arXiv subjects

Viktor Todorov

Publications and source records attributed to Viktor Todorov.

14 recordsLinked to original sources

arXiv2024

A nonparametric test for rough volatility

We develop a nonparametric test for deciding whether volatility of an asset follows a standard semimartingale process, with paths of finite quadratic variation, or a rough process with paths of infinite quadratic variation. The test utilizes the fact that volatility is rough if and only if volatility increments are negatively autocorrelated at high frequencies. It is based on the sample autocovariance of increments of spot volatility estimates computed from high-frequency asset return data. By showing a feasible CLT for this statistic under the null hypothesis of semimartingale volatility paths, we construct a test with fixed asymptotic size and an asymptotic power equal to one. The test is derived under very general conditions for the data-generating process. In particular, it is robust to jumps with arbitrary activity and to the presence of market microstructure noise. In an application of the test to SPY high-frequency data, we find evidence for rough volatility.

math.ST↗

arXiv2024

Volatility of Volatility and Leverage Effect from Options

We propose model-free (nonparametric) estimators of the volatility of volatility and leverage effect using high-frequency observations of short-dated options. At each point in time, we integrate available options into estimates of the conditional characteristic function of the price increment until the options' expiration and we use these estimates to recover spot volatility. Our volatility of volatility estimator is then formed from the sample variance and first-order autocovariance of the spot volatility increments, with the latter correcting for the bias in the former due to option observation errors. The leverage effect estimator is the sample covariance between price increments and the estimated volatility increments. The rate of convergence of the estimators depends on the diffusive innovations in the latent volatility process as well as on the observation error in the options with strikes in the vicinity of the current spot price. Feasible inference is developed in a way that does not require prior knowledge of the source of estimation error that is asymptotically dominating.

econ.EM↗

arXiv2023

Real-Time Detection of Local No-Arbitrage Violations

This paper focuses on the task of detecting local episodes involving violation of the standard Itô 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 are asymptotically exponentially distributed in the absence of Ito semimartingale violations. On the other hand, when a violation occurs, we can achieve immediate detection under infill asymptotics. A Monte Carlo study demonstrates that the asymptotic results provide a good approximation to the finite-sample behavior of the sequential detectors. An empirical application to S&P 500 index futures data corroborates the effectiveness of our detectors in swiftly identifying the emergence of an extreme return persistence episode in real time.

econ.EM↗

arXiv2023

Asymptotic Expansions for High-Frequency Option Data

We derive a nonparametric higher-order asymptotic expansion for small-time changes of conditional characteristic functions of It\^o semimartingale increments. The asymptotics setup is of joint type: both the length of the time interval of the increment of the underlying process and the time gap between evaluating the conditional characteristic function are shrinking. The spot semimartingale characteristics of the underlying process as well as their spot semimartingale characteristics appear as leading terms in the derived asymptotic expansions. The analysis applies to a general class of It\^o semimartingales that includes in particular L\'evy-driven SDEs and time-changed L\'evy processes. The asymptotic expansion results are subsequently used to construct a test for whether volatility jumps are of finite or infinite variation. In an application to high-frequency data of options written on the S\&P 500 index, we find evidence for infinite variation volatility jumps.

q-fin.ST↗

arXiv2022

Short-time expansion of characteristic functions in a rough volatility setting with applications

We derive a higher-order asymptotic expansion of the conditional characteristic function of the increment of an It\^o semimartingale over a shrinking time interval. The spot characteristics of the It\^o semimartingale are allowed to have dynamics of general form. In particular, their paths can be rough, that is, exhibit local behavior like that of a fractional Brownian motion, while at the same time have jumps with arbitrary degree of activity. The expansion result shows the distinct roles played by the different features of the spot characteristics dynamics. As an application of our result, we construct a nonparametric estimator of the Hurst parameter of the diffusive volatility process from portfolios of short-dated options written on an underlying asset.

q-fin.ST↗

arXiv2015

Jump activity estimation for pure-jump semimartingales via self-normalized statistics

We derive a nonparametric estimator of the jump-activity index $β$ of a "locally-stable" pure-jump Itô semimartingale from discrete observations of the process on a fixed time interval with mesh of the observation grid shrinking to zero. The estimator is based on the empirical characteristic function of the increments of the process scaled by local power variations formed from blocks of increments spanning shrinking time intervals preceding the increments to be scaled. The scaling serves two purposes: (1) it controls for the time variation in the jump compensator around zero, and (2) it ensures self-normalization, that is, that the limit of the characteristic function-based estimator converges to a nondegenerate limit which depends only on $β$. The proposed estimator leads to nontrivial efficiency gains over existing estimators based on power variations. In the Lévy case, the asymptotic variance decreases multiple times for higher values of $β$. The limiting asymptotic variance of the proposed estimator, unlike that of the existing power variation based estimators, is constant. This leads to further efficiency gains in the case when the characteristics of the semimartingale are stochastic. Finally, in the limiting case of $β=2$, which corresponds to jump-diffusion, our estimator of $β$ can achieve a faster rate than existing estimators.

math.ST↗

arXiv2015

Nonparametric test for a constant beta between \Ito semi-martingales based on high-frequency data

We derive a nonparametric test for constant beta over a fixed time interval from high-frequency observations of a bivariate \Ito semimartingale. Beta is defined as the ratio of the spot continuous covariation between an asset and a risk factor and the spot continuous variation of the latter. The test is based on the asymptotic behavior of the covariation between the risk factor and an estimate of the residual component of the asset, that is orthogonal (in martingale sense) to the risk factor, over blocks with asymptotically shrinking time span. Rate optimality of the test over smoothness classes is derived.

math.ST↗

arXiv2014

Limit theorems for the empirical distribution function of scaled increments of Itô semimartingales at high frequencies

We derive limit theorems for the empirical distribution function of "devolatilized" increments of an Itô semimartingale observed at high frequencies. These "devolatilized" increments are formed by suitably rescaling and truncating the raw increments to remove the effects of stochastic volatility and "large" jumps. We derive the limit of the empirical c.d.f. of the adjusted increments for any Itô semimartingale whose dominant component at high frequencies has activity index of $1<β\le2$, where $β=2$ corresponds to diffusion. We further derive an associated CLT in the jump-diffusion case. We use the developed limit theory to construct a feasible and pivotal test for the class of Itô semimartingales with nonvanishing diffusion coefficient against Itô semimartingales with no diffusion component.

math.PR↗

arXiv2014

Efficient estimation of integrated volatility in presence of infinite variation jumps

We propose new nonparametric estimators of the integrated volatility of an Itô 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" Lévy processes, that is, processes whose Lévy 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.

math.ST↗

arXiv2013

Volatility occupation times

We propose nonparametric estimators of the occupation measure and the occupation density of the diffusion coefficient (stochastic volatility) of a discretely observed Itô semimartingale on a fixed interval when the mesh of the observation grid shrinks to zero asymptotically. In a first step we estimate the volatility locally over blocks of shrinking length, and then in a second step we use these estimates to construct a sample analogue of the volatility occupation time and a kernel-based estimator of its density. We prove the consistency of our estimators and further derive bounds for their rates of convergence. We use these results to estimate nonparametrically the quantiles associated with the volatility occupation measure.

math.ST↗

arXiv2012

Realized Laplace transforms for pure-jump semimartingales

We consider specification and inference for the stochastic scale of discretely-observed pure-jump semimartingales with locally stable Lévy densities in the setting where both the time span of the data set increases, and the mesh of the observation grid decreases. The estimation is based on constructing a nonparametric estimate for the empirical Laplace transform of the stochastic scale over a given interval of time by aggregating high-frequency increments of the observed process on that time interval into a statistic we call realized Laplace transform. The realized Laplace transform depends on the activity of the driving pure-jump martingale, and we consider both cases when the latter is known or has to be inferred from the data.

math.ST↗

arXiv2011

Limit theorems for power variations of pure-jump processes with application to activity estimation

This paper derives the asymptotic behavior of realized power variation of pure-jump Itô semimartingales as the sampling frequency within a fixed interval increases to infinity. We prove convergence in probability and an associated central limit theorem for the realized power variation as a function of its power. We apply the limit theorems to propose an efficient adaptive estimator for the activity of discretely-sampled Itô semimartingale over a fixed interval.

math.PR↗

arXiv2010

Do price and volatility jump together?

We consider a process $X_t$, which is observed on a finite time interval $[0,T]$, at discrete times $0,Δ_n,2Δ_n,\ldots.$ This process is an Itô semimartingale with stochastic volatility $σ_t^2$. Assuming that $X$ has jumps on $[0,T]$, we derive tests to decide whether the volatility process has jumps occurring simultaneously with the jumps of $X_t$. There are two different families of tests for the two possible null hypotheses (common jumps or disjoint jumps). They have a prescribed asymptotic level as the mesh $Δ_n$ goes to $0$. We show on some simulations that these tests perform reasonably well even in the finite sample case, and we also put them in use on S&P 500 index data.

q-fin.ST↗

arXiv2009

Testing for common arrivals of jumps for discretely observed multidimensional processes

We consider a bivariate process $X_t=(X^1_t,X^2_t)$, which is observed on a finite time interval $[0,T]$ at discrete times $0,Δ_n,2Δ_n,....$ Assuming that its two components $X^1$ and $X^2$ have jumps on $[0,T]$, we derive tests to decide whether they have at least one jump occurring at the same time ("common jumps") or not ("disjoint jumps"). There are two different tests for the two possible null hypotheses (common jumps or disjoint jumps). Those tests have a prescribed asymptotic level, as the mesh $Δ_n$ goes to 0. We show on some simulations that these tests perform reasonably well even in the finite sample case, and we also put them in use for some exchange rates data.

math.ST↗