List of presentations and abstracts
(in alphabetical order of speaker's last name)

Sara Biagini (University of  Pisa)

Title:  Utility Maximization without  Reasonable Asymptotic Elasticity

Abstract:  For utility functions U finite only on the positive real line, Kramkov and Schachermayer showed that under a condition on U, the well- known Reasonable Asymptotic Elasticity, the associated utility maximization problem has a (unique) optimal solution, independently of the probabilistic model.What about the relaxed investor, whose utility does not satisfy RAE? This has been also addressed by Kramkov and Schachermayer, but the optimal solution is characterized only for sufficiently small initial endowments. Under a sufficient (and basically necessary) joint condition on the probabilistic model and  the utility, we show by relaxation and duality techniques that the maximization problem admits solution for any initial endowment. However, a singular part may pop up, that is the optimal investment may have a component which is concentrated on a set of probability zero. This singular part may fail to be unique. This is joint work with P. Guasoni

Andreas Hamel (Princeton)

Title: A duality theory for set--valued risk measures and applications

Abstract: Jouini et al. (Finance & Stochastics 8, 2004) proposed the concept of set–valued coherent risk measures in order to incorporate market frictions like exchange rates, transaction costs or liquidity bounds in the evaluation of the risk of a portfolio consisting of d ! 1 assets. The basic question is how to quantify the risk of a vector position (with d components) in terms of a cash vector (with m components where d " m in most cases).
We extend this concept to general risk measures with values in the set of all subsets of IRm being monotone and cash–invariant with respect to m currencies. We establish primal representation results in terms of acceptance sets as well as necessary and sufficient conditions for a set–valued risk measure to have finite values and to be (lower semi-) continuous.
For the convex case, we give a complete duality theory parallel to the scalar case including a penalty function representation. This theory is based on extensions of Convex Analysis to set–valued functions which are also new – including definitions of Fenchel conjugates for set–valued convex functions and a corresponding biconjugation theorem. The (scalar) expectation is replaced by a set–valued counterpart which seems to be a new construction in probability theory.
A list of examples is given along with ”standarized” procedures (primal and dual) how a known scalar risk easure can be transformed into a set–valued one. In many cases, there is more than one generalization (more or less risk averse) of one scalar risk measure. We shall present a set–valued counterpart for (negative) expectation, set–valued VaRs and AVaRs, generalizations of (negative) essential infimum and a set–valued entropic risk measure. Recent generalizations of scalar risk measures to vector– or set–valued ones for vector positions (Burgert & R¨uschendorf 2006, Embrechts & Puccetti 2006, among others) and so-called depth–trimmed regions (Cascos & Molchanov 2007) will turn out to be special cases of these examples.
Finally, we discuss optimization problems for set–valued risk measures like the problem of risk allocation in the presence of proportional transaction cost. In order to tackle the problem we introduce the set–valued counterpart of the infimal convolution and make use of its duality features.

Alexander Schied (Cornell University)

Title: Convex and nonlinear optimization problems for large traders

Abstract: We consider the problem of finding optimal liquidation strategies for a large portfolio in an illiquid market. Standard optimality criteria are the minimization of the expected liquidity costs or the maximization of the expected utility of the revenues. Depending on the model used to describe an illiquid market, this leads to a convex optimization problem or to even more general nonlinearities. The talk will in particular highlight the convexity aspects of this problem.

Mihai Sirbu (UT Austin)

Title:In which Financial Markets do  Mutual Fund Theorems hold true?

Abstract: The Mutual Fund Theorem (MFT) is considered in a general semimartingale financial market S with a finite time horizon T, where agents maximize expected utility of terminal wealth. The main results are:

1: Let N be the wealth process of the numeraire portfolio (i.e. the optimal portfolio for the log utility).
If any path-independent option with maturity T written on the numeraire portfolio can be replicated by trading only in N, then the (MFT) holds true for general utility functions, and the numeraire portfolio  may serve as mutual fund. This generalizes Merton's classical result on Black-Merton-Scholes markets. Conversely, under a supplementary weak completeness assumption, we show that the validity of the (MFT) for general utility functions implies the  replicability property for options on the numeraire portfolio described above.

2: If for a given class of utility functions (i.e. investors) the (MFT) holds true in all complete Brownian financial markets $S$, then all investors use the same utility function U, which must be of HARA type. This is a  result in the spirit of the classical work by Cass and Stiglitz.

This is joint work with Walter Schachermayer and Erik Taflin.

Mike Tehranchi (Cambridge University)

Title: Forward utility and consumption

Abstract: Recently, the notion of time-consistent utility functions has appeared in the mathematical finance literature.  In our framework, a forward utility is a family of adapted $(t,\omega)$-dependent utility functions which satisfy the dynamic programming principle for a Merton investment problem. Working in a fairly general (possibly incomplete) market, we present a dual characterization of the pure investment and mixed investment/consumption forward utility functions.

Mingxin Xu (UNC-Charlotte)

Title: Risk Minimizing Portfolio Optimization and Hedging with Conditional Value-at-Risk

Abstract: We look at the problem of how to find a dynamic optimal portfolio so that the Conditional Value-at-Risk (CVaR) is minimized under the condition where the returns are bounded.  CVaR is a coherent risk measure based on the popular VaR.  In a complete market setting, we derive the exact optimal conditions.  Then we provide applications in two classic complete market models: the Binomial model and the Black-Scholes model. In these cases, the procedures to find the optimal strategies are given with exact formulas.  Numerical results show, as expected, dynamic portfolio provide much lower CVaR risk than static portfolios.

Thaleia Zariphopoulou (University of Texas Austin)

Title: Bespoke portfolios and implied preferences

Abstract: In my talk I will discuss how the preferences of an investor can be inferred by his/her investment "wish-list". The analysis uses arguments from the forward utility approach and is applicable to a variety of models. Examples for the case of deterministic market price of risk will be presented.

Statistics & Applied Probability
University of California
Santa Barbara, California 93106-3110
(805) 893-2129