Empirical Volatility Research Articles

From rough to multifractal volatility: The log S-fBM model

We introduce a family of random measures MH,T(dt), namely log S-fBM, such that, for H>0, MH,T(dt)=eωH,T(t)dt where ωH,T(t) is a Gaussian process that can be considered as a stationary version of an H-fractional Brownian motion. Moreover, when H→0, one has MH,T(dt)→M˜T(dt) (in the weak sense) where M˜T(dt) is the celebrated log-normal multifractal random measure (MRM). Thus, this model allows us to consider, within the same framework, the two popular classes of multifractal (H=0) and rough volatility (0<H<1/2) models. The main properties of the log S-fBM are discussed and their estimation issues are addressed. We notably show that the direct estimation of H from the scaling properties of ln(MH,T([t,t+τ])), at fixed τ, can lead to strongly over-estimating the value of H. We propose a better GMM estimation method which is shown to be valid in the high-frequency asymptotic regime. When applied to a large set of empirical volatility data, we observe that stock indices have values around H=0.1 while individual stocks are characterized by values of H that can be very close to 0 and thus well described by a MRM. We also bring evidence that unlike the log-volatility variance ν2 whose estimation appears to be poorly reliable (though used widely in the rough volatility literature), the estimation of the so-called ”intermittency coefficient” λ2, which is the product of ν2 and the Hurst exponent H, appears to be far more reliable leading to values that seem to be universal for respectively all individual stocks and all stock indices.

Volatility measurement with pockets of extreme return persistence

Increasing evidence points towards the episodic emergence of pockets with extreme return persistence. This notion refers to intraday periods of non-trivial duration, for which stock returns are highly positively autocorrelated. Such episodes include, but are not limited to, gradual jumps and prolonged bursts in the drift component. In this paper, we develop a family of integrated volatility estimators, labeled differenced-return volatility (DV) estimators, which provide robustness to these types of Itô semimartingale violations. Specifically, we show that, by using differences in consecutive high-frequency returns, our DV estimators can reduce the non-trivial bias that all commonly-used estimators exhibit during such periods of apparent short-term intraday return predictability. A Monte Carlo study demonstrates the reliability of the newly developed volatility estimators in finite samples. In our empirical volatility forecasting application to S&P 500 index futures and individual equities, our DV-based Heterogeneous Autoregressive (HAR) model performs well relative to existing procedures according to standard out-of-sample MSE and QLIKE criteria.

Model of Continuous Random Cascade Processes in Financial Markets

This article presents a continuous cascade model of volatility formulated as a stochastic differential equation. Two independent Brownian motions are introduced as random sources triggering the volatility cascade: one multiplicatively combines with volatility; the other does so additively. Assuming that the latter acts perturbatively on the system, the model parameters are estimated by the application to an actual stock price time series. Numerical calculation of the Fokker–Planck equation derived from the stochastic differential equation is conducted using the estimated values of parameters. The results reproduce the probability density function of the empirical volatility, the multifractality of the time series, and other empirical facts.

Fractional Riccati Equation and Its Applications to Rough Heston Model Using Numerical Methods

Rough volatility models are recently popularized by the need of a consistent model for the observed empirical volatility in the financial market. In this case, it has been shown that the empirical volatility in the financial market is extremely consistent with the rough volatility. Currently, fractional Riccati equation as a part of computation for the characteristic function of rough Heston model is not known in explicit form and therefore, we must rely on numerical methods to obtain a solution. In this paper, we will be giving a short introduction to option pricing theory (Black–Scholes model, classical Heston model and its characteristic function), an overview of the current advancements on the rough Heston model and numerical methods (fractional Adams–Bashforth–Moulton method and multipoint Padé approximation method) for solving the fractional Riccati equation. In addition, we will investigate on the performance of multipoint Padé approximation method for the small u values in D α h ( u − i / 2 , x ) as it plays a huge role in the computation for the option prices. We further confirm that the solution generated by multipoint Padé (3,3) method for the fractional Riccati equation is incredibly consistent with the solution generated by fractional Adams–Bashforth–Moulton method.

Lending relationships and labor market dynamics

Motivated by a negative link between credit spreads and labor force participation (LFP) and a positive link between these spreads and unemployment over the business cycle, we study the role of LFP as an amplification mechanism of financial shocks in a labor search model with endogenous LFP, lending relationships, and credit-market disruptions. Amid aggregate productivity and financial shocks that replicate the empirical volatility of LFP and credit spreads, the model produces highly volatile unemployment and vacancies, countercyclical credit spreads and unemployment, and procyclical LFP. When we quantitatively match the cyclical behavior of credit spreads, the interaction between endogenous LFP and financial shocks gives rise to much sharper vacancy fluctuations and plays a key role in generating quantitatively factual cyclical labor market dynamics.

Joint tails impact in stochastic volatility portfolio selection models

This paper examines the impact of the joints tails of the portfolio return and its empirical volatility on the optimal portfolio choices. We assume that the portfolio return and its volatility dynamic is approximated by a bivariate Markov chain constructed on its historical distribution. This allows the introduction of a non parametric stochastic volatility portfolio model without the explicit use of a GARCH type or other parametric stochastic volatility models. We describe the bi-dimensional tree structure of the Markov chain and discuss alternative portfolio strategies based on the maximization of the Sharpe ratio and of a modified Sharpe ratio that takes into account the behaviour of a market benchmark. Finally, we empirically evaluate the impact of the portfolio and its stochastic volatility joint tails on optimal portfolio choices. In particular, we examine and compare the out of sample wealth obtained optimizing the portfolio performances conditioned on the joint tails of the proposed stochastic volatility model.

Volatility Measurement with Pockets of Extreme Return Persistence

Increasing evidence points towards the episodic emergence of pockets with extreme return persistence. This notion refers to intraday periods of non-trivial duration, for which stock returns are highly positively autocorrelated. Such episodes include, but are not limited to, gradual jumps and prolonged bursts in the drift component. In this paper, we develop a family of integrated volatility estimators, labeled differenced-return volatility (DV) estimators, which provide robustness to these types of Ito semimartingale violations. Specifically, we show that by using differences in consecutive high-frequency returns, our DV estimators can reduce the non-trivial bias, that all commonly-used estimators exhibit during such periods of apparent short-term intraday return predictability. A Monte Carlo study demonstrates the reliability of the newly developed volatility estimators in finite samples. In our empirical volatility forecast application to S&P 500 index futures and individual equities, our DV-based Heterogeneous Autoregressive (HAR) model performs well relative to existing procedures according to standard out-of-sample MSE and QLIKE criteria.

A Long History of Realized Volatility

Several unique data sets are brought together to build approximate daily realized volatility estimates back to the early 1930's. Estimators are tested extensively on modern data to see how well they line up with common estimators using high frequency pricing information. Estimators are also shown to pass several diagnostic tests from the early samples as well. Finally, well known qualitative features are tested for their stability across subsamples. This includes empirical volatility term structures which give reasonable estimates for perceived persistence to volatility shocks. Recommendations are made as to best practice for estimating long horizon realized volatility.

Labor force participation, interest rate shocks, and unemployment dynamics in emerging economies

Emerging economies (EMEs) have highly volatile unemployment and real wages. Standard models with interest rate shocks can generate higher unemployment volatility, but still fall quantitatively short of matching the empirical volatility of unemployment. A small open economy search model with endogenous participation and self-employment—a feature of EMEs—generates 90 percent of the volatility of unemployment in the data, highly volatile wages, and quantitatively-consistent cyclical aggregate dynamics amid productivity and interest rate shocks of plausible magnitudes. Endogenous participation amplifies shocks and plays a critical role in overcoming a key limitation of standard search models for EMEs.

Smooth Volatility Shifts and Spillovers in U.S. Crude Oil and Corn Futures Markets

Recent developments in biofuel technologies have resulted in heightened linkages between the petroleum and agricultural sectors. As such, a large price and/or volatility shift experienced in one sector is now more likely to spill-over into the other. In trying to capture the interrelations present in the two markets, we take seriously the importance of properly modeling smooth structural shifts. We incorporate trigonometric functions into a multivariate GARCH model of crude and corn futures prices in order to obtain the empirical volatility response functions and the time-varying correlation coefficient. Although both short-term and long-term futures exhibit shifts in the mean and volatility, volatility shifts do not manifest themselves in the same manner for different maturities. This indicates that the term structure of futures volatility changes over time.

What Goes into Risk-Neutral Volatility? EmpiricalEstimates of Risk and Subjective Risk Preferences

Under Black–Scholes (BS) assumptions, empirical volatility and risk-neutral volatility are given by a single parameter that captures all aspects of risk. Inverting the model to extract implied volatility from an option’s market price gives the market’s forecast of future empirical volatility. But real world returns are not lognormal, volatility is stochastic, and arbitrage is limited; thus, option prices embed both the market’s estimate of the empirical returns distribution and also investors’ risk attitudes, including possibly distinct preferences over different volatility-related aspects of the returns process, such as tail risk. All these influences are reflected in the risk-neutral density (RND), which can be extracted from option prices without requiring restrictive assumptions from a pricing model. The author computes daily RNDs for the SP others reflect risk attitudes, such as the level of investor confidence and the size of recent volatility forecast errors.

What Goes into Risk Neutral Volatility? Empirical Estimates of Risk and Subjective Risk Preferences

Under Black-Scholes (BS) assumptions, empirical volatility and risk neutral volatility are given by a single parameter, which captures all aspects of risk. Inverting the model to extract implied volatility from an option's market price gives the market's forecast of future empirical volatility. But real world returns are not lognormal, volatility is stochastic, and arbitrage is limited, so option prices embed both the market's estimate of the empirical returns distribution and also investors' risk attitudes, including possibly distinct preferences over different volatility-related aspects of the returns process, such as tail risk. All of these influences are reflected in the risk neutral density (RND), which can be extracted from option prices without requiring restrictive assumptions from a pricing model.We compute daily RNDs for the SP others reflect risk attitudes, such as the level of investor confidence and the size of recent volatility forecast errors.

Smooth volatility shifts and spillovers in U.S. crude oil and corn futures markets

Abstract Recent developments in biofuel technologies have resulted in heightened linkages between the petroleum and agricultural sectors. As such, a large price and/or volatility shift experienced in one sector is now more likely to spill-over into the other. In trying to capture the interrelations present in the two markets, we take seriously the importance of properly modeling smooth structural shifts. We incorporate trigonometric functions into a multivariate GARCH model of crude and corn futures prices in order to obtain the empirical volatility response functions and the time-varying correlation coefficient. Although both short-term and long-term futures exhibit shifts in the mean and volatility, volatility shifts do not manifest themselves in the same manner for different maturities. This indicates that the term structure of futures volatility changes over time.

International business cycles and risk sharing with uncertainty shocks and recursive preferences

This paper analyzes the effects of output volatility shocks on the dynamics of consumption, trade flows and the real exchange rate, in a two-country, two-good world with consumption home bias, recursive preferences, and complete financial markets. When the risk aversion coefficient exceeds the inverse of the intertemporal substitution elasticity, then an exogenous rise in a country׳s output volatility triggers a wealth transfer to that country, to compensate for the greater riskiness of the country׳s output stream. This risk sharing transfer raises the country׳s consumption, lowers its trade balance and appreciates its real exchange rate. In the recursive preferences framework here, volatility shocks account for a non-negligible share of the fluctuations of net exports, net foreign assets and the real exchange rate. These shocks help to explain the high empirical volatility of the real exchange rate and the disconnect between relative consumption and the real exchange rate.

Hedging Evidence for Unspanned Stochastic Volatility

This paper is concerned with resolving the empirical unspanned stochastic volatility (USV) question from a hedging point of view. We hedge caps that trade in the Johannesburg swap market, as well as caps simulated from two benchmarks models, which each have a USV and non-USV version. We find that the hedging results are not directly informative vis-a-vis the presence of USV, and using our simulated data, we show that the hedging results found in previous papers are compatible with both USV and non-USV models. We develop a two-tier regression test that isolates hedging errors relating to volatility, and, with substantial support from the simulation experiments, find strong evidence that USV is present in our dataset.

Risk Sharing in a World Economy with Uncertainty Shocks

This paper analyzes the effects of output volatility shocks and of risk appetite shocks on the dynamics of consumption, trade flows and the real exchange rate, in a two-country world with recursive preferences and complete financial markets. When the risk aversion coefficient exceeds the inverse of the intertemporal substitution elasticity, then an exogenous rise in a country's output volatility triggers a wealth transfer to that country, in equilibrium; this raises its consumption, lowers its trade balance and appreciates its real exchange rate. The effects of risk appetite shocks resemble those of volatility shocks. In a recursive preferences-complete markets framework, volatility and risk appetite shocks account for a noticeable share of the fluctuations of net exports, net foreign assets and the real exchange rate. These shocks help to explain the high empirical volatility of the real exchange rate and the disconnect between relative consumption growth and the real exchange rate.

Search frictions and labor market dynamics in a real business cycle model with undeclared work

We study the effects of undeclared work (UDW) on labor market dynamics in a real business cycle (RBC) model with search and matching frictions in the labor market. Distinction is made between the wages paid and the hours worked in regular and in undeclared types of activity. Calibrating the model on the US economy, we show that a greater size of UDW implies lower average employment, higher volatility of employment and lower volatility of regular wages. These volatilities are affected by the steady-state ratio between the minimum value of the regular wage that can be accepted by a worker and the maximum value that can be paid by a firm in the Nash bargaining process, which occurs when a labor match is realized. The greater the ratio, the higher the volatility of employment and the lower the volatility of wages. We demonstrate that, due to a social stigma attached to UDW, an increase in undeclared hours raises this ratio by reducing the numerator less than the denominator. This suggests that the introduction of UDW may improve the ability of RBC models to match the empirical volatilities of labor market variables.

Advanced Idiosyncratic Risk and Multi-Factor Models

A common scenario risk analysis employs a multiple factor model with assumed changes in the factors to obtain changes in non-factor variables. This analysis is sometimes designated as a “predictive stress scenario”. We choose to designate the factor model as a multifactor “CAPM” model, also related to APT models. We are not concerned with the details of the factor model. The risks of the scenarios are evaluated for a portfolio of instruments depending on changes of the variables. A systemic problem with this factor model approach is that the empirical correlations between pairs of non-factor variables are not maintained. We designate “advanced idiosyncratic risk” or “AI-Risk” as the correction to the risk of a predictive stress CAPM model scenario that includes a better approximation to the physical correlations. We have developed a new, effective, and economical AI-Risk formalism. There are two parts to AI-Risk. The first part, which is actually standard, has independent normal random variables with no correlations; we designate that as part of the CAPM model to get the variances correctly reproduced; we believe this is standard practice. The second part contains the corrections to the correlations; this is new. We evaluate AI-Risk for some stock portfolios. We find that the AI-Risk can be important, typically between 10%-50% of the predictive stress risk. The amount depends on the ratio of long/short (“L/S”) position values, with larger AI-Risk for higher L/S ratios.A technical remark is that we use correlations as the metric for model improvement. Therefore we work with z-scores or “unit” returns with the empirical volatility divided out.Finally, we indicate the generalization to cross-sectional regression factor models.In summary this work achieves two results: 1. A parsimonious formalism incorporating correlated idiosyncratic risk that gives better agreement with given empirical correlations (so risk is calculated closer to the physical world). 2. Indications of substantial addition to risk due to correlated idiosyncratic contributions, with respect to the smaller risk obtained in the standard model with uncorrelated idiosyncratic risk.

Risk Sharing in a World Economy with Uncertainty Shocks

This paper analyzes the effects of output volatility shocks and of risk appetite shocks on the dynamics of consumption, trade flows and the real exchange rate, in a two-country world with recursive preferences and complete financial markets. When the risk aversion coefficient exceeds the inverse of the intertemporal substitution elasticity, then an exogenous rise in a country’s output volatility triggers a wealth transfer to that country, in equilibrium; this raises its consumption, lowers its trade balance and appreciates its real exchange rate. The effects of risk appetite shocks resemble those of volatility shocks. In a recursive preferences-complete markets framework, volatility and risk appetite shocks account for a noticeable share of the fluctuations of net exports, net foreign assets and the real exchange rate. These shocks help to explain the high empirical volatility of the real exchange rate and the disconnect between relative consumption growth and the real exchange rate.

GARCH with omitted persistent covariate

This paper analyzes the effect of omitting a persistent covariate in the GARCH-X model. In particular, we show that if the relevant persistent covariate is omitted and the usual GARCH(1,1) model is fitted, the model will be estimated approximately as an IGARCH model. This may well explain the ubiquitous evidence of the IGARCH in empirical volatility analysis.