Memory Stochastic Volatility Model Research Articles

Estimation of Extreme Risk Measures for Stochastic Volatility Models with Long Memory and Heavy Tails

Financial data, such as returns on investments, typically exhibit some non-standard features: long memory or long range dependence (LRD) and heavy tails. Therefore, any mathematical model approximating the evolution of asset price should be able to generate these properties. This can be achieved through the use of a long memory stochastic volatility (LMSV) model. The focus is on estimation of Value-at-Risk (VaR) and Expected Shortfall (ES) for such models. While long memory has no effect on the Hill estimator of the tail index, in contrast it is shown that long memory affects the rates of convergence and asymptotic behaviour of the estimators of VaR and ES.

Estimation and forecasting of long memory stochastic volatility models

Abstract Stochastic Volatility (SV) models are an alternative to GARCH models for estimating volatility and several empirical studies have indicated that volatility exhibits long-memory behavior. The main objective of this work is to propose a new method to estimate a univariate long-memory stochastic volatility (LMSV) model. For this purpose we formulate the LMSV model in a state-space representation with non-Gaussian perturbations in the observation equation, and the estimation of parameters is performed by maximizing the likelihood written in terms derived from a Kalman filter algorithm. We also present a procedure to calculate volatility and Value-at-Risks forecasts. The proposal is evaluated by means of Monte Carlo experiments and applied to real-life time series, where an illustration of market risk calculation is presented.

The tail empirical process for long memory stochastic volatility models with leverage

We consider tail empirical processes of long memory stochastic volatility models with heavy tails and leverage. We study the limiting behaviour of the tail empirical process with both fixed and random levels. We show a dichotomous behaviour for the tail empirical process with fixed levels, according to the interplay between the long memory parameter and the tail index; leverage does not play a role. On the other hand, the tail empirical process with random levels is not affected by either long memory or leverage. The tail empirical process with random levels is used to construct a family of estimators of the tail index, including the famous Hill estimator and harmonic mean estimators. The paper can be viewed as an extension of [21] while the presence of leverage in the model creates additional theoretical problems, the limiting behaviour remains unchanged.

On generalized bivariate student-t Gegenbauer long memory stochastic volatility models with leverage: Bayesian forecasting of cryptocurrencies with a focus on Bitcoin

A Gegenbauer long memory stochastic volatility model with leverage and a bivariate Student’s t-error distribution to model the innovations of the observation and latent volatility jointly for cryptocurrency time series is presented. This is inspired by the deep rooted characteristics found in cryptocurrencies. Until recently their econometric properties have not been thoroughly investigated. Thus, a rigorous in-sample simulation is conducted to assess the performance of the model with its nested alternatives and study the behavior of many cryptocurrencies and in particular Bitcoin. The data analysis is initiated with a broad scope of 114 cryptocurrencies, then a more detailed understanding of five of the most popular cryptocurrencies and followed up with forecasts focused specifically on Bitcoin (while other forecasts are available as supplementary material). The model parameters are estimated with Bayesian approach using Markov Chain Monte Carlo sampling. In order to implement model selection, the Deviance Information Criterion (DIC) is used. Proposed models are compared with many popular models including those commonly used in industry. The models are applied in a Value-at-Risk (VaR) context and several measures are used to assess model performance.

The memory of volatility

The focus of the volatility literature on forecasting and the predominance of theconceptually simpler HAR model over long memory stochastic volatility models has led to the factthat the actual degree of memory estimates has rarely been considered. Estimates in the literaturerange roughly between 0.4 and 0.6 - that is from the higher stationary to the lower non-stationaryregion. This difference, however, has important practical implications - such as the existence or nonexistenceof the fourth moment of the return distribution. Inference on the memory order is complicatedby the presence of measurement error in realized volatility and the potential of spurious long memory.In this paper we provide a comprehensive analysis of the memory in variances of international stockindices and exchange rates. On the one hand, we find that the variance of exchange rates is subject tospurious long memory and the true memory parameter is in the higher stationary range. Stock indexvariances, on the other hand, are free of low frequency contaminations and the memory is in the lowernon-stationary range. These results are obtained using state of the art local Whittle methods that allowconsistent estimation in presence of perturbations or low frequency contaminations.

Estimating and Forecasting Generalized Fractional Long Memory Stochastic Volatility Models

This paper considers a flexible class of time series models generated by Gegenbauer polynomials incorporating the long memory in stochastic volatility (SV) components in order to develop the General Long Memory SV (GLMSV) model. We examine the corresponding statistical properties of this model, discuss the spectral likelihood estimation and investigate the finite sample properties via Monte Carlo experiments. We provide empirical evidence by applying the GLMSV model to three exchange rate return series and conjecture that the results of out-of-sample forecasts adequately confirm the use of GLMSV model in certain financial applications.

Market calibration under a long memory stochastic volatility model

ABSTRACTIn this article, we study a long memory stochastic volatility model (LSV), under which stock prices follow a jump-diffusion stochastic process and its stochastic volatility is driven by a continuous-time fractional process that attains a long memory. LSV model should take into account most of the observed market aspects and unlike many other approaches, the volatility clustering phenomenon is captured explicitly by the long memory parameter. Moreover, this property has been reported in realized volatility time-series across different asset classes and time periods. In the first part of the article, we derive an alternative formula for pricing European securities. The formula enables us to effectively price European options and to calibrate the model to a given option market. In the second part of the article, we provide an empirical review of the model calibration. For this purpose, a set of traded FTSE 100 index call options is used and the long memory volatility model is compared to a popular pricing approach – the Heston model. To test stability of calibrated parameters and to verify calibration results from previous data set, we utilize multiple data sets from NYSE option market on Apple Inc. stock.

English

This article is aimed at to derive geometric fractional Brownian motion where its volatility follow long memory stochastic volatility model, in particular the fractional Ornstein-Uhlenbech process. The innovation algorithm is utilized to simplify such derivation. A simple case of is calculated to illustrate the calculation to accompany this derivation.

Power transformations of absolute returns and long memory estimation

Abstract Different power transformations of absolute returns of various financial assets have been found to display different magnitudes of sample autocorrelations, a property referred to as the Taylor effect. In this paper, we consider the long memory stochastic volatility model for the returns, under which, the asymptotic rate of decay of the autocorrelations of powers of absolute returns is governed by their long memory parameter. Although the true long memory parameter of powers of absolute returns is the same across different powers, we show that the local Whittle estimator of the long memory parameter has finite-sample bias that differs across the power transformations chosen. A Monte-Carlo experiment provides evidence in support of our theoretical finding that the reported variation of the estimates of the long memory parameter for power transformations of returns could be due to finite-sample bias of the estimator. The local Whittle estimates of powers of absolute returns for the S&P500 index and the DM/USD exchange rate are also examined.

Relative forecasting performance of volatility models: Monte Carlo evidence

A Monte Carlo (MC) experiment is conducted to study the forecasting performance of a variety of volatility models under alternative data-generating processes (DGPs). The models included in the MC study are the (Fractionally Integrated) Generalized Autoregressive Conditional Heteroskedasticity models ((FI)GARCH), the Stochastic Volatility model (SV), the Long Memory Stochastic Volatility model (LMSV) and the Markov-switching Multifractal model (MSM). The MC study enables us to compare the relative forecasting performance of the models accounting for different characterizations of the latent volatility process: specifications that incorporate short/long memory, autoregressive components, stochastic shocks, Markov-switching and multifractality. Forecasts are evaluated by means of mean squared errors (MSE), mean absolute errors (MAE) and value-at-risk (VaR) diagnostics. Furthermore, complementarities between models are explored via forecast combinations. The results show that (i) the MSM model best forecasts volatility under any other alternative characterization of the latent volatility process and (ii) forecast combinations provide systematic improvements upon most single misspecified models, but are typically inferior to the MSM model even if the latter is applied to data governed by other processes.

Maximally Autocorrelated Power Transformations: A Closer Look at the Properties of Stochastic Volatility Models

There has been an increasing interest in the financial econometrics literature on the properties of non-linear transformations of returns. In this paper, we focus on power transformations and analyze which powers of returns provide stronger autocorrelations in the context of stochastic volatility (SV) models. We show that the sample and theoretical autocorrelations implied by short and long memory SV models peak at the same power transformation, which is close to one for realistic values of the model parameters. Consequently, we suggest that the power that maximizes the sample autocorrelations could be used as an additional descriptive tool for the adequacy of a SV model. The results are illustrated with three real series of financial returns.

Pricing European option with transaction costs under the fractional long memory stochastic volatility model

This paper deals with the problem of discrete time option pricing using the fractional long memory stochastic volatility model with transaction costs. Through the ‘anchoring and adjustment’ argument in a discrete time setting, a European call option pricing formula is obtained.

Estimating Persistence in the Volatility of Asset Returns with Signal Plus Noise Models

This paper examines the degree of persistence in the volatility of financial time series using a Long Memory Stochastic Volatility (LMSV) model. Specifically, it employs a Gaussian semiparametric (or local Whittle) estimator of the memory parameter, based on the frequency domain, proposed by Robinson (1995a), and shown by Arteche (2004) to be consistent and asymptotically normal in the context of signal plus noise models. Daily data on the NASDAQ index are analysed. The results suggest that volatility has a component of long- memory behaviour, the order of integration ranging between 0.3 and 0.5, the series being therefore stationary and mean-reverting.

Estimating persistence in the volatility of asset returns with signal plus noise models

ABSTRACTThis paper examines the degree of persistence in the volatility of financial time series using a Long Memory Stochastic Volatility (LMSV) model. Specifically, it employs a Gaussian semiparametric (or local Whittle) estimator of the memory parameter, based on the frequency domain, proposed by Robinson (Annals of Statistics 23: 1630–1661), and shown by Arteche (Journal of Econometrics 119: 131–154) to be consistent and asymptotically normal in the context of signal plus noise models. Daily data on the NASDAQ index are analysed. The results suggest that volatility has a component of long‐memory behaviour, the order of integration ranging between 0.3 and 0.5, the series being therefore stationary and mean‐reverting. Copyright © 2011 John Wiley & Sons, Ltd.

The tail empirical process for long memory stochastic volatility sequences

This paper describes the limiting behaviour of tail empirical processes associated with long memory stochastic volatility models. We show that such a process has dichotomous behaviour, according to an interplay between the Hurst parameter and the tail index. On the other hand, the tail empirical process with random levels never suffers from long memory. This is very desirable from a practical point of view, since such a process may be used to construct the Hill estimator of the tail index. To prove our results we need to establish new results for regularly varying distributions, which may be of independent interest.

Bias-Reduced Estimation of Long Memory Stochastic Volatility

We propose to use a variant of the local polynomial Whittle estimator to estimate the memory parameter in volatility for long memory stochastic volatility models with potential nonstationarity in the volatility process. We show that the estimator is asymptotically normal and capable of obtaining bias reduction as well as a rate of convergence arbitrarily close to the parametric rate, n1=2. A Monte Carlo study is conducted to support the theoretical results, and an analysis of daily exchange rates demonstrates the empirical usefulness of the estimators.

Accurate minimum capital risk requirements: A comparison of several approaches

In this paper we estimate, for several investment horizons, minimum capital risk requirements for short and long positions, using the unconditional distribution of three daily indexes futures returns and a set of short and long memory stochastic volatility and GARCH-type models. We consider the possibility that errors follow a t-Student distribution in order to capture the kurtosis of the returns’ series. The results suggest that accurate modelling of extreme observations obtained for long and short trading investment positions is possible with an autoregressive stochastic volatility model. Moreover, modelling futures returns with a long memory stochastic volatility model produces, in general, excessive volatility persistence, and consequently, leads to large minimum capital risk requirement estimates. Finally, the models’ predictive ability is assessed with the help of out-of-sample conditional tests.

Bayesian estimation and the application of long memory stochastic volatility models

A new sampling-based Bayesian approach to the long memory stochastic volatility (LMSV) process is presented; the method is motivated by the GPH-estimator in fractionally integrated autoregressive moving average (ARFIMA) processes, which was originally proposed by J. Geweke and S. Porter-Hudak [The estimation and application of long memory time series models, Journal of Time Series Analysis, 4 (1983) 221–238]. In this work, we perform an estimation of the memory parameter in the Bayesian framework; an estimator is obtained by maximizing the posterior density of the memory parameter. Finally, we compare the GPH-estimator and the Bayes-estimator by means of a simulation study and our new approach is illustrated using several stock market indices; the new estimator is proved to be relatively stable for the various choices of frequencies used in the regression.

The Local Whittle Estimator of Long-Memory Stochastic Volatility

We propose a new semiparametric estimator of the degree of persistence in volatility for long memory stochastic volatility (LMSV) models. The estimator uses the periodogram of the log squared returns in a local Whittle criterion which explicitly accounts for the noise term in the LMSV model. Finite-sample and asymptotic standard errors for the estimator are provided. An extensive simulation study reveals that the local Whittle estimator is much less biased and that the finite-sample standard errors yield more accurate confidence intervals than the widely-used GPH estimator. The estimator is also found to be robust against possible leverage effects. In an empirical analysis of the daily Deutsche Mark/US Dollar exchange rate, the new estimator indicates stronger persistence in volatility than the GPH estimator, provided that a large number of frequencies is used. , .

ON THE LOG PERIODOGRAM REGRESSION ESTIMATOR OF THE MEMORY PARAMETER IN LONG MEMORY STOCHASTIC VOLATILITY MODELS

We consider semiparametric estimation of the memory parameter in a long memory stochastic volatility model. We study the estimator based on a log periodogram regression as originally proposed by Geweke and Porter-Hudak (1983, Journal of Time Series Analysis 4, 221–238). Expressions for the asymptotic bias and variance of the estimator are obtained, and the asymptotic distribution is shown to be the same as that obtained in recent literature for a Gaussian long memory series. The theoretical result does not require omission of a block of frequencies near the origin. We show that this ability to use the lowest frequencies is particularly desirable in the context of the long memory stochastic volatility model.