Multivariate Utility Maximization with Proportional Transaction Costs and Random Endowment

Abstract

In this paper we deal with a utility maximization problem at finite horizon on a continuous-time market with conical (and time varying) constraints (particularly suited to model a currency market with proportional transaction costs). In particular, we extend the results in [CO10] to the situation where the agent is initially endowed with a random and possibly unbounded quantity of assets. We start by studying some basic properties of the value function (which is now defined on a space of random variables), then we dualize the problem following some convex analysis techniques which have proven very useful in this field of research. We finally prove the existence of a solution to the dual and (under an additional boundedness assumption on the endowment) to the primal problem. The last section of the paper is devoted to an application of our results to utility indifference pricing.

Efficient Trading Strategies in Financial Markets with Proportional Transaction Costs

“…The choice of multivariate utility functions reflects the idea that the agents will not necessarily liquidate their positions to a single numeraire at the final date (which is realistic, in particular, on a currency market). This is coherent with the recent papers [1,3] dealing with optimal investment problem under frictions. Moreover, it allows us to rely upon the duality methods developed therein.…”

“…Remark 3.1 We observe that, thanks to Remark 3.3 in Benedetti and Campi [1], this theorem can be equivalently formulated in terms of a CPS Z instead of vector-valued finitely additive measures m in D.…”

“…As in the two papers [1,3], one could include the slightly more general case of agents willing to liquidate their final portfolios to the first k assets with k ≤ d. It suffices to consider utility functions of the form U (x 1 , . .…”

Optimal Investment with Random Endowments and Transaction Costs: Duality Theory and Shadow Prices

“…We note that the optimal investment problem defined on multi-asset accounts with random endowments has been studied by [2]. In order to apply the superhedging theorem in [3], however, [2] still works with admissible portfolios and their random endowments are assumed to satisfy E T ∈ L ∞ in order to guarantee the existence of the optimal solution. Due to the unboundedness assumption on random endowments in our framework, the definition of admissible portfolios is no longer suitable and needs to be modified since the constant lower bound will become an unnatural constraint.…”

A Note on Utility-Based Pricing in Models with Transaction Costs

“…Fortunately, Campi and Owen(2010) [5] and Benedetti and Campi(2011) [1] proved the existence of the solution of the utility maximization problem using the multivariate utility function with transaction costs described by the general bid-ask processes with jumps. We believe that our method to deduce the equilibrium is applicable in the more general setting, because the form of the solution of utility maximization of Campi and Owen(2010) [5] and Benedetti and Campi(2011) [1] is essentially the same as the one presented by Bouchard(2002) [3]. Our method to deduce the equilibrium is as follows; by specifying the utility function as an exponential function, we could deduce the clear relationship between the quantity of the contingent claim and the corresponding utility.…”