Modern portfolio optimization

Abstract

[en] The objective of this thesis is to survey some of the many models studied on modern portfolio theory, one of the main branches of quantitative finance. The first part of this work is dedicated to covering some of the main results on convex optimization with special emphasis on the Lagrangian and the Karush-Kuhn-Tucker optimality conditions. The second and third chapter are dedicated to two of the first and most important optimization models: the Markowitz model and the Capital Asset Pricing Model (CAPM). These two models are of paramount importance as they are the building blocks upon which later developments stand. However these models are quite static in the sense that they only allow for one period of time so, in the fourth chapter we introduce two multi-period models. For simplicity we will only contemplate the case with one risk-free asset and one risky asset, although the ideas there exposed allow the incorporation of many risky assets. So far, all models assumed that there was only one price at which assets are sold and bought. In the final chapter we will extend the notion of optimal portfolio to the context of financial market with two prices (the bid and ask price).