Identities involving finite sums of products of hypergeometric functions and their duals have been studied since 1930s. Recently Beukers and Jouhet have used an algebraic approach to derive a very general family of duality relations. In this paper we provide an alternative way of obtaining such results. Our method is very simple and it is based on the non-local derangement identity.
Exponential functionals of Brownian motion have been extensively studied in financial and insurance mathematics due to their broad applications, for example, in the pricing of Asian options. The Black-Scholes model is appealing because of mathematical tractability, yet empirical evidence shows that geometric Brownian motion does not adequately capture features of market equity returns. One popular alternative for modeling equity returns consists in replacing the geometric Brownian motion by an exponential of a Lévy process. In this paper we use this latter model to study variable annuity guaranteed benefits and to compute explicitly the distribution of certain exponential functionals.where we have denoted (x) + = max(x, 0) andis the exponential functional of the process X. Since the no-arbitrage price of an Asian option in the Black-Scholes model is determined by the expected present value of its payoff under a risk-neutral probability measure, the key to the computation of Asian option price is the distribution of the exponential functional J T . There has been a vast amount of work in the literature devoted to the distribution of J T . To name a few, Yor [28] employs the Lamperti transformation relating the geometric Brownian motion and the exponential functional to a Bessel process. Linetsky [17] starts with an identity in distribution J t d = U t := e Xt t 0
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