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

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

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Universitat KaiserslauternFachbereich MathematikSome new aspects ofOptimal Portfolios and Option PricingMartin KrekelVom Fachbereich Mathematikder Universit at Kaiserslauternzur Verleihung des akademischen GradesDoktor der Naturwissenschaften(Doctor rerum naturalium, Dr. rer. nat.)genehmigte Dissertation1. Gutachter: Prof. Dr. Ralf Korn2. Gutachter: Prof. Dr. Mogen Ste ensenDatum der Disputation: 20.06.2003D 386.IntroductionThe main two problems of continuous-time nancial mathematics are option pricingand portfolio optimization. The rst of these problems is concerned with valuing derivativecontracts on stocks (or other underlyings) which have a non-linear payo structure such asall kind of options. The other important topic, portfolio optimization, consists of the searchfor 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 typesetting 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 surveyedin the rst chapter.The next three chapters deal with generalizations of the standard portfolio problem,also know as "Merton’s problem".

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.5 Numerical Results and Conclusion . . . . . . . . . . . . . . . . . . . . . . . 95
4.5.1 Optimal consumption for equal weights . . . . . . . . . . . . . . . . 95
4.5.2 for discounted utility . . . . . . . . . . . . . . 98
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5 An Analysis of the Pricing Methods for Baskets Options 102
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2 The Valuation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.1 Beisser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.2 Gentle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2.3 Levy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.4 Levy + Beisser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.5 Milevsky and Posner . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.6 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3.1 E ect of varying the Correlations . . . . . . . . . . . . . . . . . . . . 109
5.3.2 E ect of varying the Strikes . . . . . . . . . . . . . . . . . . . . . . . 113
5.3.3 E ect of varying the Forwards and Strikes . . . . . . . . . . . . . . . 115
5.3.4 E ect of v the Volatilities . . . . . . . . . . . . . . . . . . . . . 124
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6 Asian options on Average Spot with Average Strike 134
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3 Approximate Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.4 Final Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.4.1 Fixed Strike Option . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.4.2 Average Spot with Average Strike in Equity . . . . . . . . . . . . . . 139
6.4.3 Average Spot with Average Strike in Performance . . . . . . . . . . 140
6.5 Numerical Results and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 141
Bibliography 151
ivList of Figures
1 Optimal control with continuous consumption . . . . . . . . . . . . . . . 52
2 control with lump sum . . . . . . . . . . . . . . . 52
3 Optimal control u with continuous income . . . . . . . . . . . . . . . . . . 53
4 control u with continuous consumption . . . . . . . . . . . . . . . 53
5 Step function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6 Frequency polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7 Logistic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
S18 Parabolas M with r(.) step function and r at . . . . . . . . . . . . . . . 65
P19 Parabolas M with r(.) frequency polygon and r at . . . . . . . . . . . . 69
10 Parabolas with r(.) logistic function and r at . . . . . . . . . . . . . . . . . 70
11 Optimal control with r(:) step function and HARA utility ( = 0:5) . . . . 73
12 control with r(:) frequency polygon and HARA utility ( = 0:5) . 75
13 Absolute Optimal HARA Consumption w.r.t . . . . . . . . . . . . . . . . 95
14 Relative w.r.t . . . . . . . . . . . . . . . . 96
15e Optimal HARA w.r.t . . . . . . . . . . . . . . . . . 99
16 Relative Consumption w.r.t . . . . . . . . . . . . . . . . 100
17e Optimal LOG w.r.t . . . . . . . . . . . . . . . . . 100
18 Absolute LOG w.r.t . . . . . . . . . . . . . . . . . 101
19 Densities for the standard scenario . . . . . . . . . . . . . . . . . . . . . . . 108
20 Varying the correlations simultaneously (Table 1) . . . . . . . . . . . . . . 110
21 Varying the simultaneously (Rel. Di .) . . . . . . . . . . . . . . 110
22 Varying the correlations sym. with xed = 95% . . . . . . . . . . . . . . 11112
23 Varying the sym. with xed = 95% (Rel. Di .) . . . . . . . 11112
24 Densities for the standard scenario with = 95% . . . . . . . . . . . . . . 11212
25 for the with inhomogenous correlation . . . . . 112
26 Varying the strike (Table 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
27 Varying the strike (Rel. Di .) . . . . . . . . . . . . . . . . . . . . . . . . . . 114
28 Varying the forwards sym. with K = 100 (Table 3) . . . . . . . . . . . . . . 116
29 Varying the forwards sym. with K = 100 (Rel. Di .) . . . . . . . . . . . . . 116
T30 Varying the forwards sym. with F = 10;K = 100 . . . . . . . . . . . . . . 1171
T31 Varying the forwards sym. with F = 10;K = 100 (Rel. Di .) . . . . . . . . 1171 P4T T32 Varying the forwards sym. with F = 20 and K = 0:5 w F . . . . . . 118i1 ii=1P4T T33 Varying the forwards sym. with F = 20 and K = 0:5 w F (Rel. Di .) 118i1 ii=1P4T T34 Varying the forwards sym. with F = 20 and K = 1:5 w F . . . . . . 119i1 ii=1P4T T35 Varying the forwards sym. with F = 20 and K = 1:5 w F (Rel. Di .) 119i1 i=1 i
T36 Varying the forwards sym. with F = 200;K = 100 . . . . . . . . . . . . . . 1201
T37 Varying the forwards sym. with F = 200;K = 100 (Rel. Di .) . . . . . . . 1201 P
4T T38 Varying the forwards sym. with F = 200 and K = 0:5 w F . . . . . 121i1 i=1 iP
4T T39 Varying the forwards sym. with F = 200 and K = 0:5 w F (Rel. Di .) 121i1 i=1 iP
4T T40 Varying the forwards sym. with F = 200 and K = 1:5 w F . . . . . 122i1 i=1 i
vP4T T41 Varying the forwards sym. with F = 200 and K = 1:5 w F (Rel. Di .) 122i1 i=1 i
T42 Densities for the standard scenario with F = 10 . . . . . . . . . . . . . . . 1231
43 Varying the volatilities sym. with K = 100 (Table 4) . . . . . . . . . . . . . 125
44 V the V sym. with K = 100 (Rel. Di .) . . . . . . . . . . . . 125
45 Varying the volatilities sym. with = 5%;K = 100 . . . . . . . . . . . . . 1261
46 Varying the volatilities sym. with = 5%;K = 50 . . . . . . . . . . . . . . 1271
47 V the v sym. with = 5%;K = 150 . . . . . . . . . . . . . 1271
48 Varying the volatilities sym. with = 100%;K = 50 (Table 5) . . . . . . . 1281
49 V the v sym. with = 100%;K = 100 (Table 6) . . . . . . 1291
50 Varying the volatilities sym. with = 100%;K = 150 (Table 7) . . . . . . 1301
51 Densities for std. scenario with = 5%; = = = 100% (Figure 45) 1311 2 3 4
52 for std. with = 100%; = = = 10% (Table 6) . 1311 2 3 4
53 Densities for the standard scenario with = 90%; = = 50%; = 10% 1321 2 3 4
vi1
1 Preliminaries
1.1 The Economy and Some Basic De nitions
1.1.1 Introduction
In this section we introduce the underlying economy, modeled by a stock market and a
money-market account. This economy will be used in this doctoral thesis for both option
pricing and portfolio optimization. It is based on variants of the well-known lognormal
model.
1.1.2 The Model
We consider a security market consisting of an interest-bearing cash account and n risky
assets. The uncertainty is modeled by a probability space ( ;F;fF g ;P) . The timet t2[0;T]
period is the nite interval [0;T]. The o w of information is given by the natural ltration
fFg , i.e. the P-augmention of an (independent) n-dimensional Brownian ltration.t t2[0;T]
Without loss of generality we set F = F, so that all observable events are known. AllT
traders are assumed to be price takers, and there are no transaction costs. The cash account
is modeled by the di eren tial equation
dB(t) = B(t)r(t)dt; (1.1)
where r(t) is a bounded, positive and progressively measurable process. The price process
of the i-th (i = 1;::: ;n) risky asset is given by
2 3
nX
4 5dS (t) = S (t) (b (t) d (t))dt + (t)dW (t) (1.2)i i i i ij j
j=1
with
0 0b(t) = (b (t);::: ;b (t)) ; d(t) = (d (t);::: ;d (t)) ; (1.3)1 n 1 n
denoting the drift vector and dividend-yield vector, and
0 1
(t) (t)11 1n
B C. .. .(t) = (1.4)@ A..
(t) (t)n1 nn
the volatility matrix. Let W(t) be an n-dimensional Brownian motion, where the individual
Brownian motions are independent. The coe cien ts b (t),d (t) and (t) are assumed toi i ij
be bounded, progressively F -measurable processes. The dividend yields are tot
0be nonnegative, that means d (t) 0 for all t 2 [0;T] and i = 1;::: ;n. In addition i
has to be a strictly positive de nite n n-matrix, i.e. it exists some constant K > 0 with
0 0 0 nx (t) (t)x Kx x for all x2 IR and for all t2 [0;T] P a.s..
In the following we will present the corresponding de nitions of trading strategies and
wealth processes, which are used in portfolio optimization and option pricing.2 1 PRELIMINARIES
De nition 1.1
n+1i) A trading strategy ’ is a IR -valued, fF g -progressively measurable pro-s s2[0;T]
cess
0’(t) := (’ (t);’ (t);::: ;’ (t))0 1 n
with
Z T
j’ (s)B(s)jds <1 P a.s.;0
0
Zn TX
2(’ (s)S (s)) ds <1 P a.s.; for i = 1;::: ;n:i i
0j=1
PnThe value x := ’ (0)B(0) + ’ (0)S (0) is called initial wealth.0 0 i ii=1
ii) Let ’ be a trading strategy with initial wealth x > 0. The process0
nX
X(t) := ’ (t)B(t) + ’ (t)S (t)0 i i
i=1
is called wealth process corresponding to ’ with initial wealth x .0
iii) A nonnegative, fF g -progressively measurable, real-valued process c(s), s 2s s2[0;T]
[0;T] with
Z T
c(s)ds <1 P a.s.
0
is called consumption process.
Remark: The restrictions in De nition i) and iii) ensure, that the It^ o-integral of the
2corresponding wealth process in 1.2 below is well de ned, i.e. in H [0;T] (see
Korn&Korn (2001) for a detailed de nition).
De nition 1.2
A pair (’;c) consisting of a trading strategy ’ and a consumption process c is called self-
nancing , if the associated wealth process X(t) satis es P a.s.:
Z Znt tX
X(t) = x + ’ (s)dB(s) + ’ (s)dS (s)0 0 i i
0 0i=1
Z Zn t tX
+ ’ (s)d (s)S (s)ds c(s)dsi i i
0 0
i=1