Parametric Volatility Models Research Articles

Overnight GARCH-Itô Volatility Models

Various parametric volatility models for financial data have been developed to incorporate high-frequency realized volatilities and better capture market dynamics. However, because high-frequency trading data are not available during the close-to-open period, the volatility models often ignore volatility information over the close-to-open period and thus may suffer from loss of important information relevant to market dynamics. In this article, to account for whole-day market dynamics, we propose an overnight volatility model based on Itô diffusions to accommodate two different instantaneous volatility processes for the open-to-close and close-to-open periods. We develop a weighted least squares method to estimate model parameters for two different periods and investigate its asymptotic properties.

Overnight Garch-It\^O Volatility Models

Various parametric volatility models for financial data have been developed to incorporate high-frequency realized volatilities and better capture market dynamics. However, because high-frequency trading data are not available during the close-to-open period, the volatility models often ignore volatility information over the close-to-open period and thus may suffer from loss of important information relevant to market dynamics. In this paper, to account for whole-day market dynamics, we propose an overnight volatility model based on It\^o diffusions to accommodate two different instantaneous volatility processes for the open-to-close and close-to-open periods. We develop a weighted least squares method to estimate model parameters for two different periods and investigate its asymptotic properties. We conduct a simulation study to check the finite sample performance of the proposed model and method. Finally, we apply the proposed approaches to real trading data.

Predicting Future Implied Volatility Surface Using TDBP-Learning

Parametric volatility models can be seen as the result of some form of dimensionality reduction obtained by projecting the volatility surface into a basis of risk factors. Examples include polynomial models and stochastic volatility models having an explicit expression for the smile, such as the SVI and the SABR model. When modelling the dynamics of the risk factors, one can define some stochastic processes, consider a statistical model, or assume that they are function of some exogenous explanatory variables (such as the spot price and volume). In the first two, we are tied to our ability of capturing the true dynamics of the underlying stochastic processes, leading to computation errors, in the latter we assume known the explanatory variables. Alternatively, we can use machine learning to implicitly infer the dynamics of the risk factors, or that of their explanatory variables, from market data. We let the risk factors be function of some stochastic explanatory variables: the spot price, the set of time-to-maturity, the trading volume, some external noise factors, the estimated volatility, and financial indicators like the VIX. We use a statistical model to capture the relation between the risk factors and their explanatory variables, and estimate empirically the model parameters. The risk factors being stochastic, we estimate the future implied volatility surface on average, by solving its conditional expectation with respect to the explanatory variables. This is a multi-step prediction problem, and we propose to use temporal difference backpropagation (TDBP) models for learning to predict the value function. This way, we use market data to estimate the implicit stochastic processes driving the dynamics of the future volatility surface. We test the performance of our model on three well known parametric volatility models against Monte Carlo simulations, and we show the effects of the explanatory variables on the dynamics of the volatility surface. In all cases, we recover nearly exactly the expected volatility surface generated by the Monte Carlo simulations with only one five hundredth paths. Our framework can be applied to option pricing and hedging as well as to risk management.

Capturing the volatility smile: parametric volatility models versus stochastic volatility models

Black-Scholes option pricing model (1973) assumes that all option prices on the same underlying asset with the same expiration date, but different exercise prices should have the same implied volatility. However, instead of a flat implied volatility structure, implied volatility (inverting the Black-Scholes formula) shows a smile shape across strikes and time to maturity. This paper compares parametric volatility models with stochastic volatility models in capturing this volatility smile. Results show empirical evidence in favor of parametric volatility models. Keywords: smile volatility, parametric, stochastic, Black-Scholes. JEL Classification: C14 C68 G12 G13

Inference on co-integration parameters in heteroskedastic vector autoregressions

We consider estimation and hypothesis testing on the coefficients of the co-integrating relations and the adjustment coefficients in vector autoregressions driven by shocks which display both conditional and unconditional heteroskedasticity of a quite general and unknown form. We show that the conventional results in Johansen (1996) for the maximum likelihood estimators and associated likelihood ratio tests derived under homoskedasticity do not in general hold under heteroskedasticity. As a result, standard confidence intervals and hypothesis tests on these coefficients are potentially unreliable. Solutions based on Wald tests (using a “sandwich” estimator of the variance matrix) and on the use of the wild bootstrap are discussed. These do not require the practitioner to specify a parametric model for volatility. We establish the conditions under which these methods are asymptotically valid. A Monte Carlo simulation study demonstrates that significant improvements in finite sample size can be obtained by the bootstrap over the corresponding asymptotic tests in both heteroskedastic and homoskedastic environments. An application to the term structure of interest rates in the US illustrates the difference between standard and bootstrap inferences regarding hypotheses on the co-integrating vectors and adjustment coefficients.

Forecasting VaR models under Different Volatility Processes and Distributions of Return Innovations

ABSTRACTThis paper provides clear‐cut evidence that the out‐of‐sample VaR (value‐at‐risk) forecasting performance of alternative parametric volatility models, like EGARCH (exponential general autoregressive conditional heteroskedasticity) or GARCH, and Markov regime‐switching models, can be considerably improved if they are combined with skewed distributions of asset return innovations. The performance of these models is found to be similar to that of the EVT (extreme value theory) approach. The performance of the latter approach can also be improved if asset return innovations are assumed to be skewed distributed. The performance of the Markov regime‐switching model is considerably improved if this model allows for EGARCH effects, for all different volatility regimes considered. Copyright © 2014 John Wiley & Sons, Ltd.

Testing for Unit Roots Under Multiple Possible Trend Breaks and Non‐Stationary Volatility Using Bootstrap Minimum Dickey–Fuller Statistics

In a recent paper, Harvey et al. (2013) (HLT) propose a new unit root test that allows for the possibility of multiple breaks in trend. Their proposed test is based on the infimum of the sequence (across all candidate break points) of local GLS detrended augmented Dickey–Fuller‐type statistics. HLT show that the power of their unit root test is robust to the magnitude of any trend breaks. In contrast, HLT show that the power of the only alternative available procedure of Carrion‐i‐Silvestre et al. (2009), which employs a pretest‐based approach, can be very low indeed (even zero) for the magnitudes of trend breaks typically observed in practice. Both HLT and Carrion‐i‐Silvestre et al. (2009) base their approaches on the assumption of homoskedastic shocks. In this article, we analyse the impact of non‐stationary volatility (for example, single and multiple abrupt variance breaks, smooth transition variance breaks and trending variances) on the tests proposed in HLT. We show that the limiting null distribution of the HLT unit root test statistic is not pivotal under non‐stationary volatility. A solution to the problem, which does not require the practitioner to specify a parametric model for volatility, is provided using the wild bootstrap and is shown to perform well in practice. A number of different possible implementations of the bootstrap algorithm are discussed.

Does Anyone Need a GARCH(1,1)?

Hansen and Lunde [16] posed the question Does anything beat a GARCH(1,1)? and compared a large number of parametric volatility models in an extensive empirical study. They found that no other model provides significantly better forecasts than the GARCH(1,1) model. In contrast, this paper arrives at the conclusion that simple robust estimators such as weighted medians of past (squared) returns outperform the GARCH(1,1) model both in-sample as well as out-of-sample. This conclusion is based on theoretical arguments as well as on empirical evidence.

Inference on Co-Integration Parameters in Heteroskedastic Vector Autoregressions

It is well established that the shocks driving many key macroeconomic and financial variables display time-varying volatility. In this paper we consider estimation and hypothesis testing on the coefficients of the co-integrating relations and the adjustment coefficients in vector autoregressions driven by both conditional and unconditional heteroskedasticity of a quite general and unknown form in the shocks. We show that the conventional results in Johansen (1996) for the maximum likelihood estimators and associated likelihood ratio tests derived under homoskedasticity do not in general hold in the presence of heteroskedasticity. As a consequence, standard confidence intervals and tests of hypothesis on these coefficients are potentially unreliable. Solutions to this inference problem based on Wald tests (using a sandwich estimator of the variance matrix) and on the use of the wild bootstrap are discussed. These do not require the practitioner to specify a parametric model for volatility, or to assume that the pattern of volatility is common to, or independent across, the vector of series under analysis. We formally establish the conditions under which these methods are asymptotically valid. A Monte Carlo simulation study demonstrates that significant improvements in finite sample size can be obtained by the bootstrap over the corresponding asymptotic tests in both heteroskedastic and homoskedastic environments. An application to the term structure of interest rates in the US illustrates the difference between standard and bootstrap inferences regarding hypotheses on the co-integrating vectors and adjustment coefficients.

Inference on Co-Integration Parameters in Heteroskedastic Vector Autoregressions

It is well established that the shocks driving many key macro-economic and financial variables display time-varying volatility. In this paper we consider estimation and hypothesis testing on the coefficients of the co-integrating relations and the adjustment coefficients in vector autoregressions driven by both conditional and unconditional heteroskedasticity of a quite general and unknown form in the shocks. We show that the conventional results in Johansen (1996) for the maximum likelihood estimators and associated likelihood ratio tests derived under homoskedasticity do not in general hold in the presence of heteroskedasticity. As a consequence, standard confidence intervals and tests of hypothesis on these coefficients are potentially unreliable. Solutions to this inference problem based on Wald tests (using a sandwich estimator of the variance matrix) and on the use of the wild bootstrap are discussed. These do not require the practitioner to specify a parametric model for volatility, or to assume that the pattern of volatility is common to, or independent across, the vector of series under analysis. We formally establish the conditions under which these methods are asymptotically valid. A Monte Carlo simulation study demonstrates that significant improvements in finite sample size can be obtained by the bootstrap over the corresponding asymptotic tests in both heteroskedastic and homoskedastic environments. An application to the term structure of interest rates in the US illustrates the difference between standard and bootstrap inferences regarding hypotheses on the co-integrating vectors and adjustment coefficients.

Realized Laplace transforms for estimation of jump diffusive volatility models

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 a model-free estimate of the daily integrated empirical Laplace transform of the unobservable volatility. The estimation is then 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.

TESTING FOR UNIT ROOTS IN THE PRESENCE OF A POSSIBLE BREAK IN TREND AND NONSTATIONARY VOLATILITY

We analyze the impact of nonstationary volatility on the break fraction estimator and associated trend break unit root tests of Harris, Harvey, Leybourne, and Taylor (2009) (HHLT). We show that although HHLT’s break fraction estimator retains the same large-sample properties as demonstrated by HHLT for homoskedastic shocks, the limiting null distributions of unit root statistics based around this estimator are not pivotal under nonstationary volatility. A solution to the identified inference problem, which does not require the practitioner to specify a parametric model for volatility, is provided using the wild bootstrap and is shown to perform well in practice.

Realized Laplace Transforms for Estimation of Jump Diffusive Volatility Models

We develop a new efficient and analytically tractable method for estimation of parametric volatility models that is robust to price-level jumps and generally has good finite sample properties. The method entails first integrating intra-day data into the Realized Laplace Transform of volatility, which is a model-free and jump-robust 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 various parametric volatility models. 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.

Testing for co-integration in vector autoregressions with non-stationary volatility

Many key macroeconomic and financial variables are characterized by permanent changes in unconditional volatility. In this paper we analyse vector autoregressions with non-stationary (unconditional) volatility of a very general form, which includes single and multiple volatility breaks as special cases. We show that the conventional rank statistics computed as in Johansen (1988, 1991) are potentially unreliable. In particular, their large sample distributions depend on the integrated covariation of the underlying multivariate volatility process which impacts on both the size and power of the associated co-integration tests, as we demonstrate numerically. A solution to the identified inference problem is provided by considering wild bootstrap-based implementations of the rank tests. These do not require the practitioner to specify a parametric model for volatility, or to assume that the pattern of volatility is common to, or independent across, the vector of series under analysis. The bootstrap is shown to perform very well in practice.

Testing for Co-Integration in Vector Autoregressions with Non-Stationary Volatility

Many key macro-economic and financial variables are characterised by permanent changes in unconditional volatility. In this paper we analyse vector autoregressions with non-stationary (unconditional) volatility of a very general form, which includes single and multiple volatility breaks as special cases. We show that the conventional rank statistics computed as in Johansen (1988,1991) are potentially unreliable. In particular, their large sample distributions depend on the integrated covariation of the underlying multivariate volatility process which impacts on both the size and power of the associated co-integration tests, as we demonstrate numerically. A solution to the identified inference problem is provided by considering wild bootstrap-based implementations of the rank tests. These do not require the practitioner to specify a parametric model for volatility, nor to assume that the pattern of volatility is common to, or independent across, the vector of series under analysis. The bootstrap is shown to perform very well in practice.

Testing for Unit Roots in the Presence of a Possible Break in Trend and Non-Stationary Volatility

In this paper we analyse the impact of non-stationary volatility on the recently developed unit root tests which allow for a possible break in trend occurring at an unknown point in the sample, considered in Harris, Harvey, Leybourne and Taylor (2008) [HHLT]. HHLT's analysis hinges on a new break fraction estimator which, when a break in trend occurs, is consistent for the true break fraction at rate Op(T=1). Unlike other available estimators, however, when there is no trend break HHLT's estimator converges to zero at rate Op(T1=2). In their analysis HHLT assume the shocks to follow a linear process driven by IID innovations. Our first contribution is to show that HHLT's break fraction estimator retains the same consistency properties as demonstrated by HHLT for the IID case when the innovations display non-stationary behaviour of a quite general form, including, for example, the case of a single break in the volatility of the innovations which may or may not occur at the same time as a break in trend. However, as we subsequently demonstrate, the limiting null distribution of unit root statistics based around this estimator are not pivotal in the presence of non-stationary volatility. Associated Monte Carlo evidence is presented to quantify the impact of various models of non-stationary volatility on both the asymptotic and finite sample behaviour of such tests. A solution to the identified inference problem is then provided by considering wild bootstrap-based implementations of the HHLT tests, using the trend break estimator from the original sample data. The proposed bootstrap method does not require the practitioner to specify a parametric model for volatility, and is shown to perform very well in practice across a range of models.

Testing for multivariate volatility functions using minimum volume sets and inverse regression

We propose two new types of nonparametric tests for investigating multivariate regression functions. The tests are based on cumulative sums coupled with either minimum volume sets or inverse regression ideas; involving no multivariate nonparametric regression estimation. The methods proposed facilitate the investigation for different features such as if a multivariate regression function is (i) constant, (ii) of a bathtub shape, and (iii) in a given parametric form. The inference based on those tests may be further enhanced through associated diagnostic plots. Although the potential use of those ideas is much wider, we focus on the inference for multivariate volatility functions in this paper, i.e. we test for (i) heteroscedasticity, (ii) the so-called ‘smiling effect’, and (iii) some parametric volatility models. The asymptotic behavior of the proposed tests is investigated, and practical feasibility is shown via simulation studies. We further illustrate our methods with real financial data.

Testing for unit roots in time series models with non-stationary volatility

Abstract Many of the key macro-economic and financial variables in developed economies are characterized by permanent volatility shifts. It is known that conventional unit root tests are potentially unreliable in the presence of such behaviour, depending on a particular function (the variance profile) of the underlying volatility process. Somewhat surprisingly then, very little work has been undertaken to develop unit root tests which are robust to the presence of permanent volatility shifts. In this paper we fill this gap in the literature by proposing tests which are valid in the presence of a quite general class of permanent variance changes which includes single and multiple (abrupt and smooth-transition) volatility change processes as special cases. Our solution uses numerical methods to simulate the asymptotic null distribution of the statistics based on a consistent estimate of the variance profile which we also develop. The practitioner is not required to specify a parametric model for volatility. An empirical illustration using producer price inflation series from the Stock–Watson database is reported.

Properties of Realized Volatility and Correlation

The aim of this study is to measure and characterize one-day and one-week volatility and correlation on a number of German equities. For this purpose we measure the realized volatility from tick-by-tick data by summing up short term squared returns. Volume weighted average prices are used to filter the price series. The distributional properties of returns standardized by realized volatility are shown to be gaussian or close to gaussian. In accordance with the literature we also find that weekly returns are more normally distributed than daily returns. Also, weekly returns standardized by realized volatilities are found to be more normally distributed than daily standardized returns. In comparison to similar studies on U.S. equity data we can conclude that the distributional properties of returns and realized volatilities are alike. After examining the memory of realized volatility we find that it is highly unstable implying poor forecasting performance of parametric volatility models.

Time Inhomogeneous Multiple Volatility Modeling

Price variations at speculative markets exhibit positive autocorrelation and cross correlation. Due to large parameter spaces necessary for joint modeling of variances and covariances, multivariate parametric volatility models become easily intractable in practice. We propose an adaptive procedure that identifies periods of second-order homogeneity for each moment in time. To overcome the high dimensionality of the problem we transform the multivariate series into a set of univariate processes. We discuss thoroughly the implementation of the adaptive technique. Theoretical and Monte Carlo results are given. We provide two applications of the new method. For a bivariate exchange rate series we compare the multivariate GARCH approach with our method and find the latter to be more in line with the underlying assumption of independently distributed innovations. Analyzing a 23-dimensional vector of asset returns we underscore the case for adaptive modeling in high-dimensional systems. , .