The quantilogram: With an application to evaluating directional predictability

We propose a new diagnostic tool for time series called the quantilogram. The tool can be used formally and we provide the inference tools to do this under general conditions, and it can also be used as a simple graphical device. We apply our method to measure directional predictability and to test the hypothesis that a given time series has no directional predictability. The test is based on comparing the correlogram of quantile hits to a pointwise confidence interval or on comparing the cumulated squared autocorrelations with the corresponding critical value. We provide the distribution theory needed to conduct inference, propose some model free upper bound critical values, and apply our methods to S&P500 stock index return data. The empirical results suggest some directional predictability in returns. The evidence is strongest in mid range quantiles like 5–10% and for daily data. The evidence for predictability at the median is of comparable strength to the evidence around the mean, and is strongest at the daily frequency.     

Realized range-based estimation of integrated variance

We provide a set of probabilistic laws for estimating the quadratic variation of continuous semimartingales with the realized range-based variance—a statistic that replaces every squared return of the realized variance with a normalized squared range. If the entire sample path of the process is available, and under a set of weak conditions, our statistic is consistent and has a mixed Gaussian limit, whose precision is five times greater than that of the realized variance. In practice, of course, inference is drawn from discrete data and true ranges are unobserved, leading to downward bias. We solve this problem to get a consistent, mixed normal estimator, irrespective of non-trading effects. This estimator has varying degrees of efficiency over realized variance, depending on how many observations that are used to construct the high–low. The methodology is applied to TAQ data and compared with realized variance. Our findings suggest that the empirical path of quadratic variation is also estimated better with the realized range-based variance.     

Forecasting the volatility of asset returns: The informational gains from option prices

A new class of forecasting models is proposed that extends the realized GARCH class of models through the inclusion of option prices to forecast the variance of asset returns. The VIX is used to approximate option prices, resulting in a set of cross-equation restrictions on the model’s parameters. The full model is characterized by a nonlinear system of three equations containing asset returns, the realized variance, and the VIX, with estimation of the parameters based on maximum likelihood methods. The forecasting properties of the new class of forecasting models, as well as a number of special cases, are investigated and applied to forecasting the daily S&P500 index realized variance using intra-day and daily data from September 2001 to November 2017. The forecasting results provide strong support for including the realized variance and the VIX to improve variance forecasts, with linear conditional variance models performing well for short-term one-day-ahead forecasts, whereas log-linear conditional variance models tend to perform better for intermediate five-day-ahead forecasts.     

An investigation of long memory in various measures of stock market volatility, using wavelets and aggregate series

Using methods based on wavelets and aggregate series, long memory in the absolute daily returns, squared daily returns, and log squared daily returns of the S&P 500 Index are investigated. First, we estimate the long memory parameter in each series using a method based on the discrete wavelet transform. For each series, the variance method and the absolute value method based on aggregate series are then employed to investigate long memory. Our findings suggest that these methods provide evidence of long memory in the volatility of the S&P 500 Index. Our esteemed colleague, Robert DiSario, passed away on December 31, 2005.     

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.     

Realized volatility forecasting and option pricing

A growing literature advocates the use of microstructure noise-contaminated high-frequency data for the purpose of volatility estimation. This paper evaluates and compares the quality of several recently-proposed estimators in the context of a relevant economic metric, i.e., profits from option pricing and trading. Using forecasts obtained by virtue of alternative volatility estimates, agents price short-term options on the S&P 500 index before trading with each other at average prices. The agents’ average profits and the Sharpe ratios of the profits constitute the criteria used to evaluate alternative volatility estimates and the corresponding forecasts. For our data, we find that estimators with superior finite sample Mean-squared-error properties generate higher average profits and higher Sharpe ratios, in general. We confirm that, even from a forecasting standpoint, there is scope for optimizing the finite sample properties of alternative volatility estimators as advocated by Bandi and Russell [Bandi, F.M., Russell, J.R., 2005. Market microstructure noise, integrated variance estimators, and the accuracy of asymptotic approximations. Working Paper; Bandi, F.M., Russell, J.R., 2008b. Microstructure noise, realized variance, and optimal sampling. Review of Economic Studies 75, 339–369] in recent work.     

The VIX,the variance premium and stock market volatility

We decompose the squared VIX index, derived from US S&P500 options prices, into the conditional variance of stock returns and the equity variance premium. We evaluate a plethora of state-of-the-art volatility forecasting models to produce an accurate measure of the conditional variance. We then examine the predictive power of the VIX and its two components for stock market returns, economic activity and financial instability. The variance premium predicts stock returns while the conditional stock market variance predicts economic activity and has a relatively higher predictive power for financial instability than does the variance premium.     

No-arbitrage semi-martingale restrictions for continuous-time volatility models subject to leverage effects,jumps and i.i.d. noise: Theory and testable distributional implications

We develop a sequential procedure to test the adequacy of jump-diffusion models for return distributions. We rely on intraday data and nonparametric volatility measures, along with a new jump detection technique and appropriate conditional moment tests, for assessing the import of jumps and leverage effects. A novel robust-to-jumps approach is utilized to alleviate microstructure frictions for realized volatility estimation. Size and power of the procedure are explored through Monte Carlo methods. Our empirical findings support the jump-diffusive representation for S&P500 futures returns but reveal it is critical to account for leverage effects and jumps to maintain the underlying semi-martingale assumption.     

A reduced form framework for modeling volatility of speculative prices based on realized variation measures

Building on realized variance and bipower variation measures constructed from high-frequency financial prices, we propose a simple reduced form framework for effectively incorporating intraday data into the modeling of daily return volatility. We decompose the total daily return variability into the continuous sample path variance, the variation arising from discontinuous jumps that occur during the trading day, as well as the overnight return variance. Our empirical results, based on long samples of high-frequency equity and bond futures returns, suggest that the dynamic dependencies in the daily continuous sample path variability are well described by an approximate long-memory HAR–GARCH model, while the overnight returns may be modeled by an augmented GARCH type structure. The dynamic dependencies in the non-parametrically identified significant jumps appear to be well described by the combination of an ACH model for the time-varying jump intensities coupled with a relatively simple log-linear structure for the jump sizes. Finally, we discuss how the resulting reduced form model structure for each of the three components may be used in the construction of out-of-sample forecasts for the total return volatility.     

Trading and non-trading period realized market volatility: Does it matter for forecasting the volatility of US stocks?

We study the potential merits of using trading and non-trading period market volatilities to model and forecast the stock volatility over the next one to 22 days. We demonstrate the role of overnight volatility information by estimating heterogeneous autoregressive (HAR) model specifications with and without a trading period market risk factor using ten years of high-frequency data for the 431 constituents of the S&P 500 index. The stocks’ own overnight squared returns perform poorly across stocks and forecast horizons, as well as in the asset allocation exercise. In contrast, we find overwhelming evidence that the market-level volatility, proxied by S&P Mini futures, matters significantly for improving the model fit and volatility forecasting accuracy. The greatest model fit and forecast improvements are found for short-term forecast horizons of up to five trading days, and for the non-trading period market-level volatility. The documented increase in forecast accuracy is found to be associated with the stocks’ sensitivity to the market risk factor. Finally, we show that both the trading and non-trading period market realized volatilities are relevant in an asset allocation context, as they increase the average returns, Sharpe ratios and certainty equivalent returns of a mean–variance investor.     

Modeling dynamic conditional correlations with leverage effects and volatility spillover effects: Evidence from the Chinese and US stock markets affected by the recent trade friction

In this article, we investigate the dynamic conditional correlations (DCCs) with leverage effects and volatility spillover effects that consider time difference and long memory of returns, between the Chinese and US stock markets, in the Sino-US trade friction and previous stable periods. The widespread belief that the developed markets dominate the emerging markets in stock market interactions is challenged by our findings that both the mean and volatility spillovers are bidirectional. We do find that most of the shocks to these DCCs between the two stock markets are symmetric, and all the symmetric shocks to these DCCs are highly persistent between Shanghai’s trading return and S&P 500′s trading or overnight return, however all the shocks to these DCCs are short-lived between S&P 500′s trading return and Shanghai’s trading or overnight return. We also find clear evidence that the DCC between Shanghai’s trading return and S&P 500′s overnight return has a downward trend with a structural break, perhaps due to the “America First” policy, after which it rebounds and fluctuates sharply in the middle and later periods of trade friction. These findings have important implications for investors to pursue profits.     

A Bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation

This paper presents a Bayesian approach to bandwidth selection for multivariate kernel regression. A Monte Carlo study shows that under the average squared error criterion, the Bayesian bandwidth selector is comparable to the cross-validation method and clearly outperforms the bootstrapping and rule-of-thumb bandwidth selectors. The Bayesian bandwidth selector is applied to a multivariate kernel regression model that is often used to estimate the state-price density of Arrow–Debreu securities with the S&P 500 index options data and the DAX index options data. The proposed Bayesian bandwidth selector represents a data-driven solution to the problem of choosing bandwidths for the multivariate kernel regression involved in the nonparametric estimation of the state-price density pioneered by Aït-Sahalia and Lo [Aït-Sahalia, Y., Lo, A.W., 1998. Nonparametric estimation of state-price densities implicit in financial asset prices. The Journal of Finance, 53, 499, 547.]     

Estimating overnight volatility of asset returns by using the generalized dynamic factor model approach

This paper proposes a new approach to estimate the overnight volatility of an individual stock return. Since markets generally do not trade during the overnight period, measures of realized volatility cannot be computed on a “high-frequency” basis. Some studies have resorted to using the square overnight return as a proxy for the overnight realized volatility, but this measure is typically very noisy. The new estimator of the overnight volatility proposed is obtained using the generalized dynamic factor model. The performance of the new proxy is examined using simulated data. This is found to perform better than the squared overnight return. Empirical analysis of the S&P100 constituents confirms the potential of this proxy.     

Day-of-the-week effect on the VIX. A parsimonious representation

The study of significant deterministic seasonal patterns in financial asset returns is of high importance to academia and investors. This paper analyzes the presence of seasonal daily patterns in the VIX and S&P 500 returns series using a trigonometric specification. First, we show that, given the isomorphism between the trigonometrical and alternative seasonality representations (i.e., daily dummies), it is possible to test daily seasonal patterns by employing a trigonometrical representation based on a finite sum of weighted sines and cosines. We find a potential evolutive seasonal pattern in the daily VIX that is not in the daily S&P 500 log-returns series. In particular, we find an inverted Monday effect in the VIX level and changes in the VIX, and a U-shaped seasonal pattern in the changes in the VIX when we control for outliers. The trigonometrical representation is more robust to outliers than the one commonly used by literature, but it is not immune to them. Finally, we do not find a day-of-the-week effect in S&P 500 returns series, which suggests the presence of a deterministic seasonal pattern in the relation between VIX and S&P 500 returns.     

Functional data analysis for volatility

We introduce a functional volatility process for modeling volatility trajectories for high frequency observations in financial markets and describe functional representations and data-based recovery of the process from repeated observations. A study of its asymptotic properties, as the frequency of observed trades increases, is complemented by simulations and an application to the analysis of intra-day volatility patterns of the S&P 500 index. The proposed volatility model is found to be useful to identify recurring patterns of volatility and for successful prediction of future volatility, through the application of functional regression and prediction techniques.     

On the network topology of variance decompositions: Measuring the connectedness of financial firms

We propose several connectedness measures built from pieces of variance decompositions, and we argue that they provide natural and insightful measures of connectedness. We also show that variance decompositions define weighted, directed networks, so that our connectedness measures are intimately related to key measures of connectedness used in the network literature. Building on these insights, we track daily time-varying connectedness of major US financial institutions’ stock return volatilities in recent years, with emphasis on the financial crisis of 2007–2008.     

Dynamic estimation of volatility risk premia and investor risk aversion from option-implied and realized volatilities

This paper proposes a method for constructing a volatility risk premium, or investor risk aversion, index. The method is intuitive and simple to implement, relying on the sample moments of the recently popularized model-free realized and option-implied volatility measures. A small-scale Monte Carlo experiment confirms that the procedure works well in practice. Implementing the procedure with actual S&P500 option-implied volatilities and high-frequency five-minute-based realized volatilities indicates significant temporal dependencies in the estimated stochastic volatility risk premium, which we in turn relate to a set of macro-finance state variables. We also find that the extracted volatility risk premium helps predict future stock market returns.     

Variance trading and market price of variance risk

This paper develops a new approach for variance trading. We show that the discretely-sampled realized variance can be robustly replicated under very general conditions, including when the price can jump. The replication strategy specifies the exact timing for rebalancing in the underlying. The deviations from the optimal schedule can lead to surprisingly large hedging errors. In the empirical application, we synthesize the prices of the variance contract on S&P 500 index over the period from 01/1990 to 12/2009. We find that the market variance risk is priced, its risk premium is negative and economically very large. The variance risk premium cannot be explained by the known risk factors and option returns.     

Information transmission between small and large stocks in the National Stock Exchange in India: An empirical study

The study investigates return and volatility spillover effects between large and small stocks in the national stock exchange in India using daily index data on S&P CNX Nifty, CNX Nifty Junior and CNX Midcap. The VAR model together with the variance decomposition (VDC) and the impulse response function (IRF) analysis have been employed to uncover both casual and dynamic relationship between the large stocks and small stocks. The results show that there are very significant return spillovers from the market portfolio of large stocks to the portfolio of small stocks. To investigate the volatility spillover the study has used standard BEKK model and asymmetric BEKK model. Although, based on the standard BEKK model we have observed unidirectional volatility spillovers from the portfolio of large stocks to the portfolio of small stocks, the finding was less reliable. The more reliable finding, which is based on asymmetric BEKK model, is that there is bidirectional volatility spillover between the portfolio of large stocks and the portfolio of small stocks.     

Forecasting value at risk with intra-day return curves

Methods for incorporating high resolution intra-day asset price data into risk forecasts are being developed at an increasing pace. Existing methods such as those based on realized volatility depend primarily on reducing the observed intra-day price fluctuations to simple scalar summaries. In this study, we propose several methods that incorporate full intra-day price information as functional data objects in order to forecast value at risk (VaR). Our methods are based on the recently proposed functional generalized autoregressive conditionally heteroscedastic (GARCH) models and a new functional linear quantile regression model. In addition to providing daily VaR forecasts, these methods can be used to forecast intra-day VaR curves, which we considered and studied with companion backtests to evaluate the quality of these intra-day risk measures. Using high-frequency trading data from equity and foreign exchange markets, we forecast the one-day-ahead daily and intra-day VaR with the proposed methods and various benchmark models. The empirical results suggested that the functional GARCH models estimated based on the overnight cumulative intra-day return curves exhibited competitive performance with benchmark models for daily risk management, and they produced valid intra-day VaR curves.