Optimal Investment With an Unbounded Random Endowment and Utility‐based Pricing

Abstract

This paper studies the problem of maximizing the expected utility of terminal wealth for a financial agent with an unbounded random endowment, and with a utility function which supports both positive and negative wealth. We prove the existence of an optimal trading strategy within a class of permissible strategies -those strategies whose wealth process is a supermartingale under all pricing measures with finite relative entropy. We give necessary and sufficient conditions for the absence of utility-based arbitrage, and for the existence of a solution to the primal problem.We consider two utility-based methods which can be used to price contingent claims. Firstly we investigate marginal utility-based price processes (MUBPP's). We show that such processes can be characterized as local martingales under the normalized optimal dual measure for the utility maximizing investor. Finally, we present some new results on utility indifference prices, including continuity properties and volume asymptotics for the case of a general utility function, unbounded endowment and unbounded contingent claims.

Multivariate Utility Maximization with Proportional Transaction Costs and Random Endowment

“…The first systematic study of a utility maximization problem with a (bounded) random endowment in a general frictionless semimartingale model is due to [CSW01], where the authors considered univariate utility functions and used the duality approach based on some ideas already developed in [KrS99]. Important contributions in the same direction have later been given, among others, in [HG04] and [OZ09], where the boundedness condition on the endowment is relaxed and replaced by weaker requirements (those in [OZ09], in particular, have inspired the ones which are employed in this paper). Duality methods in a utility maximization problem with transaction costs had been introduced for the first time in [CK96] in a diffusion market model with one risky asset, constant proportional transaction costs and no random endowment (for a more complete story we refer to the Introduction in [CO10]).…”

Optimal investment and contingent claim valuation in illiquid markets

“…Mathematical theory of optimal investment with liabilities has a long history; see e.g. [14,17,25,15,29,42,4] and the references there. The purpose of this section is to study basic properties of such problems in the presence of illiquidity effects.…”

A Note on Utility Maximization with Unbounded Random Endowment

“…Other Case. Yet another approach is proposed by [14]. There the problem (1.1) is considered under the assumption that there exists x 0 ; x 00 2 R and  0 ;  00 2 V such that…”