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Non-monotone dependence modeling with copulas: an application to the volume-return relationship

Manfred Marvin Marchione, Fabio Baione

Abstract

This paper introduces an innovative method for constructing copula models capable of describing arbitrary non-monotone dependence structures. The proposed method enables the creation of such copulas in parametric form, thus allowing the resulting models to adapt to diverse and intricate real-world data patterns. We apply this novel methodology to analyze the relationship between returns and trading volumes in financial markets, a domain where the existence of non-monotone dependencies is well-documented in the existing literature. Our approach exhibits superior adaptability compared to other models which have previously been proposed in the literature, enabling a deeper understanding of the dependence structure among the considered variables.

Non-monotone dependence modeling with copulas: an application to the volume-return relationship

Abstract

This paper introduces an innovative method for constructing copula models capable of describing arbitrary non-monotone dependence structures. The proposed method enables the creation of such copulas in parametric form, thus allowing the resulting models to adapt to diverse and intricate real-world data patterns. We apply this novel methodology to analyze the relationship between returns and trading volumes in financial markets, a domain where the existence of non-monotone dependencies is well-documented in the existing literature. Our approach exhibits superior adaptability compared to other models which have previously been proposed in the literature, enabling a deeper understanding of the dependence structure among the considered variables.
Paper Structure (6 sections, 2 theorems, 37 equations, 6 figures, 1 table)

This paper contains 6 sections, 2 theorems, 37 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Copulas and measure preserving functions
  3. Non-monotonic dependence copulas
  4. Parameter estimation and model selection
  5. Data and results
  6. Conclusions

Key Result

Theorem 1

Let $X_1$ and $X_2$ be two random variables such that $X_2=\phi(X_1)$ where $\phi$ is a stricly monotonic function. Denote by $F_{X_1}$ and $F_{X_2}$ the cumulative distribution functions of $X_1$ and $X_2$ respectively. If $\phi$ is monotonically increasing then the cumulative distribution function If $\phi$ monotonically decreasing then

Figures (6)

  • Figure 1: Monte Carlo simulations of the pair $(U,g(V))$, where the variable $(U,V)$ has standard uniform marginals and a Gaussian copula with correlation $\rho$ and $g$ is the function (\ref{['scarsinipiecewiselinear']}) with parameters $\beta=0.3,\;\gamma=0.4,\;\delta=0.55,\;\epsilon=0.65,\;\zeta=0.9$. The simulation was carried out for different values of $\rho$ and for each of the graphs the number of simulated points is 1000.
  • Figure 2: Monte Carlo simulations of the pair $(U,g(V))$, where the variable $(U,V)$ has standard uniform marginals and a Gaussian copula with correlation $\rho$ and $g$ is the function (\ref{['scarsinipiecewiselinear']}) with parameters $\beta=\gamma=\delta=\epsilon=0.4$ and $\zeta=1$. The simulation was carried out for different values of $\rho$ and for each of the graphs the number of simulated points is 1000.
  • Figure 3: Monte Carlo simulations of the pair $(U,g(V))$, where the variable $(U,V)$ has standard uniform marginals and a Gaussian copula with correlation $\rho$ and $g$ is the function (\ref{['gausstypefn']}). The simulation was carried out for different values of $\rho$ and for each of the graphs the number of simulated points is 1000.
  • Figure 4: Some examples of functions that preserve the uniform distribution obtained through theorem \ref{['mainthm']} and formula (\ref{['g']}) for different choices of $f$: (a) $f(v)=\frac{1}{5}(2v^2+v+2)$ (b) $f(v)=\min\left(1,\;\frac{8}{15}v+\frac{3}{5}\right)$ (c) $f(v)=\min\left(\max\left(\frac{7}{8},\;\frac{1+v}{2}\right),\;\frac{v}{3}+\frac{3}{4}\right)$.
  • Figure 5: Graph of the points corresponding to the pseudo-observations of daily returns (on the $x$-axis) and daily trading volumes (on the $y$-axis) for the QQQ fund. The graph suggests the presence of a non-monotonic type of dependence.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • proof