Applications of semimartingales and Lévy processes in finance : duality and valuation ; Anwendungen von Semimartingalen und Lévyprozessen in der Finanzmathematik : Dualität und Bewertung
Titel:
Applications of semimartingales and Lévy processes in finance : duality and valuation ; Anwendungen von Semimartingalen und Lévyprozessen in der Finanzmathematik : Dualität und Bewertung
Anmerkungen:
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Beschreibung:
The complexity of modern financial derivatives very often leads to valuation problems that require the knowledge of the joint distribution of several random variables. This thesis aims to simplify and solve such valuation problems. Duality is related to the simplification of the valuation problem. We investigate changes of probability measures in an effort to reduce the multivariate problem to a univariate one. The asset price processes are driven either by general semimartingales or by Lévy processes and their dynamics are expressed in terms of their predictable characteristics. Imposing some very natural conditions on the driving processes, a battery of derivative products - including Asian, lookback and Margrabe options - can be simplified considerably. Valuation is related to the solution of the problem. We provide general valuation formulae for options on single and multi-asset derivatives. These formulae require the knowledge of the characteristic function, while most of the commonly used payoff functions can be treated. Using the Wiener-Hopf factorization, we provide expressions for options on the maximum of a Lévy process. Finally, we consider term structure models driven by time-inhomogeneous Lévy processes and provide duality and valuation results. ; Die Komplexität moderner Finanzderivate führt sehr oft zu Bewertungsproblemen, die die Kenntnisse der gemeinsamen Verteilung von mehreren Zufallsvariablen voraussetzt. Das Ziel dieser Arbeit ist es, solche Bewertungsprobleme zu vereinfachen und zu lösen.