernel estimator for the spot volatility of a semi-martingale at a given time point by using high frequency data, where the underlying process accommodates a jump part of infinite variation. The estimator is based on the representation of the characteristic function of Lévy processes. The consistency of the proposed estimator is established under some mild assumptions. By assuming that the jump part of the underlying process behaves like a symmetric stable Lévy process around 0, we establish the asymptotic normality of the proposed estimator. In particular, with a specific kernel function, the estimator is variance efficient. We conduct Monte Carlo simulation studies to assess our theoretical results and compare our estimator with existing ones.

Truncated Realized Covariance when prices have infinite variation jumps

The speed of convergence of the Truncated Realized Covariance (TRC) to the Integrated Covariation between the Brownian parts of two semimartingales is heavily influenced by the presence of infinite activity jumps with infinite variation (iV), through both the degree of dependence and the jump activity indices of the two small jumps processes. To show this, marginal stable small jumps with a parametric dependence structure are considered. The estimator is efficient only when the iV jumps have moderate activity.The results presented in this paper are relevant to financial economics, since through the TRC it is possible to separately estimate the common jumps between two assets, which has important implications in risk management and contagion modeling.

Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps

The implied volatility skew has received relatively little attention in the literature on short-term asymptotics for financial models with jumps, despite its importance in model selection and calibration. We rectify this by providing high order asymptotic expansions for the at-the-money implied volatility skew, under a rich class of stochastic volatility models with independent stable-like jumps of infinite variation. The case of a pure-jump stable-like Levy model is also considered under the minimal possible conditions for the resulting expansion to be well defined. Unlike recent results for “near-the-money” option prices and implied volatility, the results herein aid in understanding how the implied volatility smile near expiry is affected by important features of the continuous component, such as the leverage and vol-of-vol parameters. As intermediary results, we obtain high order expansions for at-the-money digital call option prices, which furthermore allow us to infer analogous results for the delta of at-the-money options. Simulation results indicate that our asymptotic expansions give good fits for options with maturities up to one month, underpinning their relevance in practical applications, and an analysis of the implied volatility skew in recent S&P 500 options data shows it to be consistent with the infinite variation jump component of our models.

Lack of Fit Test for Infinite Variation Jumps at High Frequencies

In this paper, we are concerned with testing for infinite variation jumps in addition to a continuous local martingale component driven by Brownian motion using high-frequency data. We developed a lack of fit type test based on the empirical distribution of the devolatized increments. Under the null that the jump component is of finite variation, the empirical process associated with the devolatized increments converges to a Gaussian process in the Skorohod topology. Under the alternative hypothesis that the jumps are of infinite variation, that empirical process explodes instead. Theoretical results show good performance of the size and power. Simulation studies justify the theory. A real financial data set is analyzed.

Truncated Realized Covariance When Prices Have Infinite Variation Jumps

The speed of convergence of the truncated realized covariance to the integrated covariation between the Brownian parts of two semimartingales is heavily influenced by the presence of infinite activity jumps with infinite variation. Namely, the two small jumps processes play a crucial role through their degree of dependence, other than through their jump activity indices. This theoretical result is established when the marginal small jumps constitute alpha-stable processes and the semimartingales are observed discretely on a finite time horizon. The estimator in many (but not all) cases is less efficient than when the model only has finite variation jumps.The result of this paper is relevant in financial economics, since by the truncated realized covariance it is possible to separately estimate the common jumps among two assets, which has important implications in risk management and contagion modeling.

Testing for pure-jump processes for high-frequency data

Pure-jump processes have been increasingly popular in modeling high-frequency financial data, partially due to their versatility and flexibility. In the meantime, several statistical tests have been proposed in the literature to check the validity of using pure-jump models. However, these tests suffer from several drawbacks, such as requiring rather stringent conditions and having slow rates of convergence. In this paper, we propose a different test to check whether the underlying process of high-frequency data can be modeled by a pure-jump process. The new test is based on the realized characteristic function, and enjoys a much faster convergence rate of order $O(n^{1/2})$ (where $n$ is the sample size) versus the usual $o(n^{1/4})$ available for existing tests; it is applicable much more generally than previous tests; for example, it is robust to jumps of infinite variation and flexible modeling of the diffusion component. Simulation studies justify our findings and the test is also applied to some real high-frequency financial data.

Estimating time-changes in noisy Lévy models

In quantitative finance, we often model asset prices as a noisy Ito semimartingale. As this model is not identifiable, approximating by a time-changed Levy process can be useful for generative modelling. We give a new estimate of the normalised volatility or time change in this model, which obtains minimax convergence rates, and is unaffected by infinite-variation jumps. In the semimartingale model, our estimate remains accurate for the normalised volatility, obtaining convergence rates as good as any previously implied in the literature.

Efficient estimation of integrated volatility in presence of infinite variation jumps

We propose new nonparametric estimators of the integrated volatility of an It\^{o} semimartingale observed at discrete times on a fixed time interval with mesh of the observation grid shrinking to zero. The proposed estimators achieve the optimal rate and variance of estimating integrated volatility even in the presence of infinite variation jumps when the latter are stochastic integrals with respect to locally "stable" L\'{e}vy processes, that is, processes whose L\'{e}vy measure around zero behaves like that of a stable process. On a first step, we estimate locally volatility from the empirical characteristic function of the increments of the process over blocks of shrinking length and then we sum these estimates to form initial estimators of the integrated volatility. The estimators contain bias when jumps of infinite variation are present, and on a second step we estimate and remove this bias by using integrated volatility estimators formed from the empirical characteristic function of the high-frequency increments for different values of its argument. The second step debiased estimators achieve efficiency and we derive a feasible central limit theorem for them.

On the Impulse Control of Jump Diffusions

Regularity of the impulse control problem for a nondegenerate $n$-dimensional jump diffusion with infinite activity and finite variation jumps was recently examined in [M. H. A. Davis, X. Guo, and G. Wu, SIAM J. Control Optim., 48 (2010), pp. 5276--5293]. Here we extend the analysis to include infinite activity and infinite variation jumps. More specifically, we show that the value function $u$ of the impulse control problem satisfies $u\in W^{2,p}_{\mathrm{loc}}(\mathbb{R}^{n})$.

Volatility Jumps

The article undertakes a nonparametric analysis of the high-frequency movements in stock market volatility using very finely sampled data on the VIX volatility index compiled from options data by the CBOE. We derive theoretically the link between pathwise properties of the latent spot volatility and the VIX index, such as presence of continuous martingale and/or jumps, and further show how to make statistical inference about them from the observed data. Our empirical results suggest that volatility is a pure jump process with jumps of infinite variation and activity close to that of a continuous martingale. Additional empirical work shows that jumps in volatility and price level in most cases occur together, are strongly dependent, and have opposite sign. The latter suggests that jumps are an important channel for generating leverage effect.

The speed of convergence of the Threshold estimator of integrated variance

In this paper we consider a semimartingale model for the evolution of the price of a financial asset, driven by a Brownian motion (plus drift) and possibly infinite activity jumps. Given discrete observations, the Threshold estimator is able to separate the integrated variance IV from the sum of the squared jumps. This has importance in measuring and forecasting the asset risks. In this paper we provide the exact speed of convergence of I V ˆ h , a result which was known in the literature only in the case of jumps with finite variation. This has practical relevance since many models used have jumps of infinite variation (see e.g. Carr et al. (2002) [4]).