Optimal investment for a large investor in a regime-switching model [Elektronische Ressource] / Michael Busch
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Optimal investment for a large investor in a regime-switching model [Elektronische Ressource] / Michael Busch

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Optimal Investment for a Large Investorin a Regime-Switching ModelMichael BuschVom Fachbereich Mathematik der Technischen Universit¨at Kaiserslauternzur Verleihung des akademischen Grades Doktor der Naturwissenschaften(Doctor rerum naturalium, Dr. rer. nat.) genehmigte Dissertation.1. Gutachter: Prof. Dr. Ralf Korn2. Gutachterin: Prof. Dr. Nicole B¨auerleDatum der Disputation: 1. M¨arz 2011D 386iiiAcknowledgementsFirst and foremost, I would like to thank my advisor Prof. Dr. RalfKorn for his support and guidance throughout my PhD studies. Healways found the time to help me with his good advice and construc-tive criticism. I am also very grateful to Prof. Dr. Nicole Ba¨uerle foraccepting to act as a referee for this thesis.I sincerely thank Dr. Frank Seifried for the good collaboration andthe fruitful discussions. ThebasicideaofthemodelthatIinvestigatein this thesis was developed by him.Further I would like to thank my colleagues and friends at the de-partment for the pleasant working atmosphere.My special thanks go to my family for their constant support duringmy studies.Finally I gratefully acknowledge the financial support by the Bun-desministerium fu¨r Bildung und Forschung in the context ofthe jointresearch project ’Alternative Investments: Modellierung, Statistik,Risikomanagement und Software’ (FKZ: 03KOPAD1).

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Optimal Investment for a Large Investor
in a Regime-Switching Model
Michael Busch
Vom Fachbereich Mathematik der Technischen 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. Gutachterin: Prof. Dr. Nicole B¨auerle
Datum der Disputation: 1. M¨arz 2011
D 386iii
Acknowledgements
First and foremost, I would like to thank my advisor Prof. Dr. Ralf
Korn for his support and guidance throughout my PhD studies. He
always found the time to help me with his good advice and construc-
tive criticism. I am also very grateful to Prof. Dr. Nicole Ba¨uerle for
accepting to act as a referee for this thesis.
I sincerely thank Dr. Frank Seifried for the good collaboration and
the fruitful discussions. ThebasicideaofthemodelthatIinvestigate
in this thesis was developed by him.
Further I would like to thank my colleagues and friends at the de-
partment for the pleasant working atmosphere.
My special thanks go to my family for their constant support during
my studies.
Finally I gratefully acknowledge the financial support by the Bun-
desministerium fu¨r Bildung und Forschung in the context ofthe joint
research project ’Alternative Investments: Modellierung, Statistik,
Risikomanagement und Software’ (FKZ: 03KOPAD1).CONTENTS v
Contents
Acknowledgements iii
Abbreviations and Symbols vii
1 Introduction 1
2 Principles of Continuous-time Portfolio Optimization 5
2.1 The Merton Investment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Mathematical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 The Optimal Investment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.3 Hamilton-Jacobi-Bellman Equation and the Verification Theorem . . . . . . . . . 7
2.1.4 Solution for crra Investors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 The Ba¨uerle-Rieder Investment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Mathematical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 The Optimal Investment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 Hamilton-Jacobi-Bellman Equations and the Verification Theorem . . . . . . . . 12
2.2.4 Solution for crra Investors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Continuous-time Portfolio Optimization for a Large Investor 17
3.1 Mathematical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 The Optimal Investment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Hamilton-Jacobi-Bellman Equations and the Verification Theorem . . . . . . . . . . . . 20
4 Solution for crra Investors with Power Utility 25
4.1 Constant Intensity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1.1 Solution of the Investment Problem . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1.2 Solution of the Merton Investment Problem . . . . . . . . . . . . . . . . . . . . . 34
4.2 Step Intensity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2.1 Portfolio-dependent Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.2 Consumption-dependent Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2.3 Portfolio- and Consumption-dependent Intensities . . . . . . . . . . . . . . . . . 61
4.3 Affine Intensity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3.1 Portfolio-dependent Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.2 Consumption-dependent Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3.3 Portfolio- and Consumption-dependent Intensities . . . . . . . . . . . . . . . . . 118vi CONTENTS
5 Solution for crra Investors with Logarithmic Utility 145
5.1 Constant Intensity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.1.1 Solution of the Investment Problem . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.1.2 Solution of the Merton Investment Problem . . . . . . . . . . . . . . . . . . . . . 147
5.2 Step Intensity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.2.1 Portfolio-dependent Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.2.2 Consumption-dependent Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.2.3 Portfolio- and Consumption-dependent Intensities . . . . . . . . . . . . . . . . . 152
5.3 Affine Intensity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.3.1 Portfolio-dependent Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.3.2 Consumption-dependent Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.3.3 Portfolio- and Consumption-dependent Intensities . . . . . . . . . . . . . . . . . 162
6 A Special Case: Two Correlated Assets 167
6.1 Step Intensity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.2 Affine Intensity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7 Model Comparisons 173
7.1 Comparison of the Different Investment Problems . . . . . . . . . . . . . . . . . . . . . . 174
7.1.1 Comparison with the Merton Investment Problem . . . . . . . . . . . . . . . . . 174
7.1.2 Comparison of Similar Investment Problems . . . . . . . . . . . . . . . . . . . . . 178
7.1.3 Comparison of Investment Problems with Identical Intensity Type . . . . . . . . 181
7.2 Parameter Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.2.1 Step Intensity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.2.2 Affine Intensity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.3 Price of Misconception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.3.1 Step Intensity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.3.2 Affine Intensity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8 Conclusion 199
References 201
Scientific Career 203
Wissenschaftlicher Werdegang 205vii
Abbreviations and Symbols
Abbreviations
a.s. almost surely
cf. compare
e.g. for example
i.e. that is
resp. respectively
w.r.t. with respect to
lim limit
max, min maximum, minimum
sup, inf supremum, infimum
Symbols
⊤v transpose of vector v
−1A inverse of matrix A
+x =max{x,0}, the positive part of x
−x =max{−x,0}, the negative part of x
x∧y =min{x,y}
x∨y =max{x,y}
|x| absolute value of x
N natural numbers
R real numbers
+R positive real numbers without 0
−R negative real numbers without 0
+R positive real numbers including 0
0
−R negative real numbers including 00
A\B relative complement of set B in set A
1 indicator function of set AA
∅ empty setviii Abbreviations and Symbols
xexp(x) =e
ln(x) natural logarithm of x
1L (P) space of P-integrable processes
2L (P) space of P-square integrable processes
C(M) space of continuous functions on M
1C (M) space of continuously differentiable functions on M
2C (M) space of twice continuously differentiable functions on M
1,2C (M×N) space of functions that are continuously differentiable w.r.t. the first component
and twice continuously differentiable w.r.t. the second component on M×N
′f (x) first-order derivative of f w.r.t. x
f (t,x), f (t,x) first-order partial derivative of f w.r.t. t, resp. xt x
f (t,x) second-order partial derivative of f w.r.t. xxx
xրx left-handed convergence of x towards x0 0
xցx right-handed convergence of x towards x0 0
E(X) expectation of the random variable X w.r.t. the probability measure P
E(X|Y) conditional expectation of the random variable X given Y
π,c π,cE (X) expectation of the random variable X w.r.t. the probability measure P
E(X) stochastic exponential of the process X1
1 Introduction
In the classical Merton investment problem of maximizing the expected utility from terminal wealth
and intermediate consumption stock prices are independent of the investor who is optimizing his
investment strategy. This is reasonable as long as the considered investor is small and thus does not
influence the asset prices. However for an investor whose actions may affect the financial market the
framework of the classical investment problem turns out to be inappropriate. Against this background
various research was done on the field of including a relation between the investor and the financial
market on which he is acting. Subsequently we present some different models.
In [Jarrow 1992] R. Jarrow discusses market manipulating trading strategies by large traders in a
discrete time setting. In this context market manipulating strategies are defined as strategies that
generate a positive real wealth without taking any risk. The financial market of the model by Jarrow
consists of a riskless money market account and a risky stock where the relative stock price is an
exogenously given function dependent on the large investor’s actual and past holdings in the money
market account and the stock, i.e.
1P 0 0 0 1 1 1 0 1 1t =G ϕ ,ϕ ,...,ϕ ,ϕ ,ϕ ,...,ϕ , t∈{1,2,...,T}, P =1, P =p ,0 t t t−1 0 t t−1 0 0 0 0Pt
0 1 iwhere P , resp. P , is the price of the money market account, resp. the risky asset, and ϕ , i = 0,1,
denotes the corresponding holdings. Jarrow presents examples for the existence of market manipulat-
ing strategies under very general conditions. Further he provides a sufficient condition on the stock
price process that excludes market manipulating strategies. The sufficient condition is that the stock
price depends only on the large investor’s actual holdings and is independent of his past portfolio
compositions.
R. Jarrow extents his aforementioned model in [Jarrow 1994] via including a derivative security into
the financial market. The relative stock price is now given by
1P 0 1 c 0 1 1t =G ϕ ,ϕ ,ϕ , t∈{1,2,...,T}, P =1, P =p ,0 t t t t 0 0 0Pt
cwhereϕ denotes the number of derivatives in the large investor’s portfolio. It turns out that the pres-
ence of the derivative security enables the existence of market manipulating strategies that would not
have been possible if there was only the money market account and the stock. Corresponding to his
results in [Jarrow 1992] Jarrow presents a sufficient condition that prevents these new market manip-
ulating strategies. Finally a theory for the valuation of options in the discussed model is introduced.
Hereby Jarrow verifies that the standard binomial option model still works, however, with random
volatilities.
Acontinuous-timehedgingproblemofaninvestorwhoseportfoliostrategyandwealthaffecttheriskless
interest rate and the drift and volatility of the stock price process is dealt with in [Cvitani´c, Ma 1996]
by J. Cvitani´c and J. Ma. In their paper the considered financial market consists of a riskless money
market account and n¯ risky assets with dynamics
0 0 0 0 0 1 1 n¯ n¯ 0dP =P r (X ,ϕ P ,ϕ P ,...,ϕ P )dt, P =1,tt t t t t t t t t 0
n¯X
n,mn n 0 0 1 1 n¯ n¯ 0 0 1 1 n¯ n¯ m n ndP =μ (P ,X ,ϕ P ,ϕ P ,...,ϕ P )dt+ σ (P ,X ,ϕ P ,ϕ P ,...,ϕ P )dW , P =p ,t t t tt t t t t t t t t t t t t t t t 0 0
m=12 1 Introduction
n nwhere X denotes the investor’s wealth process and ϕ P equals the investor’s portfolio process de-
scribing the amount of money invested in the n-th asset. Given the initial stock prices and a desired
terminal wealth the investor is searching for the hedging portfolio process of an option that goes with
the smallest initial endowment. It turns out that the problem corresponds to finding a solution of a
forward-backwardstochasticdifferentialequation(FBSDE).Cvitani´candMaprovideconditionsunder
which a solution to this FBSDE can be found.
D.CuocoandJ.Cvitani´cinvestigatein[Cuoco, Cvitani´c 1998]thecontinuous-timeoptimalinvestment
problem of a large investor whose portfolio proportions impact on the instantaneous expected returns
on the traded assets. The financial market consists of a riskless money market account and n¯ risky
assets with dynamics
0 0 0 0 0 1 1 n¯ n¯ 0dP =P r (ϕ P ,ϕ P ,...,ϕ P )dt, P =1,t t t t t t t t t 0
" #
n¯X
n,mn n n 0 0 1 1 n¯ n¯ m n ndP =P μ (ϕ P ,ϕ P ,...,ϕ P )dt+ σ dW , P =p ,t t t t t t t t t t t 0 0
m=1
n nwhere again ϕ P denotes the amount of money invested in the n-th asset. Using martingale and
duality methods they provide sufficient conditions for the existence of optimal strategies under general
assumptionsontheassetpricesandthelargeinvestor’spreferences.Inspecificexamplesoftheinvestor’s
influence Cuoco and Cvitani´c present explicit solutions for an investor with logarithmic utility.
In [Bank, Baum 2004] P. Bank and D. Baum consider a general, abstract continuous-time model for
an illiquid financial market whose asset prices can be influenced by the trades of a large investor. The
market they discuss consists of a riskless bank account and a risky asset whose dynamics are described
by a family of continuous semimartingales that depend on the large investor’s holdings in the asset,
i.e.
ϕtP =P , t∈[0,T],t t
ϕwith the family (P ) , ϕ∈R, and where ϕ denotes the investors holdings in the risky asset. Ast∈[0,T]t
opposed to [Cvitani´c, Ma 1996] and [Cuoco, Cvitani´c 1998] where the investor was solely influencing
thedriftandvolatilityofthestockprice,themodelofBankandBaumallowsimpactsonthestockprice
itself. In this model setting the authors prove the absence of arbitrage and investigate the problem
of hedging attainable claims and the utility maximization problem using the Itoˆ-Wentzell formula.
It turns out that the large investor model inherits many properties of the underlying small investor
model such as the attainability of claims, the determination of superreplication prices or the utility
maximization.
In this thesis we provide a new approach to the field of large investor models. We study the optimal
investment problem of a large investor in a jump-diffusion market which is in one of two states or
regimes. The investor’s portfolio proportions as well as his consumption rate affect the intensity of
transitions between the different regimes. Hence the asset price dynamics are given by
0 0 I 0 0t−dP =P r dt, P =p ,t t 0 0
" #
m¯X
n n I I m n nt− t−dP =P μ dt+ σ dW , P =p ,t t n n,m t 0 0
m=1
i,1−iwhere I is an {0,1}-valued state process with transition intensities ϑ (π,c), i = 0,1, that depend
on the investor’s portfolio proportions π and consumption rate c. Thus the investor is ’large’ in the