Some asymptotic results on nonstandard likelihood ratio tests, and Cox process modeling in finance
Some asymptotic results on nonstandard likelihood ratio tests, and Cox process modeling in finance
dc.contributor.advisor  Schäl, Manfred  
dc.contributor.author  Szimayer, Alexander  
dc.date.accessioned  20200405T21:04:41Z  
dc.date.available  20200405T21:04:41Z  
dc.date.issued  2002  
dc.identifier.uri  https://hdl.handle.net/20.500.11811/1815  
dc.description.abstract  This dissertation consists of two parts. In the first part, the subject of hypothesis testing is addressed. Here, nonstandard formulations of the null hypothesis are discussed, e.g., nonstationarity under the null, and boundary hypotheses. In the second part, stochastic models for financial markets are developed and studied. Particular emphasis is placed on the application of Cox processes. Part one begins with a survey of timeseries models which allow for conditional heteroscedasticity and autoregression, ARGARCH models. These models reduce to a white noise model, when some of the conditional heteroscedasticity parameters take their boundary value at zero, and the autoregressive component is in fact not present. The asymptotic distribution of the pseudologlikelihood ratio statistics for testing the presence of conditional heteroscedasticity and the autoregression term is reproduced. For financial market data, the model parameters are estimated and tests for the reduction to white noise are performed. The impact of these results on risk measurement is discussed by comparing several ValueatRisk calculations assuming the alternative model specifications. Furthermore, the power function of these tests is examined by a simulation study of the ARCH(1) and the AR(1)ARCH(1) models. First, the simulations are carried out assuming Gaussian innovations and then, the Gaussian distribution is replaced by the heavy tailed tdistribution. This reveals that a substantial loss of power is associated with the use of heavy tailed innovations. A related testing problem arises in the analysis of the OrnsteinUhlenbeck (OU) model, driven by Levy processes. This model is designed to capture mean reverting behaviour if it exists; but the data may in fact be adequately described by a pure Levy process with no OU (autoregressive) effect. For an appropriate discretized version of the model, likelihood methods are utilized to test for such a reduction of the OU process to Levy motion, deriving the distribution of the relevant pseudologlikelihood ratio statistics, asymptotically, both for a refining sequence of partitions on a fixed time interval with mesh size tending to zero, and as the length of the observation window grows large. These analyses are nonstandard in that the mean reversion parameter vanishes under the null of a pure Levy process for the data. Despite this a very general analysis is conducted with no technical restrictions on the underlying processes or parameter sets, other than a finite variance assumption for the Levy process. As a special case, for Brownian Motion as driving process, the limiting distribution is deduced in a quite explicit way, finding results which generalise the wellknown DickeyFuller ("unitroot") theory. Part two of this dissertation considers the application of Cox processes in mathematical finance. Here, a framework is discussed for the valuation of employee share options (ESO), and credit risk modeling. One popular approach for ESO valuation involves a modification of standard option pricing models, augmenting them by the possibility of departure of the executive at an exogenously given random time. Such models are called reduced form models, in contrast to structural models that require measures of the employee's utility function and other unobservable quantities. Here, an extension of the reduced form model for the valuation of ESOs is developed. This model incorporates and emphasises employee departure, company takeover, performance vesting and other exotic provisions specific to ESOs. The assumptions underlying the reduced form model are clearified, and discussed for their implications. Further, the probabilistic structure of the model is analysed which includes an explicit characterization of the set of equivalent martingale measures, as well as the computation of prominent martingale measures like, e.g., the variance optimal martingale measure and the minimal martingale measure. Particular ESO specifications are studied emphasizing different aspects of the proposed framework. In this context, also strict noarbitrage bounds for ESO prices are provided by applying optimal stopping. Furthermore, possible limitations of the proposed model are explored by examining departures from the crucial assumptions of noarbitrage, i.e. by considering the effects of the employee having inside information. In a continuous time market model, credit risk modeling and pricing of credit derivatives is discussed. In the approach it is adopted that credit risk is described by the interest rate spread between a corporate bond and a government bond. This spread is modeled in terms of explaining variables. For this purpose, a specific market model consisting of four assets is considered where the default process of the company is incorporated in a risky money market by a Cox process. It is shown that this market model has a unique equivalent martingale measure and is complete. As a consequence, contingent claim valuation can be executed in the usual way. This is illustrated with the valuation of a convertible bond which fits naturally in the given setting.  
dc.language.iso  eng  
dc.rights  In Copyright  
dc.rights.uri  http://rightsstatements.org/vocab/InC/1.0/  
dc.subject  Nuisance Parameter  
dc.subject  Cox Process  
dc.subject  Executive Stock Option  
dc.subject  Credit Risk  
dc.subject.ddc  510 Mathematik  
dc.title  Some asymptotic results on nonstandard likelihood ratio tests, and Cox process modeling in finance  
dc.type  Dissertation oder Habilitation  
dc.publisher.name  Universitäts und Landesbibliothek Bonn  
dc.publisher.location  Bonn  
dc.rights.accessRights  openAccess  
dc.identifier.urn  https://nbnresolving.org/urn:nbn:de:hbz:5n00800  
ulbbn.pubtype  Erstveröffentlichung  
ulbbnediss.affiliation.name  Rheinische FriedrichWilhelmsUniversität Bonn  
ulbbnediss.affiliation.location  Bonn  
ulbbnediss.thesis.level  Dissertation  
ulbbnediss.dissID  80  
ulbbnediss.date.accepted  17.09.2002  
ulbbnediss.institute  MathematischNaturwissenschaftliche Fakultät : Fachgruppe Mathematik / Institut für angewandte Mathematik  
ulbbnediss.fakultaet  MathematischNaturwissenschaftliche Fakultät  
dc.contributor.coReferee  Albeverio, Sergio 
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