Multivariate option pricing using copulae

Abstract

The complexity of financial products significantly increased in the past 10 years. In this paper, we investigate the pricing of basket options and more generally of complex exotic contracts depending on multiple indices. Our approach assumes that the underlying assets evolve as dependent GARCH(1, 1) processes. The dependence among the assets is modeled using a copula based on paircopula constructions. Unlike most previous studies on this topic, we do not assume that the dependence observed between historical asset prices is similar to the dependence under the risk-neutral probability. The method is illustrated with US market data on basket options written on two or three international indices.

Multivariate Option Pricing with Pair-Copulas

“…Such a problem can lead to different solutions depending on different setting. In our specific case, we are justified to consider only the Gaussian, the -Student, the Clayton and the Gumbel copulas since they are the most used types in financial applications; see, for example, [1,[5][6][7]. The particular copula is selected exploiting standard information criteria such as Akaike information criterion (AIC) and the Bayesian information criterion (BIC) respectively; see, for example, [11][12][13] and references therein, for each edge in each tree.…”

“…Since the marginal distributions, at a given time , are specified conditional to a common set of information, that is, with respect to the -algebra F , then we are allowed to exploit copula theory techniques to derive the joint conditional distribution. In what follows, inspired by [5,6], see also [1,7], we assume that the objective and the riskneutral copulas are the same.…”

“…where * , +1 are the Gaussian innovations under Q. Exploiting the transform defined in (5) it is possible to model the marginal behaviour of the marginal log-returns under the measure Q.…”

Default probability estimation via pair copula constructions

“…PCCs constitute a flexible and very appealing tool for financial analysis, see e.g. Vaz de Melo Mendes et al (2010), Min and Czado (2010), Allen et al (2013), Dißmann et al (2013), Bernard and Czado (2013), and reference therein. A collection of potentially different bivariate copulas is used to construct the joint distribution of interest via PCCs, allowing to represent different types and strengths of dependence in an easy way.…”

A Dynamic Copula Approach to Recovering the Index Implied Volatility Skew

“…Moreover, the nested structure of the C-DCC model allows us to disentangle the various sources that potentially contribute to the index skew, namely (i) alternative distributional assumptions on model innovations; (ii) uncorrelated and independent versus uncorrelated and (tail-)dependent innovations; (iii) conditionally heteroskedastic volatility coupled with symmetric versus leveraged conditional correlation dynamics. 1 Multivariate option pricing in GARCH models is also considered in Goorbergh et al (2005), Bernard and Czado (2010), and Rombouts and Stentoft (2011) though for much smaller dimensions than considered here.…”