Title: Optimization Problems in Pricing and Hedging Options

Abstract:

The purpose of this tutorial is to illustrate the most important concepts of the arbitrage pricing theory to an audience whose expertise is mostly focused in the field of optimization. The fundamental results on hedging and pricing of contingent claims will be derived through the use of basic tools of duality theory. An overview of the most significant applications of optimization's techniques to some option pricing problems will also be given.

Presentation for download.

Title: Robust Optimization in Finance

Abstract:

Many optimization problems are formulated using parameters whose true values are not known with certainty. Robust optimization refers to a modeling strategy for such problems that intends to obtain a solution that is guaranteed to perform adequately for all or most possible realizations of these uncertain parameters. During the last decade, an intensive study of robust formulations for many classes of optimization problems using various uncertainty structures have led to many successful algorithms and applications. We will review some of these results and pay special attention to current debates on the different interpretations of robustness such as robustness in constraints vs. objective function, absolute vs. relative robustness, adjustable robustness, etc. .

Optimization formulations of problems from financial mathematics often involve parameters such as returns, interest rates, and volatilities whose values will be realized in the future and therefore are uncertain. Or, these parameters may represent statistical quantities (means, covariances, etc.) associated with random events whose true distributions are unobservable. Robust optimization provides an ideal setting for addressing such uncertainties in a conservative fashion. We will review some recent applications of robust optimization approach to portfolio selection, risk management, as well as pricing and hedging of derivative securities.

Presentation for download: part 1 and part 2.