Portfolio and Optimization

Optimization methods in portfolio management and option hedging ∗
Huyˆn PHAM e Laboratoire de Probabilit´s et e Mod`les Al´atoires e e
CNRS, UMR 7599
Universit´ Paris 7 e e-mail: [e-mail] and Institut Universitaire de France

April 24, 2007

Abstract
These lecture notes give an introduction to modern, continuous-time portfolio management and option hedging. We present the stochastic control method to portfolio optimization, which covers Merton’s pioneering work. The alternative martingale approach is also exposed with a nice application on option hedging with value at risk criterion. ∗

Lectures for the CIMPA-IMAMIS school on mathematical finance, Hanoi, April-May 2007.

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Contents
1 Introduction

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2 Financial decision-making and preferences

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3 Dynamic programming and martingale methods : an illustration via a simple example
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3.1 Solution via the dynamic programming approach . . . . . . . . . . . . . . .
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3.2 Solution via the martingale approach . . . . . . . . . . . . . . . . . . . . . .
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4 Dynamic programming methods for portfolio optimization in time 4.1 Dynamic programming and Hamilton-Jacobi-Bellman equation
4.2 Merton’s portfolio selection problem . . . . . . . . . . . . . . .
4.3 Super-replication cost in an uncertain volatility model . . . . .

continuous.......
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5 Martingale approach to continuous-time portfolio problem
5.1 Utility maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Value at risk hedging criterion . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Conclusion

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Introduction

Portfolio management is a fundamental aspect in economics and finance. It is an all natural and important activity in our society for households, pension fund managers, as well as for government debt managers. One has got a certain amount of money and tries to use it in such a way that one can

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