communication: The relevance of geometry in optimal basket option bounds
speaker: Roberto Bavieraspeaker: Emanuele Nastasi
abstract: Optimal lower and upper bounds show interesting features even in the “elementary” BlackScholes framework of n assets, whose logprices are correlated via a correlation matrix ρ. These bounds can be obtained via a conditioning random variable Λ and, in general, are functions of the vector of correlations x between this variable and logprices. This vector x belongs to a ndimensional ellipsoid, a quadratic form determined by the inverse of the correlation matrix ρ. The identification of optimal x is crucial in numerical applications and in analytic approximations; in this paper we focus on nonnegative ρ, a financially relevant case that characterizes some liquid option classes (as Asian options) and most baskets with equity stocks. We show how optimal bounds are related to some characteristics in the geometry of the problem and in particular to {ρi}i=1,..n, the column vectors of ρ. In a nutshell the main results are: i) for the lower bound we prove the existence of an optimal solution on the part of the ellipsoid delimited by the positive linear span of {ρi}; we also show some sufficient conditions for uniqueness of the global maximum ii) for the ICUB upper bound, {ρi} are the only points where the bound has an angular point. In n dimensions for a ρ with a simple shape we prove that these points are local minima for the ICUB and this result looks to hold for a more general ρ. Furthermore for these points it is always possible to show that the associated partially exact comonotonic upper bound (PECUB) equals the ICUB.
timetable:
Fri 29 Jan, 9:00 - 10:30, Sala Stemmi
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