Abstract
In this article, we present an approach which allows taking into account the effect of extreme values in the modeling of financial asset returns and in the valorisation of associated options. Specifically, the marginal distribution of asset returns is modelled by a mixture of two Gaussian distributions. Moreover, we model the joint dependence structure of the returns using a copula function, the extremal one, which is suitable for our financial data, particularly the extreme values copulas. Applications are made on the Atos and Dassault Systems actions of the CAC40 index. Monte Carlo method is used to compute the values of some equity options such as the call on maximum, the call on minimum, the digital option, and the spreads option with the basket (Atos, Dassault systems) as underlying.
Highlights
Since the pioneering work of Black and Scholes [1] and Cox et al [2], option pricing has become a crucial topic in finance
Since the assumption of no arbitrage opportunity (NAO) in the markets is the basis of the fundamental results obtained in finance, it is considered by default. e advantage under this NAO assumption is that, associated with that of market completeness, there is a single risk-neutral probability for which the discounted flows are martingales
It is proved that the empirical distribution of financial asset returns has thicker tails than that of the Gaussian distribution. is indicates the presence of extreme values. is fact shows that the normal distribution does not make it possible to model rigorously the returns of financial assets because it does not take into account the extreme. is is the case with the method proposed by Black and Scholes [1]
Summary
Since the pioneering work of Black and Scholes [1] and Cox et al [2] (respectively, in the continuous and discrete case), option pricing has become a crucial topic in finance. Considering a European-type option on an underlying asset with a price St, strike K, and expiration T, Black and Scholes have made it possible to determine a formula for the price of such options under certain assumptions, the fundamentals of which are the lack of arbitrage opportunity and that on the price St of the asset underlying (St follows geometric Brownian motion), i.e., dSt μStdt + σStWt, (1). One of the most interesting results obtained in this direction on valuation is that of Breeden et al [3] It states that the second derivative (when it exists and is continuous) of the price of a standard option relative to the strike coincides with the risk-neutral density. If Dt is the price of a European option of an underlying asset with price Xt having for pay-off g(XT), T
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