LASSO-type regularization in Bernstein copula and its application in finance

Abstract

In this thesis, we propose a nonparametric Bernstein copula model to address the issue of approximating copulas flexibly and accurately. The model is penalized using the reciprocal of the empirical Bernstein copula as the weight of adaptive LASSO. To overcome the ill-posed problem of nonparametric copula estimation, we use Bernstein polynomial sieves as the sieve space and estimate it through sieve maximum likelihood (SML). We conduct extensive Monte Carlo simulations to evaluate the proposed model's performance in various scenarios. Additionally, we perform an empirical analysis and successfully capture financial contagion between four major markets.

Nonparametric density estimation offers great flexibility in modeling complex data patterns, especially when the true distribution is unknown or challenging to model parametrically. Building on our nonparametric Bernstein copula model, a semiparametric approach is developed to enhance univariate density estimation. We introduce a three-step double selection method to study the performance of individual density estimation. Results indicate that the copula-based model accurately estimates margins, but it does not capture the linear dependency between margins.

In the context of option pricing, it is often assumed that the copula function under the risk-neutral measure is the same as that under the physical measure, which can lead to misspecification of true dependence structures. To address this, we apply a nonparametric copula-based GJR-GARCH approach with generalized error distribution (GED) innovation to evaluate European bivariate rainbow options. We modify the approach to accommodate the skewness and fat-tail properties of financial data using an explicit transformation formula. We consider six pairs of price indices from both developed and emerging markets with different dependence structures as underlying assets to observe the copula's effect on option pricing.