Determining Distribution for the Product of Random Variables by Using Copulas

Abstract

Determining distributions of the functions of random variables is one of the most important problems in statistics and applied mathematics because distributions of functions have wide range of applications in numerous areas in economics, finance, risk management, science, and others. However, most studies only focus on the distribution of independent variables or focus on some common distributions such as multivariate normal joint distributions for the functions of dependent random variables. To bridge the gap in the literature, in this paper, we first derive the general formulas to determine both density and distribution of the product for two or more random variables via copulas to capture the dependence structures among the variables. We then propose an approach combining Monte Carlo algorithm, graphical approach, and numerical analysis to efficiently estimate both density and distribution. We illustrate our approach by examining the shapes and behaviors of both density and distribution of the product for two log-normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas, and estimate some common measures including Kendall’s coefficient, mean, median, standard deviation, skewness, and kurtosis for the distributions. We found that different types of copulas affect the behavior of distributions differently. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. Our results are the foundation of any further study that relies on the density and cumulative probability functions of product for two or more random variables. Thus, the theory developed in this paper is useful for academics, practitioners, and policy makers.

Review on Efficiency and Anomalies in Stock Markets

“…We have developed some theories in covariance and copulas (see, for example, Egozcue et al (2009Egozcue et al ( , 2010Egozcue et al ( , 2011aEgozcue et al ( , 2011bEgozcue et al ( , 2011cEgozcue et al ( , 2012Egozcue et al ( , 2013, Bai et al (2009aBai et al ( , 2009bBai et al ( , 2011c, and Ly et al (2019aLy et al ( , 2019b). We have also conducted some analysis by using covariance and copulas (Tang et al 2014).…”

Return and Volatility Transmission between World-Leading and Latin American Stock Markets: Portfolio Implications

“…This study employs the BEKK-GARCH model to examine the linkages between the world-leading countries and the emerging Latin American stock markets. Extensions could include other models to examine the return and volatility spillover-for example, cointegration and causality (Lv et al 2019;Demirer et al 2019), Copulas (Ly et al 2019a(Ly et al , 2019bYuan et al 2020), Stochastic Dominance (Chiang et al 2008;Abid et al 2014;Guo et al 2017;Wong et al 2018), and many others. See, for example, Chang et al (2018), Woo et al (2020), and the references therein for more information.…”

Could Mergers Become More Sustainable? A Study of the Stock Exchange Mergers of NASDAQ and OMX

“…This paper investigates the stock exchange merger of NASDAQ with OMX and examines the sustainability of co-movement between the stock markets of OMX and NASDAQ, that could affect investors' profit and decision making in their investment, changing their trading strategies, and could affect market efficiency and create arbitrage opportunity, anomaly, and additional risk. Thus, extension of our paper could include studying co-movement of other series [55,[58][59][60][61][62][63][64][65][66][67][68][69][70][71], co-movement of using different trading strategies [47,[72][73][74][75], co-movement of making use of different anomalies [76][77][78], co-movement of investing in different markets [60,64,[79][80][81][82], sustainability of making use of different market conditions [66,83], and co-movement in different types of risk [84][85][86][87][88][89][90][91][92][93][94][95]…”