Modeling Financial Returns Research Articles

Combining the MGHyp distribution with nonlinear shrinkage in modeling financial asset returns

The present paper combines nonlinear shrinkage with the multivariate generalized hyperbolic (MGHyp) distribution, thereby extending a flexible parametric model to high dimensions. An expectation–maximization (EM) algorithm is developed that is fast, stable, and applicable in high dimensions. Theoretical arguments for the monotonicity of the proposed algorithm are provided and it is shown in simulations that it is able to accurately retrieve parameter estimates. Finally, in an extensive Markowitz portfolio optimization analysis, the approach is compared to state-of-the-art benchmark models. The proposed model excels with a strong out-of-sample portfolio performance combined with a comparably low turnover.

The Interdependence of Financial Markets in Turbulent Periods: A Comparative Analysis of the China–US Cases

ABSTRACT: There has been a growing hypothesis linking the recent widespread declines in financial markets to the outbreaks of the COVID-19 pandemic. What is, however, unclear is whether the influence of the COVID-19 pandemic on the dynamics of the financial markets is as severe as those associated with traditional financial crises. Using the cases of the USA and China to represent developed and emerging economies, we examine the returns and volatility dynamics of financial market interdependence amid economic and financial crisis periods of divergent origins. Following careful consideration of relevant tests and model selection criteria, we find the VARMA-CCC-GARCH model to be the best approach for modelling financial market returns and volatility spillovers during a crisis. Employing daily financial market returns, we partitioned the study period into two sub-periods: the Great Recession (GFC) and the Great Lockdown (COVID-19), to arrive at the following empirical findings: First, we find that regardless of the type of crisis—the GFC or COVID-19—unanticipated events in the financial markets in the current period fuel high volatility in the market in the preceding period. Second, while financial markets in developed economies were by far the most affected by the 2008/2009 GFC, the great lockdown associated with COVID-19 appears to have left a permanent shock on both developed and emerging economies' financial markets. It then follows that the extent to which an economy is endowed with the ability to fend off external risks may be sensitive to the origin and nature of the crisis. In this light, it is necessary for policymakers to acknowledge that while some of the instruments used to mitigate the impact of external shocks on financial markets during the GFC may still be applicable during COVID-19 in the case of developed economies, the same appears to not be entirely true in the case of emerging financial markets during COVID-19. On the whole, financial experts and academics may want to be wary of the fact that the past realizations of both returns and volatilities matter for enhancing the forecastability of the future value of financial instruments during a crisis period.

On inverse-Gamma distribution delayed by Poisson process

The models of stochastic subordination, or random time indexing, has been recently applied to model financial returns series X(t)t≥0 exhibiting sharp, and large variations. These sharp and large variations are linked to information arrivals and/or represent sudden events, and hence we have a model with jumps. For this purpose, by substituting the usual deterministic time t as a subordinator T(t)t≥0 in a stochastic process X(t)t≥0 we obtain a new process X(T(t))t≥0 whose stochastic time is dominated by the subordinator T(t)t≥0. Therefore, we propose in this paper an alternative approach based on a combination of the continuous-time bilinear (COBL) process subordinated by a Poisson process (that it is a Levy process) which permits us to introduce further randomness for the phenomena which exhibit either a speeded up or slowed down behavior. So, the main probabilistic properties of such models are studied and the explicit expression of the higher-order moments properties are given.

Limit theory and robust evaluation methods for the extremal properties of GARCH(p, q) processes

Generalized autoregressive conditionally heteroskedastic (GARCH) processes are widely used for modelling financial returns, with their extremal properties being of interest for market risk management. For GARCH(p,q) processes with max (p,q) = 1 all extremal features have been fully characterised, but when max (p,q)ge 2 much remains to be found. Previous research has identified that both marginal and dependence extremal features of strictly stationary GARCH(p,q) processes are determined by a multivariate regular variation property and tail processes. Currently there are no reliable methods for evaluating these characterisations, or even assessing the stationarity, for the classes of GARCH(p,q) processes that are used in practice, i.e., with unbounded and asymmetric innovations. By developing a mixture of new limit theory and particle filtering algorithms for fixed point distributions we produce novel and robust evaluation methods for all extremal features for all GARCH(p,q) processes, including ARCH and IGARCH processes. We investigate our methods’ performance when evaluating the marginal tail index, the extremogram and the extremal index, the latter two being measures of temporal dependence.

Estimating the value-at-risk of JSE indices and South African exchange rate with Generalized Pareto and stable distributions

South Africa’s economy has faced many downturns in the previous decade, and to curb the spread of the novel SARS-CoV-2, the lockdown brought South African financial markets to an abrupt halt. Therefore, the implementation of risk mitigation approaches is becoming a matter of urgency in volatile markets in these unprecedented times. In this study, a hybrid generalized autoregressive conditional heteroscedasticity (GARCH)-type model combined with heavy-tailed distributions, namely the Generalized Pareto Distribution (GPD) and the Nolan’s S0-parameterization stable distribution (SD), were fitted to the returns of three FTSE/JSE indices, namely All Share Index (ALSI), Banks Index and Mining Index, as well as the daily closing prices of the US dollar against the South African rand exchange rate (USD/ZAR exchange rate). VaR values were estimated and back-tested using the Kupiec likelihood ratio test. The results of this study show that for FTSE/JSE ALSI returns, the hybrid exponential GARCH (1,1) model with SD model (EGARCH(1,1)-SD) outperforms the GARCH-GPD model at the 2.5% VaR level. At VaR levels of 95% and 97.5%, the fitted GARCH (1,1)-SD model for FTSE/JSE Banks Index returns performs better than the GARCH (1,1)-GPD. The fitted GARCH (1,1)-SD model for FTSE/JSE Mining Index returns is better than the GARCH (1,1)-GPD at 5% and 97.5% VaR levels. Thus, this study suggests that the GARCH (1,1)-SD model is a good alternative to the VaR robust model for modeling financial returns. This study provides salient results for persons interested in reducing losses or obtaining a better understanding of the South African financial industry.

Rejoinder to “Multivariate generalized hyperbolic laws for modeling financial log returns: Empirical and theoretical considerations”

Rejoinder to “Multivariate generalized hyperbolic laws for modeling financial log returns: Empirical and theoretical considerations”

Discussion of “Revisiting Multivariate generalized hyperbolic laws for modeling financial log returns by Fotopoulos, Paparas And Jandhyala”

Discussion of “Revisiting Multivariate generalized hyperbolic laws for modeling financial log returns by Fotopoulos, Paparas And Jandhyala”

Risk-Constrained Kelly Portfolios Under Alpha-Stable Laws

This paper provides a detailed framework for modeling portfolios, achieving the highest growth rate under risk constraints such as value at risk (VaR) and expected shortfall (ES) in the presence of $$\alpha $$-stable laws. Although the maximization of the expected logarithm of wealth induces outperforming any other significantly different strategy, the Kelly criterion implies larger bets than a risk-averse investor would accept. Restricting the Kelly optimization by spectral risk measures, the authors provide a generalized mapping for different measures of growth and risk. Analyzing over 30 years of S&P 500 returns for different sampling frequencies, the authors find evidence for leptokurtic behavior for all respective sampling frequencies. Given that lower sampling frequencies imply a smaller number of data points, this paper argues in favor of $$\alpha $$-stable laws and its scaling behavior to model financial market returns for a given horizon in an i.i.d. world. Instead of simulating from the class of elliptically $$\alpha $$-stable distributions, a semiparametric scaling approximation, based on hourly NASDAQ data, is proposed. Our paper also uncovers that including long put options into the portfolio optimization, improves portfolio growth for a given level of VaR or ES, leading to a new Kelly portfolio providing the highest geometric mean.

Filtering and Estimation for a Class of Stochastic Volatility Models with Intractable Likelihoods

We introduce a new approach to latent state filtering and parameter estimation for a class of stochastic volatility models (SVMs) for which the likelihood function is unknown. The α-stable stochastic volatility model provides a flexible framework for capturing asymmetry and heavy tails, which is useful when modeling financial returns. However, the α-stable distribution lacks a closed form for the probability density function, which prevents the direct application of standard Bayesian filtering and estimation techniques such as sequential Monte Carlo and Markov chain Monte Carlo. To obtain filtered volatility estimates, we develop a novel approximate Bayesian computation (ABC) based auxiliary particle filter, which provides improved performance through better proposal distributions. Further, we propose a new particle based MCMC (PMCMC) method for joint estimation of the parameters and latent volatility states. With respect to other extensions of PMCMC, we introduce an efficient single filter particle Metropolis-within-Gibbs algorithm which can be applied for obtaining inference on the parameters of an asymmetric α-stable stochastic volatility model. We show the increased efficiency in the estimation process through a simulation study. Finally, we highlight the necessity for modeling asymmetric α-stable SVMs through an application to propane weekly spot prices.

Heterogeneous tail generalized COMFORT modeling via Cholesky decomposition

A mean–variance heterogeneous tails mixture distribution is proposed for modeling financial asset returns. It captures, along with the obligatory leptokurtosis, different tail behavior among the assets. Its construction allows for joint maximum likelihood estimation of all model parameters via an expectation–maximization algorithm and thus is applicable in high dimensions. A useful and unique feature of the model is that the tail behavior of the individual assets is driven by asset-specific news effects. In the bivariate iid case, the model corresponds to the standard CAPM model, but enriched with a filter for capturing the news impact associated with both the market and asset excess returns. An empirical application using a portfolio of highly tail-heterogeneous cryptocurrencies and realistic transaction costs shows superior out-of-sample portfolio performance compared to numerous competing models. A model extension to capture asset-specific asymmetry is also discussed.

The sparse method of simulated quantiles: An application to portfolio optimization

The sparse multivariate method of simulated quantiles (S‐MMSQ) is applied to solve a portfolio optimization problem under value‐at‐risk constraints where the joint returns follow a multivariate skew‐elliptical stable distribution. The S‐MMSQ is a simulation‐based method that is particularly useful for making parametric inference in some pathological situations where the maximum likelihood estimator is difficult to compute. The method estimates parameters by minimizing the distance between quantile‐based statistics evaluated on true and synthetic data, simulated from the postulated model, penalized by adding the smoothly clipped absolute deviation ℓ1‐penalty in order to achieve sparsity. The S‐MMSQ aims to efficiently handle the problem of estimating large‐dimensional distributions with intractable likelihood, such as the stable distributions that have been widely applied in finance to model financial returns.

Bayesian Shrinkage Estimation of Time-varying Covariance Matrices in Financial Time Series

Modeling financial returns is challenging because the correlations and variance of returns are time-varying and the covariance matrices can be quite high-dimensional. In this paper, we develop a Bayesian shrinkage approach with modified Cholesky decomposition to model correlations between financial returns. We reparameterize the correlation parameters to meet their positive definite constraint for Bayesian analysis. To implement an efficient sampling scheme in posterior inference, hierarchical representation of Bayesian lasso is used to shrink unknown coefficients in linear regressions. Simulation results show good sampling properties that iterates from Markov chain Monte Carlo converge quickly. Using a real data example, we illustrate the application of the proposed Bayesian shrinkage method in modeling stock returns in Hong Kong.

Underlying Assets Distribution in Derivatives: The BRIC Case

This paper addresses one of the main issues regarding numerical derivatives valuation, particularly the search for an alternative to the normality assumption of underlying asset returns, to obtain the price by using numerical techniques. There might be difficulties in making normality assumptions, which could produce over-valuated or sub-valuated prices of derivatives. Under this consideration, the Generalized Hyperbolic family has been proven to be a proper selection to model heavy tailed distribution behavior. The Normal Inverse Gaussian (NIG) distribution is a member flexible enough to model financial returns. NIG distribution can be used to model distribution returns under different states of nature. The indexes of the Brazil, Russia, India and China (BRIC) economies were studied at different time-periods using return data series from 2002 to 2005, 2006 to 2010 and 2011 to 2015, in such a manner to demonstrate with statistical criteria that NIG fits the empirical distribution in the three periods; even throughout economic downturn. This result may be used as an improvement in derivatives valuation with indexes as underlying assets.

Self-weighted recursive estimation of GARCH models

ABSTRACTThe generalized autoregressive conditional heteroscedasticity (GARCH) processes are frequently used to investigate and model financial returns. They are routinely estimated by computationally complex off-line estimation methods, for example, by the conditional maximum likelihood procedure. However, in many empirical applications (especially in the context of high-frequency financial data), it seems necessary to apply numerically more effective techniques to calibrate and monitor such models. The aims of this contribution are: (i) to review the previously introduced recursive estimation algorithms and to derive self-weighted alternatives applying general recursive identification instruments, and (ii) to examine these methods by means of simulations and an empirical application.

Using the Generalized Pareto and Pearson Type-IV Distributions to Measure Value-At-Risk for the Daily South African Mining Index

Risk management tools, such as value-at-risk (VaR) and expected shortfall (conditional value-at-risk) are highly dependent on an appropriate set of underlying distributional assumptions being made. Identifying a distribution that best captures all aspects of financial data sets may benefit both investors and risk managers. In this study, we compare the relative performance of the GARCH-generalized Pareto distribution and the GARCH-Pearson type-IV models in estimating value-at-risk of the South African mining index returns. VaR and backtesting are performed via Kupiec likelihood ratio test. The proposed models capture some key stylized facts associated with daily index returns; e.g. heavy tails (non-normality), asymmetry, volatility clustering and their leverage effect. The advantage of the proposed models lies in their ability to capture conditional heteroscedasticity in the returns through the GARCH framework and at the same time model their heavy tail behaviour through the generalized Pareto distribution (GPD) and the Pearson type-IV distribution. The main findings indicate that the GARCH-GPD and GARCH-Pearson type-IV models give better results when compared with generalized hyperbolic distributions (GHDs), thereby providing a good alternative candidate for modelling financial returns. The accuracy of the volatility model is essential in forecasting volatility of future returns in which the predictability of volatility plays an integral role in risk management and portfolio management.

Constrained Kelly Portfolios under -Stable Laws

Investing in a portfolio strategy, which maximizes the the expected logarithm of capital implies outperforming any other significantly different strategy, given knowledge of the true underlying process (Breiman, 1961). But, the sole use of the Kelly Criterion implies larger bets than a risk-averse investor would accept (Clark and Ziemba, 1987; Hausch and Ziemba, 1985). This paper restricts the Kelly optimization by security measures Value at Risk (VaR) or Expected Shortfall as special cases of the Spectral risk measure.As financial market returns are, with large probability, non-Gaussian, this paper utilizes stable laws and its scaling behavior to model financial market returns for a given horizon in an i.i.d. world. Instead of simulating from the class of elliptically stable distributions, a nonparametric scaling approximation, based on the dataset itself, is proposed. Additionally, assets with non-linear payoff structure, long put-options, are incorporated into the nonlinear optimization to allow for asymmetric payoffs. For chosen special cases of the Spectral measure, this allows to increase the growth-criterion for a given security level.

Macro-Driven VAR Forecasts: From Very High to Very Low-Frequency Data

This paper studies in some details the joint-use of high-frequency data and economic variables to model financial returns and volatility. We extend the Realized LGARCH model by allowing for a timevarying intercept, which responds to changes in macroeconomic variables in a MIDAS framework and allows macroeconomic information to be included directly into the estimation and forecast procedure. Using more than 10 years of high-frequency transactions for 55 U.S. stocks, we argue that the combination of low-frequency exogenous economic indicators with high-frequency financial data improves our ability to forecast the volatility of returns, their full multi-step ahead conditional distribution and the multiperiod Value-at-Risk. We document that nominal corporate profits and term spreads generate accurate risk measures forecasts at horizons beyond two business weeks.

Evaluation of Ashley and Patterson model diagnostic tool in selection of ARCH/GARCH family models: case of natural rubber prices at the Colombo auction

There is a substantial amount of literature in favor of models in the ARCH/GARCH family as premier class of models for modeling financial and economic returns are of nonlinear form. However, there are many gaps in nonlinear model diagnosis and none of the nonlinearity tests are in wide acceptance. In this study, a new model diagnostic tool proposed by Ashley and Patterson in 2001 was identified as a useful diagnostic technique compared to other techniques in model selection. Moreover, it was found that the model diagnosis was straight forward with Ashley and Patterson test statistic (AP) when it is not straight forward with the use of other methods. Little work was found on the true process generating mechanism of returns of Ribbed Smoked Sheets No. 1 (RSS1) prices in the Colombo auction. In this study, by using AP, it was found that the true process generating mechanism of returns of RSS1 is nonlinear and the EGARCH (1,1) to be the most promising model that correctly generates returns of RSS1. Tropical Agricultural Research Vol. 25 (1): 109-119 (2013)

REALIZED BETA GARCH: A MULTIVARIATE GARCH MODEL WITH REALIZED MEASURES OF VOLATILITY

SUMMARYWe introduce a multivariate generalized autoregressive conditional heteroskedasticity (GARCH) model that incorporates realized measures of variances and covariances. Realized measures extract information about the current levels of volatilities and correlations from high‐frequency data, which is particularly useful for modeling financial returns during periods of rapid changes in the underlying covariance structure. When applied to market returns in conjunction with returns on an individual asset, the model yields a dynamic model specification of the conditional regression coefficient that is known as the beta. We apply the model to a large set of assets and find the conditional betas to be far more variable than usually found with rolling‐window regressions based exclusively on daily returns. In the empirical part of the paper, we examine the cross‐sectional as well as the time variation of the conditional beta series during the financial crises. Copyright © 2014 John Wiley & Sons, Ltd.

Effects of Daylight-Saving Time Changes on Stock Market Returns and Stock Market Volatility: Rebuttal

In a 2011 reply to our 2010 comment in this journal, Berument and Dogen maintained their challenge to the existence of the negative daylight-saving effect in stock returns reported by Kamstra, Kramer, and Levi in 2000. Unfortunately, in their reply, Berument and Dogen ignored all of the points raised in the comment, failing even to cite the Kamstra, et al. comment. Berument and Dogen continued to use inappropriate estimation techniques, over-parameterized models, and low-power tests and perhaps most surprisingly even failed to replicate results they themselves reported in their previous paper, written by Berument, Dogen, and Onar in 2010. The findings reported by Berument and Dogen, as well as by Berument, Dogen, and Onar, are neither well-supported nor well-reasoned. We maintain our original objections to their analysis, highlight new serious empirical and theoretical problems, and emphasize that there remains statistically significant evidence of an economically large negative daylight-saving effect in U.S. stock returns. The issues raised in this rebuttal extend beyond the daylight-saving effect itself, touching on methodological points that arise more generally when deciding how to model financial returns data.