Portfolio Management of Copula-Dependent Assets Based on P(Y < X) Reliability Models: Revisiting Frank Copula and Dagum Distributions

Modern portfolio theory indicates that portfolio optimization can be carried out based on the mean-variance model, where returns and risk are represented as the average and variance of the historical data of the stock’s returns, respectively. Several studies have been carried out to find better risk proxies, as variance was not that accurate. On the other hand, fewer papers are devoted to better model/characterize returns. In the present paper, we explore the use of the reliability measure P(Y<X) to choose between portfolios with returns given by the distributions X and Y. Thus, instead of comparing the expected values of X and Y, we will explore the metric P(Y<X) as a proxy parameter for return. The dependence between such distributions shall be modelled by copulas. At first, we derive some general results which allows us to split the value of P(Y<X) as the sum of independent and dependent parts, in general, for copula-dependent assets. Then, to further develop our mathematical framework, we chose Frank copula to model the dependency between assets. In the process, we derive a new polynomial representation for Frank copulas. To perform a study case, we considered assets whose returns’ distributions follow Dagum distributions or their transformations. We carried out a parametric analysis, indicating the relative effect of the dependency of return distributions over the reliability index P(Y<X). Finally, we illustrate our methodology by performing a comparison between stock returns, which could be used to build portfolios based on the value of the the reliability index P(Y<X).

How is price explosivity triggered in the cryptocurrency markets?

We shed light on how the price explosivity characterising Bitcoin and other major cryptocurrencies is triggered, by employing the Quantile Self-Exciting Threshold Autoregressive (QSETAR) model. Our results for Bitcoin, Ripple, and Stellar reveal that the explosive behaviour originates from the extreme upper tails of the return distributions following a price increase in the preceding day. We do not find evidence of explositivity in the price of Litecoin.

Market sentiment and heterogeneous agents in an evolutive financial model

We study a financial market populated by heterogeneous agents, whose decisions are driven by “animal spirits”. Each agent may have either correct, optimistic or pessimistic beliefs about the fundamental value, which can change from time to time based on an evolutionary mechanism. The evolutionary selection of beliefs depends on a weighted evaluation of the general market sentiment perceived by the agents and on a profitability measure of the existent strategies. As the relevance given to the sentiment index increases, a herding phenomenon in agent behavior may occur and animal spirits can drive the market toward polarized economic regimes, which coexist and are characterized by persistent high or low levels of optimism and pessimism. This conduct is detectable from agents polarized shares and beliefs, which in turn influence the price level. Such polarized regimes can consist in stable steady states or can be characterized by endogenous dynamics, generating persistent alternating waves of optimism and pessimism, as well as return distributions displaying the typical features of financial time series, such as fat tails, excess volatility and multifractality. Moreover, we show that if the sentiment has no or low relevance on belief selection, those stylized facts are abated or are missing from the simulated time series.

Financial Return Distributions: Past, Present, and COVID-19

We analyze the price return distributions of currency exchange rates, cryptocurrencies, and contracts for differences (CFDs) representing stock indices, stock shares, and commodities. Based on recent data from the years 2017–2020, we model tails of the return distributions at different time scales by using power-law, stretched exponential, and q-Gaussian functions. We focus on the fitted function parameters and how they change over the years by comparing our results with those from earlier studies and find that, on the time horizons of up to a few minutes, the so-called “inverse-cubic power-law” still constitutes an appropriate global reference. However, we no longer observe the hypothesized universal constant acceleration of the market time flow that was manifested before in an ever faster convergence of empirical return distributions towards the normal distribution. Our results do not exclude such a scenario but, rather, suggest that some other short-term processes related to a current market situation alter market dynamics and may mask this scenario. Real market dynamics is associated with a continuous alternation of different regimes with different statistical properties. An example is the COVID-19 pandemic outburst, which had an enormous yet short-time impact on financial markets. We also point out that two factors—speed of the market time flow and the asset cross-correlation magnitude—while related (the larger the speed, the larger the cross-correlations on a given time scale), act in opposite directions with regard to the return distribution tails, which can affect the expected distribution convergence to the normal distribution.

Stochastic resonance of volatility influenced by price periodic information in financial market

General researches show that all kinds of random risk information and periodic information in the financial system are mainly transmitted to the asset price through influencing the volatility, thus impacting the whole market. So can the periodic information and random factors in the price be transmitted to the volatility in reverse and cause volatility changes? Hence, in this paper, we investigate the stochastic resonance of volatility which is influenced by price periodic information in financial market, based on our proposed periodic Brownian Motion model and absolute return volatility. The parameter estimation of the periodic Brownian Motion model is obtained by minimizing the mean square deviation between the theoretical and empirical return distributions for the CSI300 data set. The good agreements of the probability density functions of the price returns, realized volatility (RV) at 5 minutes, RV at 15 minutes and absolute return volatility between theoretical and empirical calculation are found. After simulating the absolute return volatility and signal power amplification (SPA) of volatility via periodic Brownian Motion model, the results indicated that (i) single and double inverse resonance phenomena can be observed in the function of SPA versus random information intensity or economic growth rate; (ii) multiple inverse resonance phenomena can be also observed for SPA versus frequency of periodic information. The results imply that the transmission of stochastic factors and periodic information is not only from the volatility to the price, but also from the price to the volatility.

On agricultural commodities’ extreme price risk

We show how fat tails in agricultural commodity returns arise endogenously from productivity shocks in a standard macroeconomic model. Using nearly ninety years of data, we show that the eight agricultural commodities in our sample exhibit fat-tailed return distributions. Statistical tests confirm the heavy-tailedness of price spikes for agricultural commodities. We apply extreme value theory to estimate the size and likelihood of price spikes in agricultural commodities. Back-testing verifies the validity of our risk assessment methodology.

Transformational Approach to Analytical Value-at-Risk for near Normal Distributions

In this paper, we extend the parametric approach of VaR estimation that is based upon the application of two transforms, one for handling skewness and other for kurtosis. These transformations restore normality to data when applied in succession. The transforms are well defined and offer an alternative to VaR models based on the variance–covariance approach. We demonstrate the application of the technique using three pairs of uncorrelated but negatively skewed and fat-tailed stock return distributions, one pair each from recent periods in US and international market, and one from the stressed period of US economic history. Furthermore, we extend the analysis to economic domain by calculating expected shortfalls and risk capital under different estimation methods. For the sake of completion, we compare the estimation results of normal and transformation methods to non-parametric historical simulation.