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To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.ejor.2005.10.055 Byline: Silvia Muzzioli (a), Huguette Reynaerts (b) Keywords: Fuzzy sets; Fuzzy linear systems; Binary tree model Abstract: Fuzzy linear systems of equations play a major role in various financial applications. In this paper we analyse a particular fuzzy linear system: the derivation of the risk neutral probabilities in a fuzzy binary tree. This system has previously been investigated and different solutions to different forms of the same system have been proposed. The aim of this paper is twofold. First, we highlight that the different solutions proposed, arise from different forms of the same system. Second, in order to find a unique vector solution for the system, we propose a practical algorithm that boils down to the solution of a non-linear optimization problem. Author Affiliation: (a) Department of Economics, University of Modena and Reggio Emilia, V.le Berengario 51, 41100 Modena, Italy (b) Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 S9, 9000 Gent, Belgium Article History: Received 20 April 2005; Accepted 12 October 2005

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.ejor.2004.07.009 Byline: Huguette Reynaerts (a), Michele Vanmaele (a), Jan Dhaene (b), Griselda Deelstra (c) Keywords: Comonotonicity; Asian options; Superhedging strategy Abstract: Inspired by the ideas of Rogers and Shi [J. Appl. Prob. 32 (1995) 1077], Chalasani et al. [J. Comput. Finance 1(4) (1998) 11] derived accurate lower and upper bounds for the price of a European-style Asian option with continuous averaging over the full lifetime of the option, using a discrete-time binary tree model. In this paper, we consider arithmetic Asian options with discrete sampling and we generalize their method to the case of forward starting Asian options. In this case with daily time steps, the method of Chalasani et al. is still very accurate but the computation can take a very long time on a PC when the number of steps in the binomial tree is high. We derive analytical lower and upper bounds based on the approach of Kaas et al. [Insurance: Math. Econ. 27 (2000) 151] for bounds for stop-loss premiums of sums of dependent random variables, and by conditioning on the value of underlying asset at the exercise date. The comonotonic upper bound corresponds to an optimal superhedging strategy. By putting in less information than Chalasani et al. the bounds lose some accuracy but are still very good and they are easily computable and moreover the computation on a PC is fast. We illustrate our results by different numerical experiments and compare with bounds for the Black and Scholes model [J. Pol. Econ. 7 (1973) 637] found in another paper [Bounds for the price of discretely sampled arithmetic Asian options, Working paper, Ghent University, 2002]. We notice that the intervals of Chalasani et al. do not always lie within the Black and Scholes intervals. We have proved that our bounds converge to the corresponding bounds in the Black and Scholes model. Our numerical illustrations also show that the hedging error is small if the Asian option is in the money. If the option is out of the money, the price of the superhedging strategy is not as adequate, but still lower than the straightforward hedge of buying one European option with the same exercise price. Author Affiliation: (a) Department of Applied Mathematics and Computer Science, Faculty of Sciences, Ghent University, Krijgslaan 281 S9, 9000 Ghent, Belgium (b) Department of Applied Economics, Faculty of Economics and Applied Economics, Catholic University Leuven, Naamsestraat 69, 3000 Leuven, Belgium (c) Department of Mathematics, ISRO and ECARES, Universite Libre de Bruxelles, CP 120, 1050 Brussels, Belgium