Some asymptotic results on non-standard likelihood ratio tests, and Cox process modeling in finance

Abstract

This dissertation consists of two parts. In the first part, the subject of hypothesis testing is addressed. Here, non-standard formulations of the null hypothesis are discussed, e.g., non-stationarity 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. <br /> Part one begins with a survey of time-series models which allow for conditional heteroscedasticity and autoregression, AR-GARCH 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 pseudo-log-likelihood 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 Value-at-Risk 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 t-distribution. This reveals that a substantial loss of power is associated with the use of heavy tailed innovations. <br /> A related testing problem arises in the analysis of the Ornstein-Uhlenbeck (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 pseudo-log-likelihood 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 non-standard 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 well-known Dickey-Fuller ("unit-root") theory. <br /> 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 no-arbitrage 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 no-arbitrage, 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.