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Multivariate Option Pricing with Pair-Copulas

DOI: 10.1155/2014/839204

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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. 1. Introduction In what follows we will consider a European option written on 4 assets. We will assume that the risk-neutral setting holds true, according to the framework defined in [1]; that is, the price of each of the four considered underlying assets only depends on its history; moreover, it is independent form the others past behaviour. Previous condition allows us to write the price of the above mentioned option as the following discounted value: where is the maturity time of the option, is a real positive parameter usually representing the interest rate given by a bank for our cash deposit and it is assumed to be constant over the whole option’s life, is the payoff of the option written on the four assets , whose prices, at any time , are for . We would like to underline that (1) is given directly under the risk-neutral (martingale) measure and it represents the (fair or no-arbitrage) price of the option with payoff determined by the so-called martingale approach, see, for example, [2, Section??5.4]. Under suitable assumptions, the price determined in (1) can be rewritten according to the expected value definition as follows: where is the joint density probability function of the underlying assets with respect to the risk-neutral probability measure . Our aim is to apply a

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