A new realized conditional autoregressive Value-at-Risk (VaR) framework is proposed, through incorporating a measurement equation into the original quantile regression model. The framework is further extended by employing various Expected Shortfall (ES) components, to jointly estimate and forecast VaR and ES. The measurement equation models the contemporaneous dependence between the realized measure (i.e., Realized Variance and Realized Range) and the latent conditional ES. An adaptive Bayesian Markov Chain Monte Carlo method is employed for estimation and forecasting, the properties of which are assessed and compared with maximum likelihood through a simulation study. In a comprehensive forecasting study on 1% and 2.5 % quantile levels, the proposed models are compared to a range of parametric, non-parametric and semi-parametric models, based on 7 market indices and 7 individual assets. One-day-ahead VaR and ES forecasting results favor the proposed models, especially when incorporating the sub-sampled Realized Variance and the sub-sampled Realized Range in the model.
A joint conditional autoregressive expectile and Expected Shortfall framework is proposed. The framework is extended through incorporating a measurement equation which models the contemporaneous dependence between the realized measures and the latent conditional expectile. Nonlinear threshold specification is further incorporated into the proposed framework. A Bayesian Markov Chain Monte Carlo method is adapted for estimation, whose properties are assessed and compared with maximum likelihood via a simulation study. One-day-ahead VaR and ES forecasting studies, with seven market indices, provi…
A new semi-parametric Expected Shortfall (ES) estimation and forecasting framework is proposed. The proposed approach is based on a two-step estimation procedure. The first step involves the estimation of Value-at-Risk (VaR) at different quantile levels through a set of quantile time series regressions. Then, the ES is computed as a weighted average of the estimated quantiles. The quantiles weighting structure is parsimoniously parameterized by means of a Beta weight function whose coefficients are optimized by minimizing a joint VaR and ES loss function of the Fissler-Ziegel class. The proper…
Although quantile regression to calculate risk measures has been widely established in the financial literature, when considering data observed at mixed--frequency, an extension is needed. In this paper, a model is suggested built on a mixed--frequency quantile regression to directly estimate the Value--at--Risk (VaR) and the Expected Shortfall (ES) measures. In particular, the low--frequency component incorporates information coming from variables observed at, typically, monthly or lower frequencies, while the high--frequency component can include a variety of daily variables, like market ind…
Autonomous AI Agents for Option Hedging: Enhancing Financial Stability through Shortfall Aware Reinforcement Learning
Cornell Ann S. Bowers College of Computing and Information Science, Cornell University（康奈尔大学安·索斯·博尔斯计算与信息科学学院）；Department of Mathematics, University of Texas at Austin（德克萨斯大学奥斯汀分校数学系）；School of Social Sciences, Nanyang Technological University（南洋理工大学社会科学学院）；Krieger School of Arts and Sciences, Johns Hopkins University（约翰霍普金斯大学克里eger艺术与科学学院）
We present a general framework for a comparative theory of variability measures, with a particular focus on the recently introduced one-parameter families of inter-Expected Shortfall differences and inter-expectile differences, that are explored in detail and compared with the widely known and applied inter-quantile differences. From the mathematical point of view, our main result is a characterization of symmetric and comonotonic variability measures as mixtures of inter-Expected Shortfall differences, under a few additional technical conditions. Further, we study the stochastic orders induce…
In this paper, we generalize the parametric delta-VaR method from portfolios with normally distributed risk factors to portfolios with elliptically distributed ones. We treat both the expected shortfall and the Value-at-Risk of such portfolios. Special attention is given to the particular case of a multivariate t-distribution.
The joint Value at Risk (VaR) and expected shortfall (ES) quantile regression model of Taylor (2017) is extended via incorporating a realized measure, to drive the tail risk dynamics, as a potentially more efficient driver than daily returns. Both a maximum likelihood and an adaptive Bayesian Markov Chain Monte Carlo method are employed for estimation, whose properties are assessed and compared via a simulation study; results favour the Bayesian approach, which is subsequently employed in a forecasting study of seven market indices and two individual assets. The proposed models are compared to…
Value at risk and expected shortfall are increasingly popular tail risk measures in the financial risk management field. Both academia and financial institutions are working to improve tail risk forecasts in order to meet the requirements of the Basel Capital Accord; it states that one purpose of risk management and measuring risk accuracy is, since extreme movements cannot always be avoided, financial institutions can prepare for these extreme returns by capital allocation, and putting aside the appropriate amount of capital so as to avoid default in times of extreme price or index movements.…
This paper presents non-parametric estimates of spectral risk measures applied to long and short positions in 5 prominent equity futures contracts. It also compares these to estimates of two popular alternative measures, the Value-at-Risk (VaR) and Expected Shortfall (ES). The spectral risk measures are conditioned on the coefficient of absolute risk aversion, and the latter two are conditioned on the confidence level. Our findings indicate that all risk measures increase dramatically and their estimators deteriorate in precision when their respective conditioning parameter increases. Results …
Worst-case bounds on the expected shortfall risk given only limited information on the distribution of the random variables has been studied extensively in the literature. In this paper, we develop a new worst-case bound on the expected shortfall when the univariate marginals are known exactly and additional expert information is available in terms of bivariate marginals. Such expert information allows for one to choose from among the many possible parametric families of bivariate copulas. By considering a neighborhood of distance $ρ$ around the bivariate marginals with the Kullback-Leibler di…
The contour maps of the error of historical resp. parametric estimates for large random portfolios optimized under the risk measure Expected Shortfall (ES) are constructed. Similar maps for the sensitivity of the portfolio weights to small changes in the returns as well as the VaR of the ES-optimized portfolio are also presented, along with results for the distribution of portfolio weights over the random samples and for the out-of-sample and in-the-sample estimates for ES. The contour maps allow one to quantitatively determine the sample size (the length of the time series) required by the op…
This paper proposes an innovative threshold measurement equation to be employed in a Realized-GARCH framework. The proposed framework incorporates a nonlinear threshold regression specification to consider the leverage effect and model the contemporaneous dependence between the observed realized measure and hidden volatility. A Bayesian Markov Chain Monte Carlo method is adapted and employed for model estimation, with its validity assessed via a simulation study. The validity of incorporating the proposed measurement equation in Realized-GARCH type models is evaluated via an empirical study, f…
Any optimization algorithm based on the risk parity approach requires the formulation of portfolio total risk in terms of marginal contributions. In this paper we use the independence of the underlying factors in the market to derive the centered moments required in the risk decomposition process when the modified versions of Value at Risk and Expected Shortfall are considered. The choice of the Mixed Tempered Stable distribution seems adequate for fitting skewed and heavy tailed distributions. The ensuing detailed description of the optimization procedure is due to the existence of analytical…
This paper develops a Bayesian framework for the realized exponential generalized autoregressive conditional heteroskedasticity (realized EGARCH) model, which can incorporate multiple realized volatility measures for the modelling of a return series. The realized EGARCH model is extended by adopting a standardized Student-t and a standardized skewed Student-t distribution for the return equation. Different types of realized measures, such as sub-sampled realized variance, sub-sampled realized range, and realized kernel, are considered in the paper. The Bayesian Markov chain Monte Carlo (MCMC) …
A new model framework called Realized Conditional Autoregressive Expectile (Realized-CARE) is proposed, through incorporating a measurement equation into the conventional CARE model, in a manner analogous to the Realized-GARCH model. Competing realized measures (e.g. Realized Variance and Realized Range) are employed as the dependent variable in the measurement equation and to drive expectile dynamics. The measurement equation here models the contemporaneous dependence between the realized measure and the latent conditional expectile. We also propose employing the quantile loss function as the…
In this paper, we generalize the parametric Delta-VaR methods from portfolios with elliptic distributed risk factors to portfolios with mixture of elliptically distributed ones. We treat both the Expected Shortfall and the Value-at-Risk of such portfolios. Special attention is given to the particular case of the mixture of Student-t distributions.
We study the feasibility and noise sensitivity of portfolio optimization under some downside risk measures (Value-at-Risk, Expected Shortfall, and semivariance) when they are estimated by fitting a parametric distribution on a finite sample of asset returns. We find that the existence of the optimum is a probabilistic issue, depending on the particular random sample, in all three cases. At a critical combination of the parameters of these problems we find an algorithmic phase transition, separating the phase where the optimization is feasible from the one where it is not. This transition is si…
Procyclicality of historical risk measure estimation means that one tends to over-estimate future risk when present realized volatility is high and vice versa under-estimate future risk when the realized volatility is low. Out of it different questions arise, relevant for applications and theory: What are the factors which affect the degree of procyclicality? More specifically, how does the choice of risk measure affect this? How does this behaviour vary with the choice of realized volatility estimator? How do different underlying model assumptions influence the pro-cyclical effect? In this pa…
A novel forecast combination and weighted quantile based tail-risk forecasting framework is proposed, aiming to reduce the impact of modelling uncertainty in tail-risk forecasting. The proposed approach is based on a two-step estimation procedure. The first step involves the combination of Value-at-Risk (VaR) forecasts at a grid of quantile levels. A range of parametric and semi-parametric models is selected as the model universe in the forecast combination procedure. The quantile forecast combination weights are estimated by optimizing the quantile loss. In the second step, the Expected Short…
The realized GARCH framework is extended to incorporate the two-sided Weibull distribution, for the purpose of volatility and tail risk forecasting in a financial time series. Further, the realized range, as a competitor for realized variance or daily returns, is employed in the realized GARCH framework. Further, sub-sampling and scaling methods are applied to both the realized range and realized variance, to help deal with inherent micro-structure noise and inefficiency. An adaptive Bayesian Markov Chain Monte Carlo method is developed and employed for estimation and forecasting, whose proper…
Conditional forecasts of risk measures play an important role in internal risk management of financial institutions as well as in regulatory capital calculations. In order to assess forecasting performance of a risk measurement procedure, risk measure forecasts are compared to the realized financial losses over a period of time and a statistical test of correctness of the procedure is conducted. This process is known as backtesting. Such traditional backtests are concerned with assessing some optimality property of a set of risk measure estimates. However, they are not suited to compare differ…
Analyzing selected cryptocurrencies spillover effects on global financial indices: Comparing risk measures using conventional and eGARCH-EVT-Copula approaches
