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Fakultat fur Mathematik und

Wirtschaftswissenschaften

Robust consumption-investment problems

with stochastic coe cients

Dissertation

zur Erlangung des Doktorgrades Dr. rer. nat.

der Fakult at fur Mathematik und Wirtschaftswissenschaften

der Universit at Ulm

vorgelegt von

Christoph Wopperer

aus Erlangen

Ulm, November 2010Amtierender Dekan: Prof. Dr. Werner Kratz

1. Gutachter: Prof. Dr. Ulrich Rieder

2.hter: Prof. Dr. Rudiger Kiesel

Tag der Promotion: 14.02.2011Abstract

In this thesis we consider robust consumption-investment problems in a complete di u-

sion market with stochastic coe cients. We assume that the market price of risk process

is unknown. The investor tries to maximize his expected utility under the worst-case

parameter con guration. To solve robust consumption-investment problems, we derive

a stochastic version of the Bellman-Isaac equations for di erential games from the mar-

tingale optimality principle. A formal connection between a solution of these equations

and the robust optimal value function is established by a veri cation theorem. We are

able to solve the Bellman-Isaac equations for power, exponential and logarithmic utility.

In this way we can characterize a robust optimal consumption-investment strategy and a

worst-case market price of risk process in terms of the solution of a backward stochastic

di erential equation ( Bsde). The solution of this Bsde can be explicitly computed in

case of deterministic coe cients. It is given by the unique solution of a partial di erential

equation for the popular model of coe cients driven by a factor process.

Keywords: optimal consumption, stochastic Bellman-Isaac equations, backward stochas-

tic di erential equations, factor model

Zusammenfassung

In dieser Arbeit analysieren wir robuste Konsum-Investitionsprobleme in einem vollst an-

digen Di usionsmarkt mit stochastischen Koe zienten. Es wird angenommen dass der

Marktpreis des Risikos dem Investor nicht bekannt ist. Dieser versucht seinen erwarteten

Nutzen unter der fur ihn schlechtesten Parameterkon guration zu maximieren. Um ro-

buste Konsum-Investitionsprobleme zuosen,l leiten wir ausgehend von dem Martingalop-

timalit atsprinzip eine stochastische Variante der Bellman-Isaac Gleichungen fur Di eren-

tialspiele her. Ein formeller Zusammenhang zwischen einer L osung dieser Gleichungen und

der robust optimalen Wertfunktion wird durch ein Veri kationstheorem hergestellt. Im

Fall von power, exponentiellem und logarithmischem Nutzen k onnen wir die Bellman-Isaac

Gleichungenosen.l So k onnen wir eine optimale Konsum-Investitionsstrategie und einen

worst-case Marktpreis des Risikos mit Hilfe der L osung einer stochastischen Ruc kw arts-

gleichung charakterisieren. Die L osung dieser Ruc kw artsgleichung kann im Fall von deter-

ministischen Koe zienten explizit angegeben werden. In den in der Literatur popul aren

Faktormodellen ist die L osung der stochastischen Ruc kw artsgleichung durch die eindeutige

L osung einer partiellen Di erentialgleichung gegeben.

Schlagw orter : optimaler Konsum, stochastische Bellman-Isaac Gleichungen, stochastis-

che Ruckw artsgleichungen, FaktormodellContents

1 Introduction 1

1.1 Motivation and formulation of the problem . . . . . . . . . . . . . . . . . . 1

1.2 Outline and contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Maxmin problems, BSDEs and BMO-martingales 9

2.1 Some notions about maxmin problems . . . . . . . . . . . . . . . . . . . . 9

2.2 Basic results about backward stochastic di erential equations . . . . . . . . 14

2.2.1 Backward stochastic di erential equations with Lipschitz generator 15

2.2.2 BSDEs with quadratic growth . . . . . . . . . . . . . . . . . . . . . 17

2.2.3 Feynman-Kac theorems . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 BMO-martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Robust consumption-investment problems 25

3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 The martingale optimality principle . . . . . . . . . . . . . . . . . . . . . . 29

3.3 The stochastic Bellman-Isaac equations . . . . . . . . . . . . . . . . . . . . 33

3.4 Optimality shown by the comparison theorem . . . . . . . . . . . . . . . . 45

4 Special utility functions 49

4.1 Power utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Exponential utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3 Logarithmic utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Coe cients driven by a stochastic factor process 77

5.1 Power utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2 Exponential utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3 Logarithmic utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Bibliography 89

v1 Introduction

1.1 Motivation and formulation of the problem

The main focus of this thesis lies in nding optimal consumption and investment strategies

for portfolio optimization problems with an additional source of uncertainty.

Historically seen, the rst to address the question \how to invest optimally" in a mathe-

matical way, was Harry Markowitz in his seminal paper [Mar52]. He considered a single-

period problem that already contained all the features inherent in any optimal investment

problem. The problem was essentially a static optimization problem subject to a volatility

constraint. The solution could thus be found by means of quadratic programming.

The next major step was taken by Merton [Mer71] who considered dynamic problems.

Namely, he analyzed a continuous-time market model where stock prices are modeled by a

geometric Brownian motion with an investor who wants to maximize his expected utility

from intermediate consumption and terminal wealth. From a mathematical point of view,

the investment problem could then be solved with the method of dynamic programming.

The investment problems considered by Markowitz and Merton can be subdivided into

three mathematical components. The rst one is the description of a market model. More

precisely, a stochastic model that characterizes the time evolution of the di erent assets

under consideration has to be speci ed. Then, feasible strategies have to be de ned. The

investor can trade continuously or only at discrete times, he might be allowed to consume

parts or all of his wealth or not, and he might face additional practical restrictions for

example with respect to short sales or tradeable asset classes and so on. Finally, the

investor’s objective has to be formulated in terms of a mathematical optimization problem.

Mostly, this is done by using functionals that value the expected utility for a given strategy.

Obviously, all these subproblems require sophisticated modeling. The main focus of this

thesis is a closer analysis of the description of the market model. There are numerous

possible choices for the market model. Not only the investor’s asset universe has to be

limited, also the dynamics of the assets have to be speci ed.

However, classical portfolio optimization problems completely ignore the question where

to get the stochastic model for the behavior of the assets from: they postulate one and

compute the optimal consumption and investment decisions given that this model ad-

equately re ects real dynamics and given that the model parameters are fully known.

While the choice of a model seems natural, the availability of its parameters is a critical

assumption. In reality, model parameters have to be estimated from out-of-sample data in

a step preceding the optimization. It is clear that realistic portfolio optimization also has

to include the choice of the model and the estimation of its parameters. The interaction

between the choice of the model, the estimation of its parameters and the computation

of an optimal strategy is depicted schematically in Figure 1.1.

1Estimation Step Optimization Step

Input consumption-investmentestimation of model

historical data policy that is optimalparameters

according to model

possibly update of model

parameters based on

current observations

setting up parametric model

for the asset price behavior

0 T time

Figure 1.1: Estimation and optimization for portfolio optimization problems

As far as the choice of the model is concerned, the critical issue normally consists in

specifying a model for the returns of the stock prices. Indeed, return behavior is not

entirely understood and estimated returns are subject to large estimation errors. Unfor-

tunately, even small perturbations from a set of reference model parameters might lead

to drastically di erent optimal asset allocations.

In this thesis, we will assume that a model for the behavior of the stocks is chosen but that

its parameters are not exactly known. As an illustration of the di culty of estimating

returns from historical data consider the following example: suppose we observe the prices

of d shares evolving according to

1 d TdS = diag(S )( dt +dW ); S = (s ;:::;s ) ;t t t 0 0 0

where W is a d-dimensional Brownian motion and has full rank d. Here, diag(S ) ist t

1 2 dthe diagonal matrix with entriesS ;S ;:::;S . If is constant but unknown, then (event t t

if the volatility matrix is known) can be estimated in every xed nite time interval

only up to a certain precision depending on the length of the interval and the volatility

matrix only:

Lemma 1 Suppose we are given a Black-Scholes market with known and unknown .

Then, based on discrete observations (t;S ),i = 0;:::;n of thed-dimensional stock pricei ti

dprocess, a maximum likelihood estimator for the drift 2 IR is given by ^ with

k klog(S ) log(S ) 1t tk n 20^ = + k k ;k

t t 2n 0

kwhere denotes the k-th row of the volatility matrix , and ^ the k-th entry of ^.k

Furthermore, this estimator has the distribution

T

^N ; :

t tn 0

k k k

Proof LetS := log(S ) log(S ), then under the assumption of a Black-Scholest t ti i i 1

T kmarket,S is normally distributed,S N ~(t t ); (t t ) ; where ~ =t t i i 1 i i 1i i

1k 2 k k , k = 1;:::;d. Therefore, the likelihood of the observed data isk2

!

n T 1X1 ( )T

L()/ exp (S ~(t t )) (S ~(t t )) ;t i i 1 t i i 1i i2 t ti i 1

i=1

2where/ denotes the density up to a normalizing constant. The rst order condition

@L = 0 gives the maximizer ^, with

@

Pn kS 1i=1 tk i 2^ =P + k kkn

t t 2i i 1i=1

k klog(S ) log(S ) 1t tn 0 2= + k k :k

t t 2n 0

TFurthermore, sinceS N ~(t t ); (t t ) , it holds thatt i i 1 i i 1i

P n TS tii=1 N ;~

t t t tn 0 n 0

Tand that ^N ( ; ) as claimed. 2

t tn 0

The fact that it is very hard to estimate the drift of the stock price process has al-

ready received much attention in modern literature on portfolio optimization and di er-

ent approaches emerged for handling this additional source of uncertainty. The naive

approach is to choose a model for the dynamic behavior of the drift, estimate its parame-

ters and plug the point estimates into the optimization problem. This leads to the classical

consumption-investment problem where any uncertainty of the estimates is ignored.

A considerable literature exists on the partial information approach. In these models,

the risky assets are driven by a Brownian motion which is adapted to some ltration F.

This ltration is \big" in the sense that it properly includes the ltration generated by

the Brownian motion. The conditional mean return rate processes of the risky assets are

assumed to beF-adapted, while the information available to the investor is the ltration

SF generated by the asset price processes only. The problem to be solved is that of

maximizing expected utility over the class of portfolio strategies adapted to the observable

SinformationF . This leads to an optimal control problem under partial information.

Two basic references for the general theory are Lakner [Lak95, Lak98], where the rates of

returns are constant random variables or linear di usions. The linear di usion model is

studied further in Brendle [Bre06]. Using dynamic programming arguments, the case of

an underlying Markov chain for the return process is studied in B auerle and Rieder [BR04,

BR05], where also the optimal investment in the partially observable model is compared

to the one in the case of a fully observable model. The Markov chain model is also

studied, amongst others by Haussmann and Sass [SH04, HS04]. Alternative approaches

are presented in B auerle and Rieder [BR07] and Callegaro et al. [GCR06], where the asset

prices are driven by jump processes instead of the usual Wiener process. In the recent

paper of Bj ork et al. [TBL10] a quite explicit representation of the optimal terminal

wealth as well as of the optimal portfolio strategy is obtained for the general case that

the rates of returns are arbitrary semimartingales.

The approach requiring the least structure on the drift process is the robust approach.

Here, the drift is only assumed to lie in some (previously speci ed) uncertainty set. Since

the investor has no probabilistic information about the value the drift attains, he chooses a

policy maximizing the worst-case utility. Discrete versions of robust portfolio optimization

problems are considered amongst others by Goldfarb and Iyengar [GI03] and Tut unc u and

3Koenig [TK04]. An overview and an empirical study is given in the dissertation of Lutgens

[Lut04]. Continuous-time robust robust problems are treated by a duality approach in

Schied et al. [Sch05, HHS06, Sch08] and byBsde techniques in the of Muller

[Mul05].

An overview of the di erent approaches dealing with drift uncertainty in portfolio opti-

mization problems is given in Table 1.1.

Naive approach Partial information Robust approach

compute point estimate compute initial assume that the drift can

for parameters of the drift distribution for the drift, attain any value in some

model assume that driving previously speci ed set

factors of the drift model

cannot be observed

plug estimator into choose an optimal policy choose a policy

optimization problem based on the information maximizing the worst-case

of the stock prices utility

Table 1.1: Overview { approaches dealing with parameter uncertainty

1.2 Outline and contributions

As one already notices with a short glimpse on the table of contents, apart from this

introduction, this thesis is partitioned into four major chapters. We will now give a

concise outline of these chapters, in conjunction with a discussion of the most important

contributions.

Chapter 2 introduces the main mathematical tools that will be used in the subsequent

chapters to solve robust consumption-investment problems with stochastic coe cients.

They will be presented in a way that is custom-made for our speci c problem setup.

The chapter is divided into three sections: rst we provide some background on maxmin

or saddle-point problems. Then we present basic results about backward stochastic di er-

ential equations. Finally, we recall and collect a few well-known facts from the theory of

martingales of bounded mean oscillation.

The worst-case approach underlying robust problems leads in a natural way to a game

theoretic interpretation of the robust portfolio problem. One can view the investor as one

player seeking to construct an optimal p investing in the bond and the risky assets.

The other player chooses the parameters of the market model so that the investor’s utility

4