A utility maximization approach to hedging in incomplete markets

In this paper we introduce the notion of portfolio optimization by maximizing expected local utility. This concept is related to maximization of expected utility of consumption but, contrary to this common approach, the discounted financial gains are consumed immediately. In a general continuous-time market optimal portfolios are obtained by pointwise solution of equations involving the semimartingale characteristics of the underlying securities price process. The new concept is applied to hedging problems in frictionless, incomplete markets.     

Robust utility maximization with limited downside risk in incomplete markets

In this article we consider the portfolio selection problem of an agent with robust preferences in the sense of Gilboa and Schmeidler [Itzhak Gilboa, David Schmeidler, Maxmin expected utility with non-unique prior, Journal of Mathematical Economics 18 (1989) 141–153] in an incomplete market. Downside risk is constrained by a robust version of utility-based shortfall risk. We derive an explicit representation of the optimal terminal wealth in terms of certain worst case measures which can be characterized as minimizers of a dual problem. This dual problem involves a three-dimensional analogue of f$f$-divergences which generalize the notion of relative entropy.     

Asymptotic pricing in large financial markets

The problem of hedging and pricing sequences of contingent claims in large financial markets is studied. Connection between asymptotic arbitrage and behavior of the α-quantile price is shown. The large Black–Scholes model is carefully examined.      

Asymptotic arbitrage in large financial markets with friction

In the modern version of arbitrage pricing theory suggested by Kabanov and Kramkov the fundamental financially meaningful concept is an asymptotic arbitrage. The ??real world?? large market is represented by a sequence of ??models?? and, though each of them is arbitrage free, investors may obtain non-risky profits in the limit. Mathematically, absence of the asymptotic arbitrage is expressed as contiguity of envelopes of the sets of equivalent martingale measures and objective probabilities. The classical theory deals with frictionless markets. In the present paper we extend it to markets with transaction costs. Assuming that each model admits consistent price systems, we relate them with families of probability measures and consider their upper and lower envelopes. The main result concerns the necessary and sufficient conditions for absence of asymptotic arbitrage opportunities of the first and second kinds expressed in terms of contiguity. We provide also more specific conditions involving Hellinger processes and give applications to particular models of large financial markets.     

Equivalent martingale measures for large financial markets in discrete time

We show that in a discrete-time large financial market the absence of certain asymptotic arbitrage opportunities is equivalent to the existence of martingale measures in a strong sense. We also consider the Arbitrage Pricing Model with stable random variables where we are able to give explicit necessary and sufficient conditions using market parameters.     

Incomplete financial markets and contingent claim pricing in a dual expected utility theory framework

This paper investigates the price for contingent claims in a dual expected utility theory framework, the dual price, considering arbitrage-free financial markets. A pricing formula is obtained for contingent claims written on n underlying assets following a general diffusion process. The formula holds in both complete and incomplete markets as well as in constrained markets. An application is also considered assuming a geometric Brownian motion for the underlying assets and the Wang transform as the distortion function.     

Parametric network utility maximization problem

We show how to solve the parametric utility maximization problem with a continuous parameter in a finite number of steps in order to obtain a solution with given accuracy. Also, we propose a new approach for the discretization of time for the parametric utility maximization problem with Lipschitz utility function. Some numerical results are provided.     

Expected utility maximization and capital budgeting subgoals

Summary It will be shown that the capital budgeting subgoals which generally are used to determine the optimal investment and financing program of a corporation are not necessarily consistent with the objective of maximizing the expected utility of the terminal wealth of its shareholders. This holds true also if there is no conflict of interest between the shareholders. The analysis is based on a portfolio model and a stock market equilibrium model.

---

Zusammenfassung Es wird gezeigt, daß die Zielfunktionen, mit denen in der Regel bei der Bestimmung des optimalen Investitionsprogramms einer Kapitalgesellschaft gearbeitet wird, nicht zwingend im Einklang stehen mit dem Oberziel der Maximierung des erwarteten Nutzens der Anteilseigner. Dies gilt auch dann, wenn kein Interessenkonflikt zwischen den Anteilseignern besteht. Die Analyse basiert auf einem Portfolio Modell und einem darauf aufbauenden Modell zur Bestimmung der Gleichgewichtskurse von Aktien.

---

---

Vorgel. v.:H. Schneeweiss     

On admissible strategies in robust utility maximization

The existence of optimal strategy in robust utility maximization is addressed when the utility function is finite on the entire real line. A delicate problem in this case is to find a ??good definition?? of admissible strategies to admit an optimizer. Under certain assumptions, especially a kind of time-consistency property of the set ${\mathcal{P}}$ of probabilities which describes the model uncertainty, we show that an optimal strategy is obtained in the class of those whose wealths are supermartingales under all local martingale measures having a finite generalized entropy with one of ${P\in\mathcal{P}}$ .     

Informational inefficiency in financial markets

The existence of an informational inefficiency in the equity market is identified by analysing information publicly available on the internet. A large volume of blog data is used for this purpose. Informational inefficiency is established by converting company-specific blog sentiment data into a trading strategy and analysing its performance. An information-based model that approximately replicates the strategy is developed to estimate the degree of information disparity. The result shows that an efficient internet search engine can considerably enhance market efficiency, as measured in terms of the information flow rate.     

Conjugate duality in problems of constrained utility maximization

We show that a simple and elegant method of Bismut [J. Math. Analysis Appl., 44 (1973), pp. 384–404] for applying conjugate duality to convex problems of Bolza adapts directly to problems of utility maximization with portfolio constraints in mathematical finance. This gives a straightforward construction of an associated dual problem together with Euler–Lagrange and transversality relations, which are then used to establish existence of optimal portfolios in terms of solutions of the dual problem. The approach is completely synthetic, and does not require the rather difficult a priori hypothesis of a fictitious complete market for unconstrained optimization, which has been the standard approach for synthesizing optimal portfolios in problems of utility maximization with trading constraints. It also complements a duality synthesis of Rogers [Lecture Notes in Mathematics, No. LNM-1814, Springer-Verlag, New York, 2003, pp. 95–131] and Klein and Rogers [Math. Finance, 17 (2007), pp. 225–247] for general problems of utility maximization with market imperfections.     

Complexity models in financial markets

In this approach, the complexity of the self-organizing microstructure of the stock exchange is explicitly taken into consideration: the process of offers and trades as well as the adjustment of individual expectations are modelled with help of a (stochastic) jump process. Its abilities are illustrated by modelling the continuous quotations of asset prices at an auction type stock exchange. The functional form of the transition (hazard) rates is chosen to reflect the individual preferences and expectations as well as the economic environment. The model is described in detail and examples of Monte Carlo simulation results are presented.     

Risk aversion asymptotics for power utility maximization

We consider the economic problem of optimal consumption and investment with power utility. We study the optimal strategy as the relative risk aversion tends to infinity or to one. The convergence of the optimal consumption is obtained for general semimartingale models while the convergence of the optimal trading strategy is obtained for continuous models. The limits are related to exponential and logarithmic utility. To derive these results, we combine approaches from optimal control, convex analysis and backward stochastic differential equations (BSDEs).     

A control approach to robust utility maximization with logarithmic utility and time-consistent penalties

We propose a stochastic control approach to the dynamic maximization of robust utility functionals that are defined in terms of logarithmic utility and a dynamically consistent convex risk measure. The underlying market is modeled by a diffusion process whose coefficients are driven by an external stochastic factor process. In particular, the market model is incomplete. Our main results give conditions on the minimal penalty function of the robust utility functional under which the value function of our problem can be identified with the unique classical solution of a quasilinear PDE within a class of functions satisfying certain growth conditions. The fact that we obtain classical solutions rather than viscosity solutions facilitates the use of numerical algorithms, whose applicability is demonstrated in examples.     

Arbitrage and utility maximization in market models with an insider

We study arbitrage opportunities, market viability and utility maximization in market models with an insider. Assuming that an economic agent possesses an additional information in the form of an \(\mathscr {F}_T\)-measurable discrete random variable G, we give criteria for the no unbounded profits with bounded risk property to hold, characterize optimal arbitrage strategies, and prove duality results for the utility maximization problem faced by the insider. Examples of markets satisfying NUPBR yet admitting arbitrage opportunities are provided. For the case when G is a continuous random variable, we consider the notion of no asymptotic arbitrage of the first kind (NAA1) and give an explicit construction for unbounded profits if NAA1 fails.     

Regularity properties in a state-constrained expected utility maximization problem

We consider a stochastic optimal control problem in a market model with temporary and permanent price impact, which is related to an expected utility maximization problem under finite fuel constraint. We establish the initial condition fulfilled by the corresponding value function and show its first regularity property. Moreover, we can prove the existence and uniqueness of an optimal strategy under rather mild model assumptions. This will then allow us to derive further regularity properties of the corresponding value function, in particular its continuity and partial differentiability. As a consequence of the continuity of the value function, we will prove a dynamic programming principle without appealing to the classical measurable selection arguments. This permits us to establish a tight relation between our value function and a nonlinear parabolic degenerated Hamilton–Jacobi–Bellman (HJB) equation with singularity. To conclude, we show a comparison principle, which allows us to characterize our value function as the unique viscosity solution of the HJB equation.     

Rough paths in idealized financial markets

This paper considers possible price paths of a financial security in an idealized market. Its main result is that the variation index of typical price paths is at most 2; in this sense, typical price paths are not rougher than typical paths of Brownian motion. We do not make any stochastic assumptions and only assume that the price path is right-continuous. The qualification “typical” means that there is a trading strategy (constructed explicitly in the proof) that risks only one monetary unit but brings infinite capital when the variation index of the realized price path exceeds 2. The paper also reviews some known results for continuous price paths.     

Constrained nonsmooth utility maximization without quadratic inf convolution

We address a constrained utility maximization problem in an incomplete market for a utility function defined on the whole real line. We extend current research in two directions, firstly we allow for constraints on the portfolio process. Secondly we prove our results without relying on the technique of quadratic inf convolution, simplifying the proofs in this area.