Improving density forecast by modeling asymmetric features: An application to S&P500 returns

A density forecast is an estimate of the probability distribution of the possible future values of a random variable. From the current literature, an economic time series may have three types of asymmetry: asymmetry in unconditional distribution, asymmetry in conditional distribution, volatility asymmetry. In this paper, we propose three density forecasting methods under two-piece normal assumption to capture these asymmetric features. A GARCH model with two-piece normal distribution is developed to capture asymmetries in the conditional distributions. In this approach, we first estimate parameters of a GARCH model by assuming normal innovations, and then fit a two-piece normal distribution to the empirical residuals. Block bootstrap procedure, and moving average method with two-piece normal distribution are presented for volatility asymmetry and asymmetry in the conditional distributions. Application of the developed methods to the weekly S&P500 returns illustrates that forecast quality can be significantly improved by modeling these asymmetric features.     

Mixed effect models for absolute log returns of ultra high frequency data

Considering absolute log returns as a proxy for stochastic volatility, the influence of explanatory variables on absolute log returns of ultra high frequency data is analysed. The irregular time structure and time dependency of the data is captured by utilizing a continuous time ARMA(p,q) process. In particular, we propose a mixed effect model class for the absolute log returns. Explanatory variable information is used to model the fixed effects, whereas the error is decomposed in a non‐negative Lévy driven continuous time ARMA(p,q) process and a market microstructure noise component. The parameters are estimated in a state space approach. In a small simulation study the performance of the estimators is investigated. We apply our model to IBM trade data and quantify the influence of bid‐ask spread and duration on a daily basis. To verify the correlation in irregularly spaced data we use the variogram, known from spatial statistics. Copyright © 2006 John Wiley & Sons, Ltd.     

Correlated ARCH (CorrARCH): Modelling the time-varying conditional correlation between financial asset returns

Although the time variation of the conditional correlations of asset returns is a well established stylized fact (and of crucial importance for efficient financial decisions) there is no explicit general model available for its estimation and forecasting. In this paper, we propose a bivariate GARCH covariance structure in which conditional variances can follow any GARCH-type process, while conditional correlation is generated by an explicit discrete-time stochastic process, the CorrARCH process. A high order CorrARCH can parsimoniously be represented by a CorGARCH process. The model successfully generates the reported stylized facts, establishes an autocorrelation structure for correlations and thus provides an explicit framework for out-of-sample forecasting. We provide empirical evidence from the G7 Stock Market Indexes.     

Representative Agent Pricing of Financial Assets Based on L��vy Processes with Normal Inverse Gaussian Marginals

The aim of the paper is to test the assumption of normal inverse Gaussian returns from speculative investments. We construct an asset pricing model where price processes are pure jump processes having associated returns with marginal distributions of this particular type. The resulting model is not complete, and we employ a partial equilibrium framework with a representative agent. The model is confronted with some stylized facts, like the equity premium puzzle, and the results seem promising.     

Statistical modelling of asymmetric risk in asset returns

The purpose of this article is to provide a straightforward model for asset returns which captures the fundamental asymmetry in upward versus downward returns. We model this feature by using scale gamma distributions for the conditional distributions of positive and negative returns. By allowing the parameters for positive returns to differ from parameters for negative returns we can test the hypothesis of symmetry. Some applications of this process to expected utility and semi-variance calculations are considered. Finally we estimate the model using daily UK FT100 index and Futures data.     

Common volatility and correlation clustering in asset returns

We present a new multivariate framework for the estimation and forecasting of the evolution of financial asset conditional correlations. Our approach assumes return innovations with time dependent covariances. A Cholesky decomposition of the asset covariance matrix, with elements written as sines and cosines of spherical coordinates allows for modelling conditional variances and correlations and guarantees its positive definiteness at each time t. As in Christodoulakis and Satchell [Christodoulakis, G.A., Satchell, S.E., 2002. Correlated ARCH (CorrARCH): Modelling the time-varying conditional correlation between financial asset returns. European Journal of Operational Research 139 (2), 350–369] correlation is generated by conditionally autoregressive processes, thus allowing for an autocorrelation structure for correlation. Our approach allows for explicit out-of-sample forecasting and is consistent with stylized facts as time-varying correlations and correlation clustering, co-movement between correlation coefficients, correlation and volatility as well as between volatility processes (co-volatility). The latter two are shown to depend on correlation and volatility persistence. Empirical evidence on a trivariate model using monthly data from Dow Jones Industrial, Nasdaq Composite and the 3-month US Treasury Bill yield supports our theoretical arguments.     

Portfolio value-at-risk estimation in energy futures markets with time-varying copula-GARCH model

This paper combines copula functions with GARCH-type models to construct the conditional joint distribution, which is used to estimate Value-at-Risk (VaR) of an equally weighted portfolio comprising crude oil futures and natural gas futures in energy market. Both constant and time-varying copulas are applied to fit the dependence structure of the two assets returns. The findings show that the constant Student t copula is a good compromise for effectively fitting the dependence structure between crude oil futures and natural gas futures. Moreover, the skewed Student t distribution has a better fit than Normal and Student t distribution to the marginal distribution of each asset. Asymmetries and excess kurtosis are found in marginal distributions as well as in dependence. We estimate VaR of the underlying portfolio to be 95% and 99%, by using the Monte Carlo simulation. Then using backtesting, we compare the out-of-sample forecasting performances of VaR estimated by different models.     

Multiple Priors and Asset Pricing

The asset pricing implications of a statistical model consistent with multiple priors, or beliefs about return distributions, are developed. It is shown that quite generally equilibrium differences in mean returns across priors are to be explained in terms of perceived risk differences between these priors. Advances in filtering theory are employed on time series data to filter all the multiple state conditional components of risks and rewards. It is then observed that excess return differentials across priors are broadly consistent with required risk compensations under these priors, though the sharp hypothesis of zero intercept and unit slope is rejected. The filtered results also deliver numerous other interesting statistics. Here we focus on the construction of long horizon return distributions from data on daily returns using a Markov chain approach to incorporate stochasticity in elementary risk characterizations like volatility, skewness and kurtosis.      

Conditioning exceedances on covariate processes

This article concerns the statistical inference for the upper tail of the conditional distribution of a response variable Y given a covariate X = x based on n random vectors within the parametric extreme value framework. Pioneering work in this field was done by Smith (Stat Sci 4:367–393, 1989) and Smith and Shively (Atmos Environ 29:3489–3499, 1995). We propose to base the inference on a conditional distribution of the point process of exceedances given the point process of covariates. It is of importance that the conditional distribution merely depends on the conditional distribution of the response variable given the covariates. In the special case of Poisson processes such a result may be found in Reiss (1993). Our results are valid within the broader model where the response variables are conditionally independent given the covariates. It is numerically exemplified that the maximum likelihood principle leads to more accurate estimators within the conditional approach than in the previous one.     

A conditional-SGT-VaR approach with alternative GARCH models

This paper proposes a conditional technique for the estimation of VaR and expected shortfall measures based on the skewed generalized t (SGT) distribution. The estimation of the conditional mean and conditional variance of returns is based on ten popular variations of the GARCH model. The results indicate that the TS-GARCH and EGARCH models have the best overall performance. The remaining GARCH specifications, except in a few cases, produce acceptable results. An unconditional SGT-VaR performs well on an in-sample evaluation and fails the tests on an out-of-sample evaluation. The latter indicates the need to incorporate time-varying mean and volatility estimates in the computation of VaR and expected shortfall measures.     

Some properties of the bivariate lognormal distribution for reliability applications

In this paper, we study the bivariate lognormal distribution from a reliability point of view. The conditional distribution of X given Y > y is found to be log‐skew normal. The monotonicity of the hazard rates of the univariate as well as the conditional distributions is discussed. Clayton's association measure is obtained in terms of the hazard gradient, and its value in the case of our model is derived. The probability distributions, in the case of series and parallel systems, are derived, and the monotonicity of their failure rates is discussed. Three real applications of the bivariate lognormal distribution are provided, two from financial economics and one from reliability. Copyright © 2012 John Wiley & Sons, Ltd.     

Option pricing under a normal mixture distribution derived from the Markov tree model

We examine a Markov tree (MT) model for option pricing in which the dynamics of the underlying asset are modeled by a non-IID process. We show that the discrete probability mass function of log returns generated by the tree is closely approximated by a continuous mixture of two normal distributions. Using this normal mixture distribution and risk-neutral pricing, we derive a closed-form expression for European call option prices. We also suggest a regression tree-based method for estimating three volatility parameters σ, σ⁺, and σ⁻ required to apply the MT model. We apply the MT model to price call options on 89 non-dividend paying stocks from the S&P 500 index. For each stock symbol on a given day, we use the same parameters to price options across all strikes and expires. Comparing against the Black–Scholes model, we find that the MT model’s prices are closer to market prices.     

Bayesian tail‐risk forecasting using realized GARCH

A realized generalized autoregressive conditional heteroskedastic (GARCH) model is developed within a Bayesian framework for the purpose of forecasting value at risk and conditional value at risk. Student‐t and skewed‐t return distributions are combined with Gaussian and student‐t distributions in the measurement equation to forecast tail risk in eight international equity index markets over a 4‐year period. Three realized measures are considered within this framework. A Bayesian estimator is developed that compares favourably, in simulations, with maximum likelihood, both in estimation and forecasting. The realized GARCH models show a marked improvement compared with ordinary GARCH for both value‐at‐risk and conditional value‐at‐risk forecasting. This improvement is consistent across a variety of data and choice of distributions. Realized GARCH models incorporating a skewed student‐t distribution for returns are favoured overall, with the choice of measurement equation error distribution and realized measure being of lesser importance. Copyright © 2017 John Wiley & Sons, Ltd.     

Option Pricing for Log-Symmetric Distributions of Returns

We derive an option pricing formula on assets with returns distributed according to a log-symmetric distribution. Our approach is consistent with the no-arbitrage option pricing theory: we propose the natural risk-neutral measure that keeps the distribution of returns in the same log-symmetric family reflecting thus the specificity of the stock’s returns. Our approach also provides insights into the Black–Scholes formula and shows that the symmetry is the key property: if distribution of returns X is log-symmetric then 1/X is also log-symmetric from the same family. The proposed options pricing formula can be seen as a generalization of the Black–Scholes formula valid for lognormal returns. We treat an important case of log returns being a mixture of symmetric distributions with the particular case of mixtures of normals and show that options on such assets are underpriced by the Black–Scholes formula. For the log-mixture of normal distributions comparisons with the classical formula are given.      

Nested Conditional Value-at-Risk portfolio selection: A model with temporal dependence driven by market-index volatility

In a multistage stochastic programming framework, we develop a new method for finding an approximated portfolio allocation solution to the nested Conditional Value-at-Risk model when asset log returns are stagewise dependent. We describe asset log returns through a single-factor model where the driving factor is the market-index log return modeled by a Generalized Autoregressive Conditional Heteroskedasticity process to take into account the serial dependence usually observed. To solve the nested Conditional Value-at-Risk model, we implement a backward induction scheme coupled with cubic spline interpolation that reduces the computational complexity of the optimal portfolio allocation and allows to treat problems otherwise unmanageable.     

Random Flights in Higher Spaces

We consider in this paper random flights in ℝ ^d performed by a particle changing direction of motion at Poisson times. Directions are uniformly distributed on hyperspheres S ₁ ^d . We obtain the conditional characteristic function of the position of the particle after n changes of direction. From this characteristic function we extract the conditional distributions in terms of (n+1)−fold integrals of products of Bessel functions. These integrals can be worked out in simple terms for spaces of dimension d=2 and d=4. In these two cases also the unconditional distribution is determined in explicit form. Some distributions connected with random flights in ℝ³ are discussed and in some special cases are analyzed in full detail. We point out that a strict connection between these types of motions with infinite directions and the equation of damped waves holds only for d=2. Related motions with random velocity in spaces of lower dimension are analyzed and their distributions derived.     

Measuring financial risk and portfolio optimization with a non-Gaussian multivariate model

In this paper, we propose a multivariate market model with returns assumed to follow a multivariate normal tempered stable distribution. This distribution, defined by a mixture of the multivariate normal distribution and the tempered stable subordinator, is consistent with two stylized facts that have been observed for asset distributions: fat-tails and an asymmetric dependence structure. Assuming infinitely divisible distributions, we derive closed-form solutions for two important measures used by portfolio managers in portfolio construction: the marginal VaR and the marginal AVaR. We illustrate the proposed model using stocks comprising the Dow Jones Industrial Average, first statistically validating the model based on goodness-of-fit tests and then demonstrating how the marginal VaR and marginal AVaR can be used for portfolio optimization using the model. Based on the empirical evidence presented in this paper, our framework offers more realistic portfolio risk measures and a more tractable method for portfolio optimization.     

On Order Types of Systems of Segments in the Plane

Let r(n) denote the largest integer such that every family C\mathcal{C} of n pairwise disjoint segments in the plane in general position has r(n) members whose order type can be represented by points. Pach and Tóth gave a construction that shows r(n) < n ^log8/log9 (Pach and Tóth 2009). They also stated that one can apply the Erdős–Szekeres theorem for convex sets in Pach and Tóth (Discrete Comput Geom 19:437–445, 1998) to obtain r(n) > log₁₆ n. In this note, we will show that r(n) > cn ^1/4 for some absolute constant c.     

Asset pricing and portfolio selection based on the multivariate extended skew-Student-<Emphasis Type="Italic">t</Emphasis> distribution

The returns on most financial assets exhibit kurtosis and many also have probability distributions that possess skewness as well. In this paper a general multivariate model for the probability distribution of assets returns, which incorporates both kurtosis and skewness, is described. It is based on the multivariate extended skew-Student-t distribution. Salient features of the distribution are described and these are applied to the task of asset pricing. The paper shows that the market model is non-linear in general and that the sensitivity of asset returns to return on the market portfolio is not the same as the conventional beta, although this measure does arise in special cases. It is shown that the variance of asset returns is time varying and depends on the squared deviation of market portfolio return from its location parameter. The first order conditions for portfolio selection are described. Expected utility maximisers will select portfolios from an efficient surface, which is an analogue of the familiar mean-variance frontier, and which may be implemented using quadratic programming.     

Stable distributions and the term structure of interest rates

In an abundance of cases asset returns fit stable distributions better than normal distributions. We examine a model where the term structure of interest rates follows a subordinate process directed by a stable process and indicate how to fix the parameters of the model.