We propose a new and flexible non-parametric framework for estimating the jump tails of Itˆo semimartingale processes. The approach is based on a relatively simple-to-implement set of estimating equations associated with the compensator for the jump measure, or its “intensity”, that only utilizes the weak assumption of regular variation in the jump tails, along with in-fill asymptotic arguments for directly estimating the “large” jumps. The procedure assumes that the “large” sized jumps are identically distributed, but otherwise allows for very general dynamic dependencies in jump occurrences, and importantly does not restrict the behavior of the “small” jumps, nor the continuous part of the process and the temporal variation in the stochastic volatility. On implementing the new estimation procedure with actual high-frequency data for the S&P 500 aggregate market portfolio, we find strong evidence for richer and more complex dynamic dependencies in the jump tails than hitherto entertained in the literature.

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.

We provide a new framework for estimating the systematic and idiosyncratic jump tail risks in financial asset prices. Our estimates are based on in-fill asymptotics for directly identifying the jumps, together with Extreme Value Theory (EVT) approximations and methods-of-moments for assessing the tail decay parameters and tail dependencies. On implementing the procedures with a panel of intraday prices for a large cross-section of individual stocks and the S&P 500 market portfolio, we find that the distributions of the systematic and idiosyncratic jumps are both generally heavy-tailed and close to symmetric, and show how the jump tail dependencies deduced from the high-frequency data together with the day-to-day variation in the diffusive volatility account for the “extreme” joint dependencies observed at the daily level.

We show that the compensation for rare events accounts for a large fraction of the average equity and variance risk premia. Exploiting the special structure of the jump tails and the pricing thereof we identify and estimate a new Investor Fears index. The index reveals large time-varying compensations for fears of disasters. Our empirical investigations involve new extreme value theory approximations and high-frequency intraday data for estimating the expected jump tails under the statistical probability measure, and short maturity out-of-the money options and new model-free implied variation measures for estimating the corresponding risk neutral expectations.

We develop an efficient and analytically tractable method for estimation of parametric volatility models that is robust to price-level jumps. The method entails first integrating intra-day data into the Realized Laplace Transform of volatility, which is model-free estimate of daily integrated empirical Laplace transform of the unobservable volatility. The estimation then is done by matching moments of the integrated joint Laplace transform with those implied by the parametric volatility model. In the empirical application, the best fitting volatility model is a non-diffusive two-factor model where low activity jumps drive its persistent component and more active jumps drive the transient one.

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.

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.

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.

We provide a new theoretical framework for disentangling and estimating sensitivity towards systematic diffusive and jump risks in the context of factor pricing models. Our estimates of the sensitivities towards systematic risks, or betas, are based on the notion of increasingly finer sampled returns over fixed time intervals. In addition to establishing consistency of our estimators, we also derive Central Limit Theorems characterizing their asymptotic distributions. In an empirical application of the new procedures using high-frequency data for forty individual stocks and an aggregate market portfolio, we find the estimated diffusive and jump betas with respect to the market to be quite different for many of the stocks. Our findings have direct and important implications for empirical asset pricing finance and practical portfolio and risk management decisions.

Using high-frequency stock market data and (synthetic) variance swap rates, this paper identifies and investigates the temporal variation in the market variance risk-premium. The variance risk is manifest in two salient features of financial returns: stochastic volatility and jumps. The pricing of these two components is analyzed in a general semiparametric framework. The key empirical results imply that investors' fears of future jumps are especially sensitive to recent jump activity and that their willingness to pay for protection against jumps increases significantly immediately after the occurrence of jumps. This in turn suggests that time-varying risk aversion, as previously documented in the literature, is primarily driven by large, or extreme, market moves. The dynamics of risk-neutral jump intensity extracted from deep out-of-the-money put options confirms these findings.

This paper proposes a method of inference for general stochastic volatility models containing price jumps. The estimation is based on treating realized multipower variation statistics calculated from high-frequency data as their unobservable (fill-in) asymptotic limits. The paper provides easy-to-check conditions under which the error in estimation resulting from this approximation is o_p(1) and additional ones under which it is , where T is the number of days in the sample. Extensive Monte Carlo analysis shows that the proposed estimation method works well in finite samples, provided asymptotic approximations are used. The estimation technique is applied to the estimation of two semiparametric models.

This paper introduces and studies the econometric properties of a general new class of models, which I refer to as jump-driven stochastic volatility models, in which the volatility is a moving average of past jumps. I focus attention on two particular semiparametric classes of jump-driven stochastic volatility models. In the first the price has a continuous component with time-varying volatility and time-homogenous jumps. The second jump-driven stochastic volatility model analyzed here has only jumps in the price, which have time-varying size. In the empirical application I model the memory of the stochastic variance with a CARMA(2,1) kernel and set the jumps in the variance to be proportional to the squared price jumps. The estimation, which is based on matching moments of certain realized power variation statistics calculated from high-frequency foreign exchange data, shows that the jump-driven stochastic volatility model containing continuous component in the price performs best. It outperforms a standard two-factor affine jump-diffusion model, but also the pure-jump jump-driven stochastic volatility model for the particular jump specification.

We introduce and derive the asymptotic behavior of a new measure constructed from high-frequency data which we call the Realized Laplace Transform of volatility. The statistic provides a nonparametric estimate for the empirical Laplace transform function of the latent stochastic volatility process over a given interval of time and is robust to presence of jumps in the price process. With a long span of data, i.e., under joint long-span and infill asymptotics, the statistic can be used to construct a nonparametric estimate of the volatility Laplace transform as well as of the integrated joint Laplace transform of volatility over di®erent points of time. We derive feasible functional limit theorems for our statistic both under fixed span and infill asymptotics as well as under joint long span and infill asymptotics which allow to quantify the precision in estimation under both sampling schemes.