Hedging Dependence Risk with Spread Options via the Power Frank and Power Student <i>t</i> Copulas

Abstract

In this paper we consider the problem of pricing and hedging European derivatives written on two underlying assets, when individual marginal distributions are known. Our aim is twofold. First, we conduct a parallel analysis between implied volatility and implied correlation for spread options in order to analyze the available dependence models in terms of produced implied correlation smile and we propose asymmetric extensions of the Frank and Student t copulas, respectively named the Power Frank (PF) and Power Student t (PST) copulas. Second, we address the problem of hedging portfolios made of two-asset derivatives, of various payoffs and strikes, with a limited number of spread options. To do so, we propose two methods based on the PF and PST copulas that allow a projection of the different aspects of dependence risk onto spread options elementary trades. The proposed hedging methods for dependence risk are compared to alternatives in a numerical analysis in which we compute the hedges of realistic portfolios of two-asset options and the associated profit and loss occurring under different scenarios affecting the risk-neutral dependence structure. The method based on the PST copula is found to markedly outperform its alternatives, while remaining parsimonious.