## Fachbereich Mathematik

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#### Fachbereich / Organisatorische Einheit

- Fachbereich Mathematik (1041)
- Fraunhofer (ITWM) (2)

In this thesis, we focus on the application of the Heath-Platen (HP) estimator in option
pricing. In particular, we extend the approach of the HP estimator for pricing path dependent
options under the Heston model. The theoretical background of the estimator
was first introduced by Heath and Platen [32]. The HP estimator was originally interpreted
as a control variate technique and an application for European vanilla options was
presented in [32]. For European vanilla options, the HP estimator provided a considerable
amount of variance reduction. Thus, applying the technique for path dependent options
under the Heston model is the main contribution of this thesis.
The first part of the thesis deals with the implementation of the HP estimator for pricing
one-sided knockout barrier options. The main difficulty for the implementation of the HP
estimator is located in the determination of the first hitting time of the barrier. To test the
efficiency of the HP estimator we conduct numerical tests with regard to various aspects.
We provide a comparison among the crude Monte Carlo estimation, the crude control
variate technique and the HP estimator for all types of barrier options. Furthermore, we
present the numerical results for at the money, in the money and out of the money barrier
options. As numerical results imply, the HP estimator performs superior among others
for pricing one-sided knockout barrier options under the Heston model.
Another contribution of this thesis is the application of the HP estimator in pricing bond
options under the Cox-Ingersoll-Ross (CIR) model and the Fong-Vasicek (FV) model. As
suggested in the original paper of Heath and Platen [32], the HP estimator has a wide
range of applicability for derivative pricing. Therefore, transferring the structure of the
HP estimator for pricing bond options is a promising contribution. As the approximating
Vasicek process does not seem to be as good as the deterministic volatility process in the
Heston setting, the performance of the HP estimator in the CIR model is only relatively
good. However, for the FV model the variance reduction provided by the HP estimator is
again considerable.
Finally, the numerical result concerning the weak convergence rate of the HP estimator
for pricing European vanilla options in the Heston model is presented. As supported by
numerical analysis, the HP estimator has weak convergence of order almost 1.

Multifacility location problems arise in many real world applications. Often, the facilities can only be placed in feasible regions such as development or industrial areas. In this paper we show the existence of a finite dominating set (FDS) for the planar multifacility location problem with polyhedral gauges as distance functions, and polyhedral feasible regions, if the interacting facilities form a tree. As application we show how to solve the planar 2-hub location problem in polynomial time. This approach will yield an ε-approximation for the euclidean norm case polynomial in the input data and 1/ε.

In this article a new numerical solver for simulations of district heating networks is presented. The numerical method applies the local time stepping introduced in [11] to networks of linear advection equations. In combination with the high order approach of [4] an accurate and very efficient scheme is developed. In several numerical test cases the advantages for simulations of district heating networks are shown.

In this paper, we demonstrate the power of functional data models for a statistical analysis of stimulus-response experiments which is a quite natural way to look at this kind of data and which makes use of the full information available. In particular, we focus on the detection of a change in the mean of the response in a series of stimulus-response curves where we also take into account dependence in time.

Following the ideas presented in Dahlhaus (2000) and Dahlhaus and Sahm (2000) for time series, we build a Whittle-type approximation of the Gaussian likelihood for locally stationary random fields. To achieve this goal, we extend a Szegö-type formula, for the multidimensional and local stationary case and secondly we derived a set of matrix approximations using elements of the spectral theory of stochastic processes. The minimization of the Whittle likelihood leads to the so-called Whittle estimator \(\widehat{\theta}_{T}\). For the sake of simplicity we assume known mean (without loss of generality zero mean), and hence \(\widehat{\theta}_{T}\) estimates the parameter vector of the covariance matrix \(\Sigma_{\theta}\).
We investigate the asymptotic properties of the Whittle estimate, in particular uniform convergence of the likelihoods, and consistency and Gaussianity of the estimator. A main point is a detailed analysis of the asymptotic bias which is considerably more difficult for random fields than for time series. Furthemore, we prove in case of model misspecification that the minimum of our Whittle likelihood still converges, where the limit is the minimum of the Kullback-Leibler information divergence.
Finally, we evaluate the performance of the Whittle estimator through computational simulations and estimation of conditional autoregressive models, and a real data application.

In this thesis we address two instances of duality in commutative algebra.
In the first part, we consider value semigroups of non irreducible singular algebraic curves
and their fractional ideals. These are submonoids of Z^n closed under minima, with a conductor and which fulfill special compatibility properties on their elements. Subsets of Z^n
fulfilling these three conditions are known in the literature as good semigroups and their ideals, and their class strictly contains the class of value semigroup ideals. We examine
good semigroups both independently and in relation with their algebraic counterpart. In the combinatoric setting, we define the concept of good system of generators, and we
show that minimal good systems of generators are unique. In relation with the algebra side, we give an intrinsic definition of canonical semigroup ideals, which yields a duality
on good semigroup ideals. We prove that this semigroup duality is compatible with the Cohen-Macaulay duality under taking values. Finally, using the duality on good semigroup ideals, we show a symmetry of the Poincaré series of good semigroups with special properties.
In the second part, we treat Macaulay’s inverse system, a one-to-one correspondence
which is a particular case of Matlis duality and an effective method to construct Artinian k-algebras with chosen socle type. Recently, Elias and Rossi gave the structure of the inverse system of positive dimensional Gorenstein k-algebras. We extend their result by establishing a one-to-one correspondence between positive dimensional level k-algebras and certain submodules of the divided power ring. We give several examples to illustrate
our result.

We continue in this paper the study of k-adaptable robust solutions for combinatorial optimization problems with bounded uncertainty sets. In this concept not a single solution needs to be chosen to hedge against the uncertainty. Instead one is allowed to choose a set of k different solutions from which one can be chosen after the uncertain scenario has been revealed. We first show how the problem can be decomposed into polynomially many subproblems if k is fixed. In the remaining part of the paper we consider the special case where k=2, i.e., one is allowed to choose two different solutions to hedge against the uncertainty. We decompose this problem into so called coordination problems. The study of these coordination problems turns out to be interesting on its own. We prove positive results for the unconstrained combinatorial optimization problem, the matroid maximization problem, the selection problem, and the shortest path problem on series parallel graphs. The shortest path problem on general graphs turns out to be NP-complete. Further, we present for minimization problems how to transform approximation algorithms for the coordination problem to approximation algorithms for the original problem. We study the knapsack problem to show that this relation does not hold for maximization problems in general. We present a PTAS for the corresponding coordination problem and prove that the 2-adaptable knapsack problem is not at all approximable.

This paper presents a case study of duty rostering for physicians at a department of orthopedics and trauma surgery. We provide a detailed description of the rostering problem faced and present an integer programming model that has been used in practice for creating duty rosters at the department for more than a year. Using real world data, we compare the model output to a manually generated roster as used previously by the department and analyze the quality of the rosters generated by the model over a longer time span. Moreover, we demonstrate how unforeseen events such as absences of scheduled physicians are handled.

Order-semi-primal lattices
(1994)