Abstract
Linear price systems, typically used to model “perfect” markets, are widely known not to accommodate most of the typical frictions featured in “actual” ones. Since some years, “proportional” frictions (taxes, bid-ask spreads, and so on) are modeled by means of sublinear price functionals, which proved to give a more “realistic” description. In this paper, we want to introduce two more classes of functionals, not yet widely used in Mathematical Finance, which provide a further improvement and an even closer adherence to actual markets, namely the class of granular functionals, obtained when the unit prices of traded assets are increasing w.r.t. the traded amount; and the class of star-shaped functionals, obtained when the average unit prices of traded assets are increasing w.r.t. the traded amount. A characterisation of such functionals, together with their relationships with arbitrages and other (more significant) market inefficiencies, is explored.
Highlights
One of the first and biggest concerns of Mathematical Finance is to study the prices of a suitable set of risky financial assets of any type, including stocks, indexed bonds, variable rate deposits, derivative securities, and so on
Financial assets are modeled as random variables on some state sets, which are supposed to be the same for every asset in the considered market
From a mathematical point of view, L is the subdifferential of π at 0. According to such a characterisation, if π does not allow for convenient super-hedgings, every φ ∈ L can be represented as the expected value with respect to a suitable measure
Summary
One of the first and biggest concerns of Mathematical Finance is to study the prices of a suitable set of risky financial assets of any type, including stocks, indexed bonds, variable rate deposits, derivative securities, and so on. This may be the case, for instance, when an agent can choose to buy an asset on several different markets, featuring different increasing unit prices: the purchase will be conducted in such a way that the overall price (or, which is the same, the average unit price) is as low as possible This leads to a totally new class of price functions, which we name star-shaped because their epigraph turns out to be a star-shaped set with respect to the origin, in the sense of Stewart & Tall (1983). We show that such functionals are the result of pricing by superhedging by means of assets whose average price is increasing, and show that such a requirement is a proper generalisation of the previous, convex case.
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