Spot Variance Research Articles

The rough Hawkes Heston stochastic volatility model

AbstractWe study an extension of the Heston stochastic volatility model that incorporates rough volatility and jump clustering phenomena. In our model, named the rough Hawkes Heston stochastic volatility model, the spot variance is a rough Hawkes‐type process proportional to the intensity process of the jump component appearing in the dynamics of the spot variance itself and the log returns. The model belongs to the class of affine Volterra models. In particular, the Fourier‐Laplace transform of the log returns and the square of the volatility index can be computed explicitly in terms of solutions of deterministic Riccati‐Volterra equations, which can be efficiently approximated using a multi‐factor approximation technique. We calibrate a parsimonious specification of our model characterized by a power kernel and an exponential law for the jumps. We show that our parsimonious setup is able to simultaneously capture, with a high precision, the behavior of the implied volatility smile for both S&P 500 and VIX options. In particular, we observe that in our setting the usual shift in the implied volatility of VIX options is explained by a very low value of the power in the kernel. Our findings demonstrate the relevance, under an affine framework, of rough volatility and self‐exciting jumps in order to capture the joint evolution of the S&P 500 and VIX.

Variance swaps with mean reversion and multi-factor variance

This paper proposes a novel model for the valuation of variance swaps that incorporates a multi-factor stochastic spot variance and a multi-factor stochastic long-term variance, while allowing mean reversion in the asset price and a co-jump structure in the model. We propose a general analytical approach for pricing discretely monitored variance swaps via a moment-based method and confirm its accuracy and efficiency using Monte Carlo simulations. Our empirical results indicate that the model with a three-factor spot variance and a one-factor long-term variance significantly outperforms other nested models. The incorporation of multiple factors in our model is essential not only for fitting market data, but also for reconciling the term structure of variance swaps.

From zero-intelligence to queue-reactive: limit-order-book modeling for high-frequency volatility estimation and optimal execution

Estimators of integrated and spot variance are tested using the queue-reactive model of the limit order book

Jump-diffusion volatility models for variance swaps: An empirical performance analysis

This paper studies a class of tractable jump-diffusion models, including stochastic volatility models with various specifications of jump intensity for stock returns and variance processes. We employ the Markov chain Monte Carlo (MCMC) method to implement model estimation, and investigate the performance of all models in capturing the term structure of variance swap rates and fitting the dynamics of stock returns. It is evident that the stochastic volatility models, equipped with self-exciting jumps in the spot variance and linearly-dependent jumps in the central-tendency variance, can produce consistent model estimates, aptly explain the stylized facts in variance swaps, and boost pricing performance. Moreover, our empirical results show that large self-exciting jumps in the spot variance, as an independent risk source, facilitate term structure modeling for variance swaps, whilst the central-tendency variance may jump with small sizes, but signaling substantial regime changes in the long run. Both types of jumps occur infrequently, and are more related to market turmoils over the period from 2008 to 2021.

Bootstrapping Laplace transforms of volatility

This paper studies inference for the realized Laplace transform (RLT) of volatility in a fixed‐span setting using bootstrap methods. Specifically, since standard wild bootstrap procedures deliver inconsistent inference, we propose a local Gaussian (LG) bootstrap, establish its first‐order asymptotic validity, and use Edgeworth expansions to show that the LG bootstrap inference achieves second‐order asymptotic refinements. Moreover, we provide new Laplace transform‐based estimators of the spot variance as well as the covariance, correlation, and beta between two semimartingales, and adapt our bootstrap procedure to the requisite scenario. We establish central limit theory for our estimators and first‐order asymptotic validity of their associated bootstrap methods. Simulations demonstrate that the LG bootstrap outperforms existing feasible inference theory and wild bootstrap procedures in finite samples. Finally, we illustrate the use of the new methods by examining the coherence between stocks and bonds during the global financial crisis of 2008 as well as the COVID‐19 pandemic stock sell‐off during 2020, and by a forecasting exercise.

High-dimensional estimation of quadratic variation based on penalized realized variance

In this paper, we develop a penalized realized variance (PRV) estimator of the quadratic variation (QV) of a high-dimensional continuous Itô semimartingale. We adapt the principle idea of regularization from linear regression to covariance estimation in a continuous-time high-frequency setting. We show that under a nuclear norm penalization, the PRV is computed by soft-thresholding the eigenvalues of realized variance (RV). It therefore encourages sparsity of singular values or, equivalently, low rank of the solution. We prove our estimator is minimax optimal up to a logarithmic factor. We derive a concentration inequality, which reveals that the rank of PRV is—with a high probability—the number of non-negligible eigenvalues of the QV. Moreover, we also provide the associated non-asymptotic analysis for the spot variance. We suggest an intuitive data-driven subsampling procedure to select the shrinkage parameter. Our theory is supplemented by a simulation study and an empirical application. The PRV detects about three–five factors in the equity market, with a notable rank decrease during times of distress in financial markets. This is consistent with most standard asset pricing models, where a limited amount of systematic factors driving the cross-section of stock returns are perturbed by idiosyncratic errors, rendering the QV—and also RV—of full rank.

Bias-optimal vol-of-vol estimation: the role of window overlapping

The simplest and most natural vol-of-vol estimator, the pre-estimated spot variance-based realized variance, is typically plagued by a large finite-sample bias. In this paper, we analytically show that allowing for the overlap of consecutive local windows to pre-estimate the spot variance may correct for this bias. In particular, we provide a feasible rule for the bias-optimal selection of the length of local windows when the volatility is a CKLS process. The effectiveness of this rule for practical applications is supported by numerical and empirical analyses.

CUSUM-Based Monitoring for Explosive Episodes in Financial Data in the Presence of Time-Varying Volatility

Abstract We generalize the Homm and Breitung (2012) CUSUM-based procedure for the real-time detection of explosive autoregressive episodes in financial price data to allow for time-varying volatility. Such behavior can heavily inflate the false positive rate (FPR) of the CUSUM-based procedure to spuriously signal the presence of an explosive episode. Our modified procedure involves replacing the standard variance estimate in the CUSUM statistics with a nonparametric kernel-based spot variance estimate. We show that the sequence of modified CUSUM statistics has a joint limiting null distribution which is invariant to any time-varying volatility present in the innovations and that this delivers a real-time monitoring procedure whose theoretical FPR is controlled. Simulations show that the modification is effective in controlling the empirical FPR of the procedure, yet sacrifices only a small amount of power to detect explosive episodes, relative to the standard procedure, when the shocks are homoskedastic. An empirical illustration using Bitcoin price data is provided.

High-Dimensional Estimation of Quadratic Variation Based on Penalized Realized Variance

In this paper, we develop a penalized realized variance (PRV) estimator of the quadratic variation (QV) of a high-dimensional continuous Ito semimartingale. We adapt the principle idea of regularization from linear regression to covariance estimation in a continuous-time high-frequency setting. We show that under a nuclear norm penalization, the PRV is computed by soft-thresholding the eigenvalues of realized variance (RV). It therefore encourages sparsity of singular values or, equivalently, low rank of the solution. We prove our estimator is minimax optimal up to a logarithmic factor. We derive a concentration inequality, which reveals that the rank of PRV is-with a high probability-the number of non-negligible eigenvalues of the QV. Moreover, we also provide the associated non-asymptotic analysis for the spot variance. We suggest an intuitive data-driven bootstrap procedure to select the shrinkage parameter. Our theory is supplemented by a simulation study and an empirical application. The PRV detects about three-five factors in the equity market, with a notable rank decrease during times of distress in financial markets. This is consistent with most standard asset pricing models, where a limited amount of systematic factors driving the cross-section of stock returns are perturbed by idiosyncratic errors, rendering the QV-and also RV-of full rank.

Time-Varying Crash Risk Embedded in Index Options: The Role of Stock Market Liquidity*

Abstract We estimate a continuous-time model for the stock market index where the stochastic volatility and crash probability depend on the realized spot variance and the stock market illiquidity. We find that market illiquidity is a useful economic covariate in the modeling of time-varying stock market crash risk embedded in index options. The relative contribution of spot variance in the time-varying crash risk is weakened once the market illiquidity variable is added to the model, and out-of-sample option pricing error also improves. Examining the relationship between market illiquidity and option-implied crash risk, we find that the availability of arbitrage capital and adverse selection facing liquidity providers are potential economic links. Our study highlights the benefits of adding a market illiquidity measure to index return models with time-varying crash risk.

Supplementary Material of 'On the Estimation of Jump-Diffusion Models Using High-Frequency Data: A Filtering-Based Approach'

We adopt a flexible filtering procedure to extract information from high-frequency data. Specifically, we provide a parsimonious framework to integrate realized measures from high frequency index and derivative prices. In a simulation study, we document the incremental information offered by realized measures and show that even though high-frequency index prices help identify spot variance and jump price dynamics, it is the addition of high-frequency option prices that enables variance jumps to be identified. A series of empirical studies based on the S&P 500 index and options show that estimation precision improves with the addition of information from intraday option prices.

Is the Variance Swap Rate Affine in the Spot Variance? Evidence from S&P500 Data

ABSTRACT We empirically investigate the functional link between the variance swap rate and the spot variance. Using S&P500 data over the period 2006–2018, we find overwhelming empirical evidence supporting the affine link implied by exponential affine stochastic volatility models. Tests on yearly subsamples suggest that exponential mean-reverting variance models provide a good fit during periods of extreme volatility, while polynomial modelsare suited for years characterized by more frequent price jumps.

Does OVX affect WTI and Brent oil spot variance? Evidence from an entropy analysis

Abstract This paper examines the flow of information and its direction between the oil volatility index (OVX) and the spot variance of WTI and Brent returns. Since OVX is an indicator of the investor sentiment about oil market performance, we aim first at evidencing whether there is an exchange of information between OVX and the spot variance of the two crudes. Moreover, since OVX is linked to WTI crude oil, it is useful to investigate whether it shares an information content with Brent. To this purpose, we propose an entropy-based approach which exploits two non-parametric methods: the mutual information and the transfer entropy. The results show an increase in the information flow between OVX and the spot variance of Brent returns and a corresponding decrease in the information flow with WTI. Furthermore, the direction of the information flow comes from OVX to both oil spot variances, thus investor sentiment about oil market performance drives uncertainty in the corresponding spot market. However, the information flow from the oil spot variances to OVX is more statistically significant for WTI than for Brent and, since transfer entropy is a measure of resolution of uncertainty, we demonstrate that the spot variance of WTI returns helps more in reducing uncertainty about OVX (than Brent).

Volatility regressions with fat tails

Nowadays, a common method to forecast integrated variance is to use the fitted value of a simple OLS autoregression of the realized variance. However, non-parametric estimates of the tail index of this realized variance process reveal that its second moment is possibly unbounded. In this case, the behavior of the OLS estimators and the corresponding statistics are unclear. We prove that when the second moment of the spot variance is unbounded, the slope of the spot variance’s autoregression converges to a random variable as the sample size diverges. The same result holds when one uses the integrated or realized variance instead of the spot variance. We then consider the class of diffusion variance models with an affine drift, a class which includes GARCH and CEV processes, and we prove that IV estimation with adequate instruments provide consistent estimators of the drift parameters as long as the variance process has a finite first moment regardless of the existence of the second moment. In particular, for the GARCH diffusion model with fat tails, an IV estimation where the instrument equals the sign of the centered lagged value of the variable of interest provides consistent estimators. Simulation results corroborate the theoretical findings of the paper.

Bootstrapping Laplace Transforms of Volatility

This paper studies inference for the realized Laplace transform (RLT) of volatility in a fixed-span setting using bootstrap methods. Specifically, since standard wild bootstraps provide inconsistent inference, we propose local Gaussian (LG) and modified wild (MW) bootstrap procedures, and establish their first-order asymptotic validity. Moreover, motivated by its superior finite sample performance in simulations, we use Edgeworth expansions to show that the LG inference achieves second-order asymptotic refinements. To further broaden the scope of the bootstraps, we provide new Laplace transform-based estimators of the spot variance as well as the covariance, correlation and beta between two semimartingales, and adapt our inference procedures to the requisite scenario. We establish central limit theory for our estimators and show first-order asymptotic validity of their associated bootstraps. Not surprisingly, a Monte Carlo study shows that the LG bootstrap outperforms the MW bootstrap and existing (first-order) feasible inference theory in finite samples. Moreover, it demonstrates that our new spot measure estimators and inference procedures are very accurate. Finally, we illustrate the use of the new methods by examining the volatility of, and the coherence between, stocks and bonds during the large equity sell-off in December 2018.

Is the Variance Swap Rate Affine in the Spot Variance? Evidence From S&P500 Data

We empirically investigate the functional link between the variance swap rate and the spot variance. Using S&P500 data over the period 2006-2018, we find overwhelming empirical evidence supporting the affine link analytically found by Kallsen et al. (2011) in the context of exponentially affine stochastic volatility models. Tests on yearly subsamples suggest that exponentially mean-reverting variance models provide a good fit during periods of extreme volatility, while polynomial models, introduced in Cuchiero (2011), are suited for years characterized by more frequent price jumps.

Roughness in Spot Variance? A GMM Approach for Estimation of Fractional Log-Normal Stochastic Volatility Models Using Realized Measures

In this paper, we develop a generalized method of moments approach for joint estimation of the parameters of a fractional log-normal stochastic volatility model. We show that with an arbitrary Hurst exponent an estimator based on integrated variance is consistent. Moreover, under stronger conditions we also derive a central limit theorem. These results stand even when integrated variance is replaced with a realized measure of volatility calculated from discrete high-frequency data. However, in practice a realized estimator contains sampling error, the effect of which is to skew the fractal coeffcient toward roughness. We construct an analytical approach to control this error. In a simulation study, we demonstrate convincing small sample properties of our approach based both on integrated and realized variance over the entire memory spectrum. We show that the bias correction attenuates any systematic deviance in the estimated parameters. Our procedure is applied to empirical high-frequency data from numerous leading equity indexes. With our robust approach the Hurst index is estimated around 0.05, confirming roughness in integrated variance.

The realized empirical distribution function of stochastic variance with application to goodness-of-fit testing

We propose a nonparametric estimator of the empirical distribution function (EDF) of the latent spot variance of the log-price of a financial asset. We show that over a fixed time span our realized EDF (or REDF) – inferred from noisy high-frequency data – is consistent as the mesh of the observation grid goes to zero. In a double-asymptotic framework, with time also increasing to infinity, the REDF converges to the cumulative distribution function of volatility, if it exists. We exploit these results to construct some new goodness-of-fit tests for stochastic volatility models. In a Monte Carlo study, the REDF is found to be accurate over the entire support of volatility. This leads to goodness-of-fit tests that are both correctly sized and relatively powerful against common alternatives. In an empirical application, we recover the REDF from stock market high-frequency data. We inspect the goodness-of-fit of several two-parameter marginal distributions that are inherent in standard stochastic volatility models. The inverse Gaussian offers the best overall description of random equity variation, but the fit is less than perfect. This suggests an extra parameter (as available in, e.g., the generalized inverse Gaussian) is required to model stochastic variance.

Thin-layer chromatogram development with a moving pipette delivering the mobile phase onto the surface of the adsorbent layer

The new approach to the development of thin-layer chromatograms is presented. For the first time we show flexible mobile phase dosage onto the surface of the adsorbent layer by moving pipette combined with precise syringe pumps. The pipette is driven into movement by computer controlled 3D machine (modified 3D printer mechanism). Delivery of the mobile phase to the adsorbent layer is equal to or lower than that of conventional development. Therefore chromatograms can be developed with optimal mobile phase velocity, adjusted to its absorption rate by the adsorbent layer. Under such conditions there is no excess of eluent on the surface of the adsorbent layer so higher performance of the chromatographic system can be obtained. Moreover chromatograms can be developed with constant linear mobile phase velocity and therefore the relationships the plate height vs. mobile phase linear velocity obtained with planar chromatography driven by capillary forces are investigated and reported. In addition the contribution of starting spot variance in total peak variance and the influence of narrowing of starting spots on performance of the chromatographic system have been studied. The results confirm a very significant starting spot variance contribution to total peak variance and consequently considerable influence of starting spot size on plate height of the separation system, when chromatogram is developed on a short distance.In the paper the advantages and disadvantages of the prototype device and its possible application are discussed.

The Realized Empirical Distribution Function of Stochastic Variance with Application to Goodness-of-Fit Testing

We propose a nonparametric estimator of the empirical distribution function (EDF) of the latent spot variance of the log-price of a financial asset. We show that over a fixed time span our realized EDF (or REDF)-inferred from noisy high-frequency data-is consistent as the mesh of the observation grid goes to zero. In a double-asymptotic framework, with time also increasing to infinity, the REDF converges to the cumulative distribution function of volatility, if it exists. We exploit these results to construct some new goodness-of-fit tests for stochastic volatility models. In a Monte Carlo study, the REDF is found to be accurate over the entire support of volatility. This leads to goodness-of-fit tests that are both correctly sized and relatively powerful against common alternatives. In an empirical application, we recover the REDF from stock market high-frequency data. We inspect the goodness-of-fit of several two-parameter marginal distributions that are inherent in standard stochastic volatility models. The inverse Gaussian offers the best overall description of random equity variation, but the fit is less than perfect. This suggests an extra parameter (as available in, e.g., the generalized inverse Gaussian) is required to model stochastic variance.