Multivariate Option Pricing with Pair-Copulas

Abstract

We propose a copula-based approach to solve the option pricing problem in the risk-neutral setting and with respect to a structured derivative written on several underlying assets. Our analysis generalizes similar results already present in the literature but limited to the trivariate case. The main difficulty of such a generalization consists in selecting the appropriate vine structure which turns to be of D-vine type, contrary to what happens in the trivariate setting where the canonical vine is sufficient. We first define the general procedure for multivariate options and then we will give a concrete example for the case of an option written on four indexes of stocks, namely, the S&P 500 Index, the Nasdaq 100 Index, the Nasdaq Composite Index, and the Nyse Composite Index. Moreover, we calibrate the proposed model, also providing a comparison analysis between real prices and simulated data to show the goodness of obtained estimates. We underline that our pair-copula decomposition method produces excellent numerical results, without restrictive assumptions on the assets dynamics or on their dependence structure, so that our copula-based approach can be used to model heterogeneous dependence structure existing between market assets of interest in a rigorous and effective way.

Multivariate Joint Probability Function of Earthquake Ground Motion Prediction Equations Based on Vine Copula Approach

“…e structure of D-vine is defined by using equation 4, and the pair copula families have been restricted to the bivariate t copula for the reason of comparison. Since the five-dimensional t copula is nested in the D-vine copula structure, the likelihood ratio test can be performed between these two copulas [22]. e results of D-vine copula are reported in Table 2.…”

“…Multivariate elliptical copulas, such as normal copula and t copula, have been used for modeling multivariate dependence [22][23][24], including for earthquake GMPs [21,25]. However, multivariate normal copula is unable to capture extreme dependence because of its independent property in the tail region.…”

Multivariate Option Pricing with Gaussian Mixture Distributions and Mixed Copulas

“…Church (2011) used a similar idea and devised a copulabased algorithm for pricing path dependent basket options and demonstrated how the marginals selected can significantly affect the price of the options. Barban and Di Persio (2014); Berton and Mercuri (2021) presented an alternative to the Monte Carlo simulation method to price multivariate options by using a copula GARCH model. Hassane et al (2021) presented an approach to price options that provide for the modeling of financial asset returns to take the impact of extreme values into consideration by using a combination of two gaussian distributions and an extreme copula to model the returns' combined dependence structure.…”

Pricing Multivariate European Equity Option Using Gaussians Mixture Distributions and EVT-Based Copulas

“…For valuation in the multivariate framework, this riskneutral formula is a simple generalization (for example, [4,5] are used this generalization). Talponen and Viitasaari [6] recently gave the multivariate version of the univariate result.…”

“…However, all this work did not take into account the effects of extreme values in the marginal, which is not without effect on valuation (risk of overvaluation or undervaluation). However, there are other copula modeling approaches based on volatility dynamics as in van den Goorbergh et al [13], Bernard and Czado [14], and Barban and Di Persio [4]. e reader can consult them for full details on the literature on this approach.…”