Dynamics of dependency: finite difference solutions to the Black-Scholes PDE with copulas
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Abstract
The copula function is a tool for modeling any form of dependency that may exist between several random variables. Given n random variables, there exists an n- dimensional copula that captures their dependence (Sklar's Theorem). In this paper, we present a method for obtaining a class of copulas that models the dependence between the underlyings of a bivariate option over time until the option expires. This class of copulas verifies the 2-dimensional Black-Scholes partial differential equation. Given the difficulty of obtaining an analytical solution to such an equation, We use a Finite Difference approximation to obtain an approximated solution. We show that the numerical scheme obtained is conditionally convergent. Numerical results are also presented using MATLAB software.