Some new aspects of optimal portfolios and option pricing [Elektronische Ressource] / Martin Krekel

Sujets

• Mathematics

Informations

• Publié par: technische_universitat_kaiserslautern
• Publié le: 01 janvier 2003
• Nombre de lectures: 29
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Universitat Kaiserslautern
Fachbereich Mathematik
Some new aspects of
Optimal Portfolios and Option Pricing
Martin Krekel
Vom Fachbereich Mathematik
der Universit at Kaiserslautern
zur Verleihung des akademischen Grades
Doktor der Naturwissenschaften
(Doctor rerum naturalium, Dr. rer. nat.)
genehmigte Dissertation
1. Gutachter: Prof. Dr. Ralf Korn
2. Gutachter: Prof. Dr. Mogen Ste ensen
Datum der Disputation: 20.06.2003
D 386.Introduction
The main two problems of continuous-time nancial mathematics are option pricing
and portfolio optimization. The rst of these problems is concerned with valuing derivative
contracts on stocks (or other underlyings) which have a non-linear payo structure such as
all kind of options. The other important topic, portfolio optimization, consists of the search
for the best investment strategy of an investor who is trading securities at a nancial market.
In this thesis, various new aspects of the above major topics of nancial mathemat-
ics will be discussed. In all our considerations we will assume the standard di usion type
setting for securitiy prices which is today well-know under the term "Black-Scholes model".
This setting and the basic results of option pricing and portfolio optimization are surveyed
in the rst chapter.
The next three chapters deal with generalizations of the standard portfolio problem,
also know as "Merton’s problem". Here, we will always use the stochastic control approach
as introduced in the seminal papers by Merton (1969, 1971, 1990). Although thish
is known for some time now, there are a lot of natural generalizations of the problem which
are not treated in the literature.
One such problem is the very realistic setting of an investor who is faced with xed
monetary streams. More precisely, in addition to maximizing the utility from nal wealth
via choosing an investment strategy, the investor also has to ful ll certain consumption
needs (such as paying a monthly rent) that can be deterministic or even stochastic.Also the
opposite situation, an additional income stream (such as a payin scheme) can now be taken
into account in our portfolio optimization problem. We consider various such examples
and solve them on one hand via classical stochastic control methods (such as setting up a
corresponding Hamilton- Jacobi-Bellman equation and proving a corresponding veri cation
theorem (see Korn and Korn (2001)) and on the other hand show by means of a general
separation theorem how the problem solution can be reduced to that of well-examined
subproblems. This together with some numerical examples forms Chapter 2.
Chapter 3 is mainly concerned with the portfolio problem if the investor has di er-
ent lending and borrowing rates. Even more, the borrowing rate depends on the percentage
of his holdings which is already nanced by a credit. Again, this is a very natural problem
and is not yet treated in the literature in the form we consider. We give explicit solutions
(where possible) and numerical methods to calculate the optimal strategy in the cases of
log utility and HARA utility for three di eren t modelling approaches of the dependence of
the borrowing rate on the fraction of wealth nanced by a credit.
A further generalization of the standard Merton problem consists in considering si-
multaneously the possibilities for continuous and discrete consumption (with respect to
itime). In our general approach there is a possibility for assigning the di eren t consumption
times di eren t weights which is a generalization of the usual way of making them compa-
rable via discounting. To solve this problem some new veri cation theorems have to be set
up and have to be proved. Also, the martingale optimality principle of stochastic control
(see Korn (2003)) proves to be very usefull in this chapter and is adapted to the special
problems we are looking at. Again, all our ndings are illustrated by some numerical
examples.
The nal two chapters of this thesis look at numerical methods for calculating op-
tion prices. Although, the option pricing problem in a complete market setting such
as the one we are considering here is fully understood, there often remain numerical
problems with the only remaining task, the computation of the expectation of the
discounted nal option payo . Very often the payo of so-called exotic options is highly
complicated and can depend on the whole path of the underlying’s price over the whole
life time of the option. This makes it very di cult and sometimes impossible to have an
explicit analytical formula for the option price. In such a situation, numerical methods
are needed. Besides the classical candidates such as Monte Carlo simulation, tree meth-
ods or solving a corresponding partial di eren tial equation, typically methods which are
tailored to the exact speci cation of the option come into the game and prove to be e cien t.
Chapter 5 deals with the special case of pricing basket options. Here, the main
problem is not path-dependence but the multi-dimensionality which makes it impossible
to give usuefull analytical representations of the option price. We review the literature and
compare six di eren t numerical methods in a systematic way. Thereby we also look at the
in uence of various parameters such as strike, correlation, forwards or volatilities on the
performance of the di eren t numerical methods.
The problem of pricing Asian options on average spot with average strike is the
topic of Chapter 6. We here apply the bivariate normal distribution to obtain an approxi-
mate option price. This method proves to be very reliable and e cien t for the valuation of
di eren t variants of Asian options on average spot with average strike.
Acknowledgements
First and mostly I would like to thank my supervisor Prof. Dr. Ralf Korn, who made it
possible for me to write this thesis and always gave me helpful advice throughout the
whole creation process. Secondly I would like to thank Prof. Dr. Mogens Ste ensen, who
supported me with break-through ideas in the Optimal Portfolios chapters. Many other
people at the ITWM helped in quite various ways to nish this thesis. I therefore thank
Dr. Susanne Kruse, Dr. Holger Kraft, Tin-Kwai Man, Kalina Natcheva, Mesrop Janunts
and Johan de Kock.
iiContents
Introduction i
Table of Contents iii
List of Figures v
1 Preliminaries 1
1.1 The Economy and Some Basic De nitions . . . . . . . . . . . . . . . . . . . 1
1.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 The Hamilton Jacobi Bellman - Theorem . . . . . . . . . . . . . . . 11
1.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3 Pricing Derivatives with Martingale Methods . . . . . . . . . . . . . . . . . 21
1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Optimal Portfolios with Fixed Monetary Streams 30
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 The Model and Some Basic De nitions . . . . . . . . . . . . . . . . . . . . . 31
2.3 Problems with Fixed Consumption/Income: the HJB-Solution . . . . . . . . 32
2.3.1 Constant continuous consumption requirements . . . . . . . . . . . . 32
2.3.2 Lump Sum Consumption . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.3 Generalized and Income . . . . . . . . . . . . . . . . . 39
2.4 A Separation Theorem for Requirements . . . . . . . . . . . . . . . . . . . . 43
2.5 Numerical Illustration and Conclusions . . . . . . . . . . . . . . . . . . . . . 51
3 Optimal Portfolios with loan-dependent Interest Rates 55
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Logarithmic Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.1 Step function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.2 Frequency polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3.3 Logistic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4 HARA Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.1 Step function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.4.2 Frequency Polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
iii4 Optimal Continuous and Discrete Consumption 77
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 HARA Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Logarithmic Utility . . . .