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: pham@math.jussieu.fr

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.

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

1

Contents

1 Introduction

3

2 Financial decision-making and preferences

4

3 Dynamic programming and martingale methods : an illustration via a simple example

7

3.1 Solution via the dynamic programming approach . . . . . . . . . . . . . . . 8

3.2 Solution via the martingale approach . . . . . . . . . . . . . . . . . . . . . . 9

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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11

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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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17

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6 Conclusion

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2

1

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 draw the maximum possible utility from the results of the corresponding activities. This principle covers numerous and various situations of daily life. For example, imagine you are thinking of buying a house and are oﬀered two diﬀerent ones you can aﬀord. One close to your oﬃce, but without a garden and close to a motor way, the other one with a nice landscape but requiring a long distance to work everyday. The decision about which one is more convenient for you (or has your preference) is in principle a portfolio problem. In a ﬁnancial terminology, the problem of portfolio optimization of an investor trading in diﬀerent assets is to choose an optimal investment, that is how many shares of which asset he should hold at any trading time, in order to maximize some subjective (depending on his preferences) criterion relying on his total wealth and/or consumption.

The earliest approach to solving a portfolio problem is the so-called mean-variance approach pioneered by H. Markowitz [9] in a one-period decision model. It still has great importance in real-life applications, and is widely applied in the risk management departments of banks. The main reasons for this is being the simplicity with which the algorithm can be implemented, and that it requires no special knowledge on probability (only expectation and covariances of random variables are enough to know). Markowitz was awarded the 1990 Nobel prize in economics for the importance of his contribution on the mean-variance approach.

However, the main drawback of this approach is the static nature of the problem : after the decision concerning the allocation of initial wealth to the diﬀerent assets has been made at the beginning of the period, no further actions is allowed until the end of the period. Once the initial portfolio is chosen, the investor’s job is complete and his only feasible action is to watch the prices move without the possibility to intervene. This is an extreme oversimpliﬁcation of reality and totally...