Tail Risk

Broken Symmetry of Stock Returns – a Modified Jones-Faddy Skew t-Distribution ArXiv ID: 2512.23640 “View on arXiv” Authors: Siqi Shao, Arshia Ghasemi, Hamed Farahani, R. A. Serota Abstract We argue that negative skew and positive mean of the distribution of stock returns are largely due to the broken symmetry of stochastic volatility governing gains and losses. Starting with stochastic differential equations for stock returns and for stochastic volatility we argue that the distribution of stock returns can be effectively split in two – for gains and losses – assuming difference in parameters of their respective stochastic volatilities. A modified Jones-Faddy skew t-distribution utilized here allows to reflect this in a single organic distribution which tends to meaningfully capture this asymmetry. We illustrate its application on distribution of daily S&P500 returns, including analysis of its tails. ...

Deep Hedging to Manage Tail Risk ArXiv ID: 2506.22611 “View on arXiv” Authors: Yuming Ma Abstract Extending Buehler et al.’s 2019 Deep Hedging paradigm, we innovatively employ deep neural networks to parameterize convex-risk minimization (CVaR/ES) for the portfolio tail-risk hedging problem. Through comprehensive numerical experiments on crisis-era bootstrap market simulators – customizable with transaction costs, risk budgets, liquidity constraints, and market impact – our end-to-end framework not only achieves significant one-day 99% CVaR reduction but also yields practical insights into friction-aware strategy adaptation, demonstrating robustness and operational viability in realistic markets. ...

Comparing Bitcoin and Ethereum tail behavior via Q-Q analysis of cryptocurrency returns ArXiv ID: 2507.01983 “View on arXiv” Authors: A. H. Nzokem Abstract The cryptocurrency market presents both significant investment opportunities and higher risks relative to traditional financial assets. This study examines the tail behavior of daily returns for two leading cryptocurrencies, Bitcoin and Ethereum, using seven-parameter estimates from prior research, which applied the Generalized Tempered Stable (GTS) distribution. Quantile-quantile (Q-Q) plots against the Normal distribution reveal that both assets exhibit heavy-tailed return distributions. However, Ethereum consistently shows a greater frequency of extreme values than would be expected under its Bitcoin-modeled counterpart, indicating more pronounced tail risk. ...

Predicting Stock Market Crash with Bayesian Generalised Pareto Regression ArXiv ID: 2506.17549 “View on arXiv” Authors: Sourish Das Abstract This paper develops a Bayesian Generalised Pareto Regression (GPR) model to forecast extreme losses in Indian equity markets, with a focus on the Nifty 50 index. Extreme negative returns, though rare, can cause significant financial disruption, and accurate modelling of such events is essential for effective risk management. Traditional Generalised Pareto Distribution (GPD) models often ignore market conditions; in contrast, our framework links the scale parameter to covariates using a log-linear function, allowing tail risk to respond dynamically to market volatility. We examine four prior choices for Bayesian regularisation of regression coefficients: Cauchy, Lasso (Laplace), Ridge (Gaussian), and Zellner’s g-prior. Simulation results suggest that the Cauchy prior delivers the best trade-off between predictive accuracy and model simplicity, achieving the lowest RMSE, AIC, and BIC values. Empirically, we apply the model to large negative returns (exceeding 5%) in the Nifty 50 index. Volatility measures from the Nifty 50, S&P 500, and gold are used as covariates to capture both domestic and global risk drivers. Our findings show that tail risk increases significantly with higher market volatility. In particular, both S&P 500 and gold volatilities contribute meaningfully to crash prediction, highlighting global spillover and flight-to-safety effects. The proposed GPR model offers a robust and interpretable approach for tail risk forecasting in emerging markets. It improves upon traditional EVT-based models by incorporating real-time financial indicators, making it useful for practitioners, policymakers, and financial regulators concerned with systemic risk and stress testing. ...

Multivariate Distributions in Non-Stationary Complex Systems I: Random Matrix Model and Formulae for Data Analysis ArXiv ID: 2412.11601 “View on arXiv” Authors: Unknown Abstract Risk assessment for rare events is essential for understanding systemic stability in complex systems. As rare events are typically highly correlated, it is important to study heavy-tailed multivariate distributions of the relevant variables, especially in the presence of non-stationarity. We use a generalized scalar product between correlation matrices to clearly demonstrate this non-stationarity. Further, we present a model that we recently put forward, which captures how the non-stationary fluctuations of correlations make the tails of multivariate distributions heavier. Here, we provide the resulting formulae including Gaussian or Algebraic features. Compared to our previous results, we manage to remove in the Algebraic cases one out of the two, respectively three, fit parameters which considerably facilitates applications. We demonstrate the usefulness of these results by deriving joint distributions for linear combinations of amplitudes and validating them with financial data. Furthermore, we explicitly work out the moments of our model distributions. In a forthcoming paper we apply the model to financial markets. ...

Elicitability and identifiability of tail risk measures ArXiv ID: 2404.14136 “View on arXiv” Authors: Unknown Abstract Tail risk measures are fully determined by the distribution of the underlying loss beyond its quantile at a certain level, with Value-at-Risk, Expected Shortfall and Range Value-at-Risk being prime examples. They are induced by law-based risk measures, called their generators, evaluated on the tail distribution. This paper establishes joint identifiability and elicitability results of tail risk measures together with the corresponding quantile, provided that their generators are identifiable and elicitable, respectively. As an example, we establish the joint identifiability and elicitability of the tail expectile together with the quantile. The corresponding consistent scores constitute a novel class of weighted scores, nesting the known class of scores of Fissler and Ziegel for the Expected Shortfall together with the quantile. For statistical purposes, our results pave the way to easier model fitting for tail risk measures via regression and the generalized method of moments, but also model comparison and model validation in terms of established backtesting procedures. ...

Sparse spanning portfolios and under-diversification with second-order stochastic dominance ArXiv ID: 2402.01951 “View on arXiv” Authors: Unknown Abstract We develop and implement methods for determining whether relaxing sparsity constraints on portfolios improves the investment opportunity set for risk-averse investors. We formulate a new estimation procedure for sparse second-order stochastic spanning based on a greedy algorithm and Linear Programming. We show the optimal recovery of the sparse solution asymptotically whether spanning holds or not. From large equity datasets, we estimate the expected utility loss due to possible under-diversification, and find that there is no benefit from expanding a sparse opportunity set beyond 45 assets. The optimal sparse portfolio invests in 10 industry sectors and cuts tail risk when compared to a sparse mean-variance portfolio. On a rolling-window basis, the number of assets shrinks to 25 assets in crisis periods, while standard factor models cannot explain the performance of the sparse portfolios. ...

Bitcoin versus S&P 500 Index: Return and Risk Analysis ArXiv ID: 2310.02436 “View on arXiv” Authors: Unknown Abstract The S&P 500 index is considered the most popular trading instrument in financial markets. With the rise of cryptocurrencies over the past years, Bitcoin has also grown in popularity and adoption. The paper aims to analyze the daily return distribution of the Bitcoin and S&P 500 index and assess their tail probabilities through two financial risk measures. As a methodology, We use Bitcoin and S&P 500 Index daily return data to fit The seven-parameter General Tempered Stable (GTS) distribution using the advanced Fast Fractional Fourier transform (FRFT) scheme developed by combining the Fast Fractional Fourier (FRFT) algorithm and the 12-point rule Composite Newton-Cotes Quadrature. The findings show that peakedness is the main characteristic of the S&P 500 return distribution, whereas heavy-tailedness is the main characteristic of the Bitcoin return distribution. The GTS distribution shows that $80.05%$ of S&P 500 returns are within $-1.06%$ and $1.23%$ against only $40.32%$ of Bitcoin returns. At a risk level ($α$), the severity of the loss ($AVaR_α(X)$) on the left side of the distribution is larger than the severity of the profit ($AVaR_{“1-α”}(X)$) on the right side of the distribution. Compared to the S&P 500 index, Bitcoin has $39.73%$ more prevalence to produce high daily returns (more than $1.23%$ or less than $-1.06%$). The severity analysis shows that at a risk level ($α$) the average value-at-risk ($AVaR(X)$) of the bitcoin returns at one significant figure is four times larger than that of the S&P 500 index returns at the same risk. ...