Variance Jumps Research Articles

Modeling Price and Variance Jump Clustering Using the Marked Hawkes Process

Abstract We examine the clustering behavior of price and variance jumps using high-frequency data, modeled as a marked Hawkes process (MHP) embedded in a bivariate jump-diffusion model with intraday periodic effects. We find that the jumps of both individual stocks and a broad index exhibit self-exciting behavior. The three dimensions of the model, namely positive price jumps, negative price jumps, and variance jumps, impact one another in an asymmetric fashion. We estimate model parameters using Bayesian inference by Markov Chain Monte Carlo, and find that the inclusion of the jump parameters improves the fit of the model. When we quantify the jump intensity and study the characteristics of jump clusters, we find that in high-frequency settings, jump clustering can last between 2.5 and 6 hours on average. We also find that the MHP generally outperforms other models in terms of reproducing two cluster-related characteristics found in the actual data.

Crash Risk in Currency Returns

We develop an empirical model of bilateral exchange rates. It includes normal shocks with stochastic variance and jumps in an exchange rate and in its variance. The probability of a jump in an exchange rate corresponding to depreciation (appreciation) of the U.S. dollar is increasing in the domestic (foreign) interest rate. The probability of a jump in variance is increasing in the variance only. Jumps in exchange rates are associated with announcements; jumps in variance are not. On average, jumps account for 25% of currency risk. The dollar carry index retains these features. Options suggest that jump risk is priced.

The Variance Process Implied in VIX Options: Affine vs. Non-Affine Models

We use the informational content of VIX derivatives to infer implications on the non-affine modeling of the stock returns' variance dynamics. We find that both a non-affine diffusion and variance jumps are necessary to capture the short- and long-term implied volatility distribution. In- and out-of-sample, a linear CEV jump-diffusion model leads to valuation errors that are negligibly affected by the VIX options' maturity and moneyness. For this setup, we obtain parameter estimates that are comparable to the stock option pricing literature. Compared to a linear specification, we find that estimating the diffusion exponent freely does not lead to significantly better results.

The Volatility-of-Volatility Term Structure

We apply the model-free implied risk-neutral measure of variance to VIX options to investigate the volatility-of-volatility (VVIX) term structure. We find that the information content of the VVIX term structure is fairly distinct from the VIX term structure. Both carry different pieces of information about market uncertainty. The second PCA component (SlopeVVIX) of the VVIX term structure is a significant risk factor. In joint regressions SlopeVVIX predicts straddle returns of S&P500 and VIX options incremental to the VIX term structure and the variance risk premium. We propose an affine approximation method of the VVIX and use parameter estimates of a VIX option pricing model to identify three factors that describe most of the term structure. Continuous volatility-of-volatility adds most to the term structure, followed by variance jumps and a lower bound component. These systematic changes of the risk-factors can be put in economic context by the variance-of-variance to variance ratio.

Time-Varying Uncertainty and Jump Intensities: Why Should Variance Jumps Be Different?

A major assumption of many financial models is that return jump intensities are proportional in the stochastic uncertainty of the underlying. Transferring this intuition to volatility jumps requires that in affine models the variance jump intensity is associated with changes in the stochastic variance-of-variance (q), not with changes in the local variance (V). A model-free analysis shows that the local variance-of-variance describes risk-neutral variance jump expectations well across different maturities, whereas the local variance has almost no explanatory power. The analysis suggests further that q relates to short-term jump expectations on stock return level. The two competing hypotheses of V- and q-associated variance jumps are included separately and jointly in VIX option pricing models. The results show that a single upward variance jump specification with an intensity that is affine in q leads to the best pricing results. This alternative modeling of the jump intensity has two implications for the variance dynamics. First, the local variance is estimated to be much higher in times of crises, whereas its long-run stochastic mean remains at lower levels. Second, the vol-of-vol risk-premium increases to 6-7%, which is else close to zero.

Efficiently pricing continuously monitored barrier options under stochastic volatility model with jumps

ABSTRACTStochastic volatility model with jumps fits almost perfectly the empirical implied volatility surface. Under this model, this paper considers continuously monitored barrier options pricing by Monte Carlo simulation. Based on quadratic exponential scheme, this paper develops an algorithm for pricing barrier options and provides convergence of the algorithm by moment-matching techniques. Variance reduction technique based on control variates further improves the efficiency of the algorithm. The algorithm is also extended to stochastic volatility model with contemporaneous jumps in variance and stock price. Simulations show that the proposed algorithm is efficient and easy to implement. Compared to contemporaneous jumps in variance and stock price, only jumps in stock price produce more profound impact on barrier options prices.

Predictive Power of Option-Implied Densities from High-Frequency Data

Duffie, Pan, and Singleton's (2000) model is used to estimate implied densities using daily and high-frequency FTSE 100 index option contracts from 2005 to 2009. The empirical results suggest the following phenomena during the financial crisis: (1) more negative relationships between variance jumps and price jumps; (2) a larger magnitude of the negative mean of price jumps; (3) a larger variance of price jumps and a larger mean of variance jumps; and (4) a higher jump intensity. Further findings are as follows: (1) high-frequency data provide superior predictive power; and (2) RNDs exhibit satisfactory predictive power for option expiration dates.

A Journey into the Dark Arts of Quantitative Finance

A Journey into the Dark Arts of Quantitative Finance

Hedging Surprises, Jumps, and Model Misspecification: A Risk Management Perspective on Hedging S&P 500 Options*

This article provides comprehensive tests of alternative jump-diffusion models for the purpose of hedging S&P 500 options. We explicitly take into account the risk arising from price and variance jumps and assess the hedging performance by focusing on the ability of competing specifications to forecast hedging errors. To this end, we devise density prediction tests and find evidence that jumps are important features of S&P 500 index dynamics. All jump-diffusion models tested in this article show signs of misspecification, but the inclusion of jumps can improve the hedging performance and risk assessment, especially in two-instrument hedges.

The Fine Structure of Variance: Consistent Pricing of VIX Derivatives

This paper provides a tractable framework for consistently modeling and pricing the two most actively traded options on the Chicago Board Options Exchange (CBOE), namely SPX and VIX options. We derive the dynamics of the CBOE volatility index and give semi-closed form solutions for derivatives on it in a general ane jump-diusion setup. We compare the implications of several special cases of the general model with the major empirically observed properties of VIX derivatives and the time-series behavior of the VIX. We show that commonly used ane jump-diusion models cannot reproduce the basic patterns observed in the data. The ne structure of the variance process is essential to reconcile the empirical regularities with the theoretical models. We nd that both variance jumps and a stochastic volatility of variance seem to be important factors in this respect.

Crash Risk in Currency Returns

We quantify crash risk in currency returns. To accomplish this task, we develop and estimate an empirical model of exchange rate dynamics using daily data for four currencies relative to the US dollar: the Australian dollar, the British pound, the Swiss franc, and the Japanese yen. The model includes (i) normal shocks with stochastic variance, (ii) jumps up and down in the exchange rate, and (iii) jumps in the variance. We identify these components using data on exchange rates and at-the-money implied variances. We find that the probability of an upward (downward) jump in the exchange rate, associated with depreciation (appreciation) of the US dollar, is increasing in the domestic (foreign) interest rate. The probability of jumps in variance is increasing in the variance but not related to interest rates. Many of the jumps in exchange rates are associated with macroeconomic and political news, but jumps in variance are not. On average, jumps account for 25% (and can be as high as 40%) of total currency risk, as measured by the entropy of exchange rate changes, over horizons of one to three months. Preliminary analysis suggests that these properties of currency returns correspond to observed option smiles and that jump risk is priced.

Time-varying leverage effects

Vast empirical evidence points to the existence of a negative correlation, named ”leverage effect”, between shocks to variance and shocks to returns. We provide a nonparametric theory of leverage estimation in the context of a continuous-time stochastic volatility model with jumps in returns, jumps in variance, or both. Leverage is defined as a flexible function of the state of the firm, as summarized by the spot variance level. We show that its point-wise functional estimates have asymptotic properties (in terms of rates of convergence, limiting biases, and limiting variances) which crucially depend on the likelihood of the individual jumps and co-jumps as well as on the features of the jump size distributions. Empirically, we find economically important time-variation in leverage with more negative values associated with higher variance levels.

Time-Varying Leverage Effects

Vast empirical evidence points to the existence of a negative correlation, named “leverage effect”, between shocks in variance and shocks in returns. We provide a nonparametric theory of leverage estimation in the context of a continuous-time stochastic volatility model with jumps in returns, jumps in variance, or both. Leverage is defined as a flexible function of the state of the firm, as summarized by the spot variance level. We show that its point-wise functional estimates have asymptotic properties (in terms of rates of convergence, limiting biases, and limiting variances) which crucially depend on the likelihood of the individual jumps and co-jumps as well as on the features of the jump size distributions. Empirically, we find economically important time-variation in leverage with more negative values associated with higher variance levels.

Variance dynamics: Joint evidence from options and high-frequency returns

This paper analyzes the S&P 500 index return variance dynamics and the variance risk premium by combining information in variance swap rates constructed from options and quadratic variation estimators constructed from tick data on S&P 500 index futures. Estimation shows that the index return variance jumps. The jump arrival rate is not constant over time, but is proportional to the variance rate level. The variance jumps are not rare events but arrive frequently. Estimation also identifies a strongly negative variance risk premium, the absolute magnitude of which is proportional to the variance rate level.

A note on Bayesian identification of change points in data sequences

Recent research in mathematical methods for finance suggests that time series for financial data should be studied with non-stationary models and with structural changes that include both jumps and heteroskedasticity (with jumps in variance). It has been recognized that discriminating between variations caused by the continuous motion of Brownian shocks and the genuine discontinuities in the path of the process constitutes a challenge for existing computational procedures. This issue is addressed here, using the product partition model (PPM), for performing such discrimination and the estimation of process jump parameters. Computational implementation aspects of PPM applied to the identification of change points in data sequences are discussed. In particular, we analyze the use of a Gibbs sampling scheme to compute the estimates and show that there is no significant impact of such use on the quality of the results. The influence of the size of the data sequence on the estimates is also of interest, as well as the efficiency of the PPM to correctly identify atypical observations occurring in close instants of time. Extensive Monte Carlo simulations attest to the effectiveness of the Gibbs sampling implementation. An illustrative financial time series example is also presented.

Variance Dynamics: Joint Evidence from Options and High-Frequency Returns

With a portfolio of options on S&P 500 index, the Chicago Board of Options Exchange constructs a volatility index named VIX that approximates the 30-day return variance swap rate on the index. Using high-frequency return data, researchers have proposed various return quadratic variation estimators. This paper estimates the index return variance dynamics and variance risk premium jointly from the VIX index and various quadratic variation estimators constructed from tick data on S&P 500 index futures. Estimation shows that the index return variance jumps. The jump arrival rate is not constant over time, but proportional to the variance rate level. Furthermore, jumps in the index return variance are not rare events, but arrive frequently and generate sample paths that show infinite variation. Estimation also identifies a strongly negative variance risk premium, the absolute magnitude of which is proportional to the variance rate level. Finally, the estimation highlights the importance and necessity for removing microstructure noise in estimating quadratic variations.

Volatility in Emerging Stock Markets Revisited

This study examines the issue of detecting permanent shifts in the volatility of emerging stock market indexes returns. We show that standard tests have no power in disentangling conditional heteroscedasticity versus jumps in the variance of stock returns. We propose two methods to detect jumps in the variance when there is conditional heteroscedasticity. The first one is based on the properties of temporal aggregation of GARCH (1, 1) models. The second one consists in filtering the stock returns series with a GARCH (1, 1) model in conjunction with the ICSS algorithm. We show that jumps in variance are less frequent than previously believed. Moreover, the jumps are country specific so that they can be diversified.