Identification in financial models with time-dependent volatility and stochastic drift components [Elektronische Ressource] / vorgelegt von Romy Krämer

Sujets

• Mathematics

Informations

• Publié par: technische_universitat_chemnitz
• Publié le: 01 janvier 2007
• Nombre de lectures: 12
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Identification in Financial Models with
Time-Dependent Volatility and
Stochastic Drift Components
DISSERTATION
zur Erlangung des akademischen Grades
Doctor rerum naturalium
(Dr. rer. nat.)
TECHNISCHE UNIVERSITÄT CHEMNITZ
Fakultät für Mathematik
vorgelegt von Dipl.-Math. Romy Krämer
geb. am 08.02.1980 in Karl-Marx-Stadt (Chemnitz)
Betreuer: Prof. Dr. Bernd Hofmann (TU Chemnitz)
Gutachter: Prof. Dr. Bernd Hofmann (TU Chemnitz)
Dr. P. Mathé (WIAS Berlin)
Prof. Dr. W. Grecksch (MLU Halle)
Tag der Verteidigung: 31. Mai 2007
Verfügbar im MONARCH der TU Chemnitz:
http://archiv.tu-chemnitz.de/pub/2007/0080Contents
1 Introduction 7
2 Stochastic Preliminaries 10
3 The Bivariate Ornstein-Uhlenbeck model 22
3.1 Solution of the stochastic differential equation . . . . . . . . . . . . . . . . . 24
3.2 Pricing of European Call Options . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Volatility estimation by wavelet methods 28
4.1 Introduction to wavelet analysis . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 The situation and the estimator . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Asymptotic study of the estimator . . . . . . . . . . . . . . . . . . . . . . . 36
4.3.1 Weak convergence of the estimator . . . . . . . . . . . . . . . . . . . 36
4.3.2 Mean integrated square error . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Numerical case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.1 The L-method as criteria for the choice of the resolution level . . . . 62
4.5 Outlook: Wavelet thresholding . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Operator equations with Nemytskii operators 70
5.1 Inverse problems and regularization methods . . . . . . . . . . . . . . . . . . 70
5.2 Nemytskii operators: Acting conditions and continuity . . . . . . . . . . . . 76
5.3 Nemytskii operators with monotone generator functions . . . . . . . . . . . . 79
2CONTENTS 3
6 Identification of the time-dependent volatility using option prices 89
6.1 Inverse option pricing: Tikhonov-Regularization . . . . . . . . . . . . . . . . 89
6.2 The outer problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2.1 Analytical studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.2 Numerical case studies concerning ill-conditioning effects . . . . . . . 102
6.2.3 Regularization by monotonization – Algorithm . . . . . . . . . . . . . 108
6.2.4 Discrete Variant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7 Identification of the drift parameters 121
7.1 Preliminary considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2 State space representation and Kalman filter . . . . . . . . . . . . . . . . . . 125
7.3 Likelihood function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.4 Numerical Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Notations
Spaces
P polynomials of degree ≤nn
p p dL (I), L (I;R ) Lebesgue spaces of p-power integrable functions
pL (I) contains functions f :I →R, whereas
p d dL (I;R ) contains functions f :I →R
kC (I) space of k-times continuously differentiable functions f :I →R
If I = [a,b] we also writeC[a,b].
Furthermore, we use the abbreviationC when appropriate. P
2 2l (Z) := {x } :x ∈R and x <∞k k∈Z k k∈Z k P
2 2l (N) := {x } :x ∈R and x <∞k k∈N k kk∈N
sH (R) Sobolev spaces cf. (4.5)
<,> scalar product in the Hilbert space HH
Functions
[r] entier function ([r] is the largest integer that is less or equal than r)
1 s∈A
χ indicator function of the set A, i.e.χ (s) :=
A A 0 s∈/ A
Operators
R∞ −iξxˆFf(ξ) =f(ξ) := e f(x)dx : Fourier transform of f−∞
∗F adjoint operator of F
4CONTENTS 5
Norms
Pm,n 2 2|| For a matrix A = (a ) we set|A| := |a |i,j i,ji=1,j=1
i,j
kfk := max |f(x)|C(I) x∈I
1R
p p
pkfk := |f(x)| dxL (I) I
Miscellaneous
∧ r ∧r := min{r ,r }1 2 1 2
⊕ orthogonal sum
# cardinality
Vectors and Matrices
nI identity matrix, I = (δ )i,j i,j=1
0 vector that contains only zeros
Stochastics
Eξ Expectation of the random vector ξ
T2D ξ :=E(ξ−Eξ)(ξ−Eξ)
T
Cov(ξ,η) :=E(ξ−Eξ)(η−Eη)
A({ξ } ) The sigma-algebra generated by thec c∈C
random variables ξ (c∈C)c
ξ∼N(m,R) ξ is normally distributed with expectation m
and covariance matrix R
F (x) distribution function of the random vector ξξ R
i<t,x>φ (t) = e dF (x) characteristic function of the random vector ξξ n ξR
ξ =o (1) {ξ } converges to zero in probability, i.e.n P n
∀ε> 0 it holdsP(|ξ |>ε)→ 0 for n→∞.n
Equivalences
If A(u) and B(u) are positive functions of a set of parameters, the notation
A(u).B(u)6 CONTENTS
means, that there exists a constant C > 0 such that A(u) ≤ CB(u) independently of u.
Furthermore, the notation
A(u)hB(u)
means A(u). B(u) and B(u). A(u). Furthermore, we use the Landau symbols O and o
to describe the asymptotic behavior of functions. To be precise, for two functions f and g
we write
f(x) =O(g(x)) for x→∞
if and only if there exists an x and a constant M > 0 such that0
|f(x)|≤ M|g(x)| for x>x .0
Besides, we write
f(x) =o(g(x)) for x→∞
f(x)
if and only if → 0 for x→∞.
g(x)
In order to distinguish between results that are cited from the literature and own contribu-
tions we use the term proposition when we reformulate facts that are found in the literature.
As opposed to that lemmas and theorems state and prove assertions that we could not find
in the literature. As usual we use the term lemma for auxiliary results that are mainly used
to prove a theorem.Chapter 1
Introduction
During the last decades a great diversity of price models for financial assets has been de-
veloped. It is well-known that as long as it is only possible to observe asset prices (or the
corresponding returns) in a discrete scheme, it is always possible to find a model based on
a geometric Brownian motion with constant volatility coefficient and stochastic drift term
which has identical distributions as the observed returns (cf., e.g. [46]). Clearly, due to this
fact one must not argue that the empirically observed returns which fail to have indepen-
dent normal distributions require extensions of the classical model in order to price options
accurately.
On the other hand it is obvious that by introducing further random effects into the cor-
responding models via a drift for a given (fixed) behaviour of the observed data there are
changes in option prices, even though the option price formula itself is unaffected by changes
in the drift. Consequently, the study of corresponding models is meaningful. In this context
the estimate of volatility has to be reinterpreted in the light of the specific model which is
assumed.
Speaking generally, these models are based on stochastic processes which are specified by
several model parameters and these parameters have to be calibrated to observed market
data. Obviously,acorrectidentificationoftheseparametersisofcoreimportanceasotherwise
the models do not yield a good approximation of the real price processes. Moreover, the
model parameters are also necessary for pricing derivatives. A computation of these prices
with wrong parameters can lead to results which are far away from the prices observed on
real markets even if the correct formulas have been used.
Here and in what follows the term parameter means either a finite dimensional vector or a
function that specifies a model. This manner of speech is common in the literature of inverse
problems but it differs from the statistical literature. There, this term is generally used in
the meaning of a finite dimensional parameter vector. Consequently, the branch of statistics
which is concerned with the identification of finite dimensional parameter vectors is called
parametric statistics, whereas nonparametric statistics aims at the calibration of models
containing unknown functions (for example a volatility function). If we want to stress that
certain parameters are finite dimensional, we speak of finite dimensional parameter vectors
or real-valued parameters.
78 1. INTRODUCTION
The focus of this thesis is on parameter identification in market models with partial infor-
mation in which the stochastic drift of the logarithmic asset price process depends on an
unobservable state process. The asset price process is assumed to have a time-dependent but
deterministic volatility, which has to be identified. Furthermore, the stochastic drift and the
underlying stateprocess arecharacterised byafinite number ofreal-valued parameterswhich
are assumed tobeconstant with respect totime. The aim isan analysisofseveral calibration
techniques which are suitable for the identification of the described parameters. As a toy ex-
ample we consider a slightly modified version of the Bivariate Trending Ornstein-Uhlenbeck
model which has been introduced by Lo and Wang in [37].
Intheliteratureseveral calibrationtechniques arediscussed. Speakinggenerally, there areon
the one hand statistical approaches which aim at estimating the parameters from observed
asset prices. On the other hand there are approaches which use prices of observed derivatives
(e.g. observed option price data). In general, the last approach leads