Viktor Todorov

Todorov, Viktor and George Tauchen. 2011. Volatility Jumps. Journal of Business and Economic Statistics. 29(3): 356-371.

The paper undertakes a non-parametric 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.

Todorov, Viktor and George Tauchen. 2011. Limit Theorems for Power Variations of Pure-Jump Processes with Application to Activity Estimation. Annals of Applied Probability. 21(2): 546-588.

We define a new concept termed the activity signature function, which is constructed from discrete observations of a process evolving continuously in time. Under quite general regularity conditions, we derive the asymptotic properties of the function as the sampling frequency increases and show that it is a useful device for making inferences about the activity level of an Ito semimartingale. Monte Carlo work confirms the theoretical results. One empirical application is from finance. It indicates that the classical model comprised of a continuous component plus jumps is more plausible than a pure-jump model for the spot $/DM exchange rate over 1986-1999. A second application pertains to internet traffic data at NASA servers. We find that a pure-jump model with no continuous component and paths of infinite variation is appropriate for modeling this data set. In both cases the evidence obtained from the signature functions is quite convincing, and these two very disparate empirical outcomes illustrate the discriminatory power of the methodology.

Todorov, Viktor and George Tauchen. 2010. Activity Signature Functions for High-Frequency Data Analysis. Journal of Econometrics. 154: 125-138.

We define a new concept termed the activity signature function, which is constructed from discrete observations of a process evolving continuously in time. Under quite general regularity conditions, we derive the asymptotic properties of the function as the sampling frequency increases and show that it is a useful device for making inferences about the activity level of an Ito semimartingale. Monte Carlo work confirms the theoretical results. One empirical application is from finance. It indicates that the classical model comprised of a continuous component plus jumps is more plausible than a pure-jump model for the spot exchange rate over 1986-1999. A second application pertains to internet traffic data at NASA servers. We find that a pure-jump model with no continuous component and paths of infinite variation is appropriate for modeling this data set. In both cases the evidence obtained from the signature functions is quite convincing, and these two very disparate empirical outcomes illustrate the discriminatory power of the methodology. Money exchange rate over 1986-1999. A second application pertains to internet traffic data at NASA servers. We find that a pure-jump model with no continuous component and paths of infinite variation is appropriate for modeling this data set. In both cases the evidence obtained from the signature functions is quite convincing, and these two very disparate empirical outcomes illustrate the discriminatory power of the methodology.

Jacod, Jean and Viktor Todorov. 2009. Testing for Common Arrival of Jumps in Discretely-Observed Multidimensional Processes. Annals of Statistics. 37: 1792-1838.

We consider a bivariate process Xt=(Xt1, Xt2), which is observed on a finite time interval [0,T] at discrete times 0, n, 2n. Assuming that its two components X1 and X2 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 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.