American options are classical financial derivative contracts which lead to free boundary problems. The objective of this article is to give some qualitative properties of the exercise region of American options by means of analytic techniques. We prove that the price of an American option is the unique viscosity solution of an obstacle problem. We also prove comparison principles and strict comparison principles. These results enable us to localize the exercise region and to prove the propagation of convexity for American options. As a result, we study the influence of the volatility parameter on the price of American options.
In this paper, we use copulas to define multivariate risk-neutral distributions. We can then derive general pricing formulas for multi-asset options and best possible bounds with given volatility smiles. Finally, we apply the copula framework to define 'forward-looking' indicators of the dependence function between asset returns.
IntroductionCopulas have been introduced in finance for risk management purposes. For derivatives pricing, Rosenberg [1999] proposes to use Plackett distributions for the following reason:
[...] This method allows for completely general marginal risk-neutral densities and is compatible with all univariate risk-neutral density estimation techniques. Multivariate contingent claim prices using this method are consistent with current market prices of univariate contigent claims.A Plackett distribution is actually a special case of the copula construction of multidimensional probability distribution. Cherubini and Luciano [2000] extend then Rosenberg's original work by using general copula functions. At the same time, Bikos [2000] uses the same framework to estimate multivariate RND for monetary policy purposes.Our paper follows these previous works. After defining multivariate risk-neutral distributions with copulas, we derive pricing formulas for some multi-asset options. We study then best possible bounds with given volatility smiles. Finally, we apply the copula framework to define 'forward-looking' indicators of the dependence function between asset returns. * We are very grateful to Jean-Frédéric Jouanin for his helpful comments.
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