Title: Portfolio Optimization with Consumption and Trading Constraints

Abstract:

Practical portfolio management often includes a variety of constraints surrounding the consumption of resources and liquidity in certain asset classes. In isolation, each of these forms of constraints may have relatively little impact on portfolio allocations, but their combined effect can be substantial. This talk will describe various approaches to including these constraints and will present results on their joint effects, especially in terms of investments in alternative asset classes.

Title: Optimizing a Multi-Strategy Hedge Fund

Abstract:

Hedge fund managers have much greater flexibility to invest in novel investments and strategies than traditional portfolio managers. Our objective is to develop a modeling language, by which we can efficiently analyze/optimize a portfolio of hedge fund investment strategies. The primary focus is multi-stage stochastic programs. To illustrate the approach, we show the advantages of a multi-strategy fund as compared with the typical fund-of-hedge-funds.

Title: Risk Measures and Safeguarding in Optimization Under Uncertainty

Abstract:

Coping with the uncertainties in future outcomes is a fundamental theme of optimization in a stochastic environment. It enters in the treatment of constraints as well as the treatment of objectives.

In the field of stochastic programming, which has grown from the traditions of linear and quadratic programming, constraints on future outcomes have commonly been relaxed by penalty expressions, unless they can be satisfied almost surely through recourse actions. Probabilistic constraints, requiring that a condition only to be satisfied up to a given probability, have sometimes utilized instead, but with the drawback that convexity and even continuity in a problem formulation can be lost, except in special circumstances. Objectives have usually taken the form of maximizing expected utility or minimizing an expected cost which may come in part from constraint penalties. Some extensions involving information and entropy have also been explored.

In financial optimization, where uncertainties are likewise unavoidable, approaches other than stochastic programming have prevailed. Although traditional portfolio theory was focused on minimizing variance of return subject to a constraint on expected return, other schemes have more recently gained popularity. An important example is constraints and objectives based on the notion of value-at-risk, which relates closely to probabilistic constraints and unfortunately, therefore, suffers from similar mathematical shortcomings. Value-at-risk suffers even from financial inconsistencies, which have led to the axiomatic development of "coherent risk measures", including the robust alternative called conditional value-at-risk.

Title: Hedging under Gamma Constraints, Second Order BSDE's, and Fully Non-Linear PDE's

Abstract:

We provide a quasi-explicit solution to the super-replication problem with gamma constraints. In particular, the upper bound constraint on the gamma implies that the optimal strategy consists in hedging a conveniently face-lifted payoff function, while the lower bound induces an optimal stopping problem.

Motivated by this problem, we introduce the notion of second order backward stochastic differential equations for which we prove existence and uniqueness under some conditions. This result provides an extension of the Feynman-Kac representation theorem to the case of parabolic fully non-linear PDE's.

Title: Pricing Options in Incomplete Market

Abstract:

The paper considers a regression approach to pricing options in incomplete markets. The algorithm replicates an option by a portfolio consisting of an underlying security and a risk-free bond. We applied the linear regression framework and quadratic programming with linear constraints (input = sample paths of underlying security; output = table of option prices as function of time and price of the underlying security). Risk neutral processes or probabilities are not needed in this framework. We populated the model with historical prices of the underlying security ("massaged" to the present volatility). We evaluated numerical performance of the algorithm with several real life datasets: options on S&P500, on natural gas futures, and on crude oil futures.

Title: Financial Products with Guarantees: Applications, Models and Internet-Based Services

Abstract:

Endowments with a minimum guaranteed rate of return appear in insurance policies, pension plans and social security plans. In several cases, especially in the insurance industry, such endowments also participate in the business and receive bonuses from the firm's asset portfolio. In this paper we develop a scenario based stochastic optimization model for asset and liability management of participating insurance policies with minimum guarantees. The model allows the analysis of the tradeoffs facing an insurance firm in structuring its policies as well as the choices in covering their cost. The model is applied to the analysis of policies offered by insurance firms in Italy and the UK. While the optimized model results are in general agreement with current industry practices, inefficiencies are still identified and potential improvements are suggested.

The modeling tools developed for the management of insurance policies are also used to develop a web-based system for individual investors. Investors goals and risk profiles are addressed in an integrated fashion. The requirements for real-time modeling by the average investor must be reflected in the model, and this issue will be discussed as well. The practical experience with this model will be discussed.