Sublinear and Locally Sublinear Prices

1 Introduction The purpose of this work is to analyze extensions of linear pricing models which incorporate a great deal of characteristics of real life prices, which can be generally referred to as transaction costs. Linear prices can not account for the presence of frictions in the assets traded on the market, and the presence of prices that allow for arbitrages creates the possibility to set up trading strategies resulting in unbounded profits. The absence of any form of transaction costs and the possibility to scale the size of a trade without any effect on the price of the transaction are two assumptions that oversimplify the price system and can be relaxed. The properties of subadditivity and positive homogeneity that characterize sub- linear prices allow for the construction of a price system in which it is possible to differentiate between bid and ask prices, leading to a more realistic model in which arbitrages are more difficult because of frictions, and a weaker form of price incon- sistency arises, namely the possibility of convenient super-replications. We expand the concept of internal consistency of a price system to the case where the riskless asset is affected by frictions and impose conditions on prices quoted on the market so that arbitrages and super-replications are not allowed. While a sublinear price system represents an improvement over the standard lin- ear case, it still suffers from the limitation that derives from the property of positive homogeneity. The absence of any dependence of prices from the size of the trade, which is a corollary of positive homogeneity, automatically excludes any possible modeling of important issues such as liquidity of the securities and transparency of prices. In order to extend sublinear models we allow prices to vary at different thresholds determined by portfolios of securities traded on the market. In this way, prices are increasing with the size and direction of the trade, thus making the price system as a whole not positive homogeneous. We show that a model of this kind is characterized by sublinear price increments and derive the pricing functional as the maximum of a family of convex functionals, which in turn can be represented as maxima of affine functionals. We conclude with interpretations of this model based mainly on the possibility to accommodate the issues of liquidity, transparency, and market depth. The analysis is organized as follows: in Section 2 we make a brief introduction about linear prices, their usual representation, and the shortcomings of a linear pricing model. In Section 3 we show the mathematical properties of sublinear functionals. Section 4 contains a discussion about the possibility to extend a linear pricing model by introducing sublinear prices for non-replicable contingent claims. A fully sublinear model is presented in Section 5 and more insights into the problems of internal coherence of prices in the sublinear case can be found in Section 6, where the riskless asset is affected by frictions. In Section 7 we develop a model for locally sublinear prices, where the removal of positive homogeneity allows us to limit the possibility of arbitrages and convenient super-replications to a subset of the whole price system and also to introduce the issue of liquidity affecting the securities traded on the market. In the concluding remarks in Section 8 we provide some comments about possible developments of future research on this topic. 5