Polytopes associated with lattices of subsets and maximising expectation\n of random variables Article Swipe
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· 2020
· Open Access
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· DOI: https://doi.org/10.48550/arxiv.2002.06253
· OA: W3122684934
The present paper originated from a problem in Financial Mathematics\nconcerned with calculating the value of a European call option based on\nmultiple assets each following the binomial model. The model led to an\ninteresting family of polytopes $P(b)$ associated with the power-set\n$\\mathcal{L} = \\wp\\{1,\\dots,m\\}$ and parameterized by $b \\in \\mathbb{R}^m$,\neach of which is a collection of probability density function on $\\mathcal{L}$.\nFor each non-empty $P(b)$ there results a family of probability measures on\n$\\mathcal{L}^n$ and, given a function $F \\colon \\mathcal{L}^n \\to \\mathbb{R}$,\nour goal is to find among these probability measures one which maximises (resp.\nminimises) the expectation of $F$. In this paper we identify a family of such\nfunctions $F$, all of whose expectations are maximised (resp. minimised under\nsome conditions) by the same {\\em product} probability measure defined by a\ndistinguished vertex of $P(b)$ called the supervertex (resp. the subvertex).\nThe pay-offs of European call options belong to this family of functions.\n
