Value at Risk (VaR)

A novel scaling approach for unbiased adjustment of risk estimators ArXiv ID: 2312.05655 “View on arXiv” Authors: Unknown Abstract The assessment of risk based on historical data faces many challenges, in particular due to the limited amount of available data, lack of stationarity, and heavy tails. While estimation on a short-term horizon for less extreme percentiles tends to be reasonably accurate, extending it to longer time horizons or extreme percentiles poses significant difficulties. The application of theoretical risk scaling laws to address this issue has been extensively explored in the literature. This paper presents a novel approach to scaling a given risk estimator, ensuring that the estimated capital reserve is robust and conservatively estimates the risk. We develop a simple statistical framework that allows efficient risk scaling and has a direct link to backtesting performance. Our method allows time scaling beyond the conventional square-root-of-time rule, enables risk transfers, such as those involved in economic capital allocation, and could be used for unbiased risk estimation in small sample settings. To demonstrate the effectiveness of our approach, we provide various examples related to the estimation of value-at-risk and expected shortfall together with a short empirical study analysing the impact of our method. ...

Generative Machine Learning for Multivariate Equity Returns ArXiv ID: 2311.14735 “View on arXiv” Authors: Unknown Abstract The use of machine learning to generate synthetic data has grown in popularity with the proliferation of text-to-image models and especially large language models. The core methodology these models use is to learn the distribution of the underlying data, similar to the classical methods common in finance of fitting statistical models to data. In this work, we explore the efficacy of using modern machine learning methods, specifically conditional importance weighted autoencoders (a variant of variational autoencoders) and conditional normalizing flows, for the task of modeling the returns of equities. The main problem we work to address is modeling the joint distribution of all the members of the S&P 500, or, in other words, learning a 500-dimensional joint distribution. We show that this generative model has a broad range of applications in finance, including generating realistic synthetic data, volatility and correlation estimation, risk analysis (e.g., value at risk, or VaR, of portfolios), and portfolio optimization. ...

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. ...

Combining Deep Learning and GARCH Models for Financial Volatility and Risk Forecasting ArXiv ID: 2310.01063 “View on arXiv” Authors: Unknown Abstract In this paper, we develop a hybrid approach to forecasting the volatility and risk of financial instruments by combining common econometric GARCH time series models with deep learning neural networks. For the latter, we employ Gated Recurrent Unit (GRU) networks, whereas four different specifications are used as the GARCH component: standard GARCH, EGARCH, GJR-GARCH and APARCH. Models are tested using daily logarithmic returns on the S&P 500 index as well as gold price Bitcoin prices, with the three assets representing quite distinct volatility dynamics. As the main volatility estimator, also underlying the target function of our hybrid models, we use the price-range-based Garman-Klass estimator, modified to incorporate the opening and closing prices. Volatility forecasts resulting from the hybrid models are employed to evaluate the assets’ risk using the Value-at-Risk (VaR) and Expected Shortfall (ES) at two different tolerance levels of 5% and 1%. Gains from combining the GARCH and GRU approaches are discussed in the contexts of both the volatility and risk forecasts. In general, it can be concluded that the hybrid solutions produce more accurate point volatility forecasts, although it does not necessarily translate into superior VaR and ES forecasts. ...

Theoretical Economics as Successive Approximations of Statistical Moments ArXiv ID: 2310.05971 “View on arXiv” Authors: Unknown Abstract This paper studies the links between the descriptions of macroeconomic variables and statistical moments of market trade, price, and return. The randomness of market trade values and volumes during the averaging interval Δ results in the random properties of price and return. We describe how averages and volatilities of price and return depend on the averages, volatilities, and correlations of market trade values and volumes. The averages, volatilities, and correlations of market trade, price, and return can behave randomly during the long interval Δ2»Δ. To describe their statistical properties during the long interval Δ2, we introduce the secondary averaging procedure of trade, price, and return. We explain why, in the coming years, predictions of market-based probabilities of price and return will be limited by Gaussian distributions. We discuss the roots of the internal weakness of the commonly used hedging tool, Value-at-Risk, that cannot be solved and remains the source of additional risks and losses. One should consider theoretical economics as a set of successive approximations, each of which describes the next array of the n-th statistical moments of market trades, price, return, and macroeconomic variables, which are repeatedly averaged during the sequence of increasing time intervals. ...

Sizing Strategies for Algorithmic Trading in Volatile Markets: A Study of Backtesting and Risk Mitigation Analysis ArXiv ID: 2309.09094 “View on arXiv” Authors: Unknown Abstract Backtest is a way of financial risk evaluation which helps to analyze how our trading algorithm would work in markets with past time frame. The high volatility situation has always been a critical situation which creates challenges for algorithmic traders. The paper investigates different models of sizing in financial trading and backtest to high volatility situations to understand how sizing models can lower the models of VaR during crisis events. Hence it tries to show that how crisis events with high volatility can be controlled using short and long positional size. The paper also investigates stocks with AR, ARIMA, LSTM, GARCH with ETF data. ...

Real-time VaR Calculations for Crypto Derivatives in kdb+/q ArXiv ID: 2309.06393 “View on arXiv” Authors: Unknown Abstract Cryptocurrency market is known for exhibiting significantly higher volatility than traditional asset classes. Efficient and adequate risk calculation is vital for managing risk exposures in such market environments where extreme price fluctuations occur in short timeframes. The objective of this thesis is to build a real-time computation workflow that provides VaR estimates for non-linear portfolios of cryptocurrency derivatives. Many researchers have examined the predictive capabilities of time-series models within the context of cryptocurrencies. In this work, we applied three commonly used models - EMWA, GARCH and HAR - to capture and forecast volatility dynamics, in conjunction with delta-gamma-theta approach and Cornish-Fisher expansion to crypto derivatives, examining their performance from the perspectives of calculation efficiency and accuracy. We present a calculation workflow which harnesses the information embedded in high-frequency market data and the computation simplicity inherent in analytical estimation procedures. This workflow yields reasonably robust VaR estimates with calculation latencies on the order of milliseconds. ...

The Effect of COVID-19 on Cryptocurrencies and the Stock Market Volatility – A Two-Stage DCC-EGARCH Model Analysis ArXiv ID: 2307.09137 “View on arXiv” Authors: Unknown Abstract This research examines the correlations between the return volatility of cryptocurrencies, global stock market indices, and the spillover effects of the COVID-19 pandemic. For this purpose, we employed a two-stage multivariate volatility exponential GARCH (EGARCH) model with an integrated dynamic conditional correlation (DCC) approach to measure the impact on the financial portfolio returns from 2019 to 2020. Moreover, we used value-at-risk (VaR) and value-at-risk measurements based on the Cornish-Fisher expansion (CFVaR). The empirical results show significant long- and short-term spillover effects. The two-stage multivariate EGARCH model’s results show that the conditional volatilities of both asset portfolios surge more after positive news and respond well to previous shocks. As a result, financial assets have low unconditional volatility and the lowest risk when there are no external interruptions. Despite the financial assets’ sensitivity to shocks, they exhibit some resistance to fluctuations in market confidence. The VaR performance comparison results with the assets portfolios differ. During the COVID-19 outbreak, the Dow (DJI) index reports VaR’s highest loss, followed by the S&P500. Conversely, the CFVaR reports negative risk results for the entire cryptocurrency portfolio during the pandemic, except for the Ethereum (ETH). ...

Failure of Fourier pricing techniques to approximate the Greeks ArXiv ID: 2306.08421 “View on arXiv” Authors: Unknown Abstract The Greeks Delta and Gamma of plain vanilla options play a fundamental role in finance, e.g., in hedging or risk management. These Greeks are approximated in many models such as the widely used Variance Gamma model by Fourier techniques such as the Carr-Madan formula, the COS method or the Lewis formula. However, for some realistic market parameters, we show empirically that these three Fourier methods completely fail to approximate the Greeks. As an application we show that the Delta-Gamma VaR is severely underestimated in realistic market environments. As a solution, we propose to use finite differences instead to obtain the Greeks. ...

Modeling and evaluating conditional quantile dynamics in VaR forecasts ArXiv ID: 2305.20067 “View on arXiv” Authors: Unknown Abstract We focus on the time-varying modeling of VaR at a given coverage $τ$, assessing whether the quantiles of the distribution of the returns standardized by their conditional means and standard deviations exhibit predictable dynamics. Models are evaluated via simulation, determining the merits of the asymmetric Mean Absolute Deviation as a loss function to rank forecast performances. The empirical application on the Fama-French 25 value-weighted portfolios with a moving forecast window shows substantial improvements in forecasting conditional quantiles by keeping the predicted quantile unchanged unless the empirical frequency of violations falls outside a data-driven interval around $τ$. ...