Black-scholes type equations

In this work we are concerned with the analysis and numerical solution of Black-Scholes type equations arising in the modeling of incomplete financial markets and an inverse problem of determining the local volatility function in a generalized Black-Scholes model from observed option prices. In the first chapter a fully nonlinear Black-Scholes equation which models transaction costs arising in option pricing is discretized by a new high order compact scheme. The compact scheme is proved to be unconditionally stable and non-oscillatory and is very efficient compared to classical schemes. Moreover, it is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. In the next chapter we turn to the calibration problem of computing local volatility functions from market data in a generalized Black-Scholes setting. We follow an optimal control approach in a Lagrangian framework. We show the existence of a global solution and study first- and second-order optimality conditions. Furthermore, we propose an algorithm that is based on a globalized sequential quadratic programming method and a primal-dual active set strategy, and present numerical results. In the last chapter we consider a quasilinear parabolic equation with quadratic gradient terms, which arises in the modeling of an optimal portfolio in incomplete markets. The existence of weak solutions is shown by considering a sequence of approximate solutions. The main difficulty of the proof is to infer the strong convergence of the sequence. Furthermore, we prove the uniqueness of weak solutions under a smallness condition on the derivatives of the covariance matrices with respect to the solution, but without additional regularity assumptions on the solution. The results are illustrated by a numerical example.

Parameterschätzung in zeitdiskreten ergodischen Markov-Prozessen am Beispiel des Cox-Ingersoll-Ross-Modells

In dieser Arbeit geht es um die Schätzung von Parametern in zeitdiskreten ergodischen Markov-Prozessen im allgemeinen und im CIR-Modell im besonderen. Beim CIR-Modell handelt es sich um eine stochastische Differentialgleichung, die von Cox, Ingersoll und Ross (1985) zur Beschreibung der Dynamik von Zinsraten vorgeschlagen wurde. Problemstellung ist die Schätzung der Parameter des Drift- und des Diffusionskoeffizienten aufgrund von äquidistanten diskreten Beobachtungen des CIR-Prozesses. Nach einer kurzen Einführung in das CIR-Modell verwenden wir die insbesondere von Bibby und Sørensen untersuchte Methode der Martingal-Schätzfunktionen und -Schätzgleichungen, um das Problem der Parameterschätzung in ergodischen Markov-Prozessen zunächst ganz allgemein zu untersuchen. Im Anschluss an Untersuchungen von Sørensen (1999) werden hinreichende Bedingungen (im Sinne von Regularitätsvoraussetzungen an die Schätzfunktion) für die Existenz, starke Konsistenz und asymptotische Normalität von Lösungen einer Martingal-Schätzgleichung angegeben. Angewandt auf den Spezialfall der Likelihood-Schätzung stellen diese Bedingungen zugleich lokal-asymptotische Normalität des Modells sicher. Ferner wird ein einfaches Kriterium für Godambe-Heyde-Optimalität von Schätzfunktionen angegeben und skizziert, wie dies in wichtigen Spezialfällen zur expliziten Konstruktion optimaler Schätzfunktionen verwendet werden kann. Die allgemeinen Resultate werden anschließend auf das diskretisierte CIR-Modell angewendet. Wir analysieren einige von Overbeck und Rydén (1997) vorgeschlagene Schätzer für den Drift- und den Diffusionskoeffizienten, welche als Lösungen quadratischer Martingal-Schätzfunktionen definiert sind, und berechnen das optimale Element in dieser Klasse. Abschließend verallgemeinern wir Ergebnisse von Overbeck und Rydén (1997), indem wir die Existenz einer stark konsistenten und asymptotisch normalen Lösung der Likelihood-Gleichung zeigen und lokal-asymptotische Normalität für das CIR-Modell ohne Einschränkungen an den Parameterraum beweisen.

Weathering of Fe-bearing minerals under extraterrestrial conditions, investigated by Mössbauer spectroscopy

The weathering of Fe-bearing minerals under extraterrestrial conditions was investigated by Mössbauer (MB) spectroscopy to gain insights into the role of water on the planet Mars. The NASA Mars Exploration Rovers Spirit and Opportunity each carry a miniaturized Mössbauer spectrometer MIMOS II for the in situ investigation of Martian soils and rocks as part of their payload. The MER flight instruments had to be modified in order to work over the Martian diurnal temperature range (180 K – 290 K) and within the unique electronic environment of the rovers. The modification required special calibration procedures. The integration time necessary to obtain a good quality Mössbauer spectrum with the MIMOS II flight instruments was reduced by 30 % through the design of a new collimator. The in situ investigation of rocks along the rover Spirit's traverse in Gusev crater revealed weakly altered olivine basalt on the plains and pervasively altered basalt in the Columbia Hills. Correlation plots of primary Fe-bearing minerals identified by MB spectroscopy such as olivine versus secondary Fe-bearing phases such as nanophase Fe oxides showed that olivine is the mineral which is primarily involved in weathering reactions. This argues for a reduced availability of water. Identification of the Fe-oxyhydroxide goethite in the Columbia Hills is unequivocal evidence for aqueous weathering processes in the Columbia Hills. Experiments in which mineral powders were exposed to components of the Martian atmosphere showed that interaction with the atmosphere alone, in the absence of liquid water, is sufficient to oxidize Martian surface materials. The fine-grained dust suspended in the Martian atmosphere may have been altered solely by gas-solid reactions. Fresh and altered specimens of Martian meteorites were investigated with MIMOS II. The study of Martian meteorites in the lab helped to identify in Bounce Rock the first rock on Mars which is similar in composition to basaltic shergottites, a subgroup of the Martian meteorites. The field of astrobiology includes the study of the origin, evolution and distribution of life in the universe. Water is a prerequisite for life. The MER Mössbauer spectrometers identified aqueous minerals such as jarosite and goethite. The identification of jarosite was crucial to evaluate the habitability of Opportunity's landing site at Meridiani Planum during the formation of the sedimentary outcrop rocks, because jarosite puts strong constrains on pH levels. The identification of olivine in rocks and soils on the Gusev crater plains provide evidence for the sparsity of water under current conditions on Mars. Ratios of Fe2+/Fe3+ were obtained with Mössbauer spectroscopy from basaltic glass samples which were exposed at a deep sea hydrothermal vent. The ratios were used as a measure of potential energy for use by a microbial community. Samples from Mars analogue field sites on Earth exhibiting morphological biosignatures were also investigated.