Boundary Value Problems for Elliptic Differential Operators of First Order

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.Comment: 79 pages, 6 figures, minor corrections, references adde

Time Variation in the Tail Behaviour of Bund Futures Returns

The literature on the tail behaviour of asset prices focuses mainly on the foreign exchange and stock markets, with only a few papers dealing with bonds or bond futures. The present paper addresses this omission. We focus on three questions: (i) Are heavy tails a relevant feature of the distribution of BUND futures returns? (ii) Is the tail behaviour constant over time? (iii) If it is not, can we use the tail index as an indicator for financial market risk and does it add value in addition to classical indicators? The answers to these questions are (i) yes, (ii) no, and (iii) yes. We find significant heaviness of the tails of the Bund future returns. The tail index is on average around 3, implying the nonexistence of the forth moments. With the aid of a recently developed test for changes in the tail behaviour we identify several breaks in the degree of heaviness of the return tails. Interestingly, the tails of the return distribution do not move in parallel to realised volatility. This suggests that the tails of futures returns contain information for risk management that complements those gained from more standard statistical measures. -- Die Literatur über Extreme der Renditeverteilung hat sich bisher überwiegend mit Wechselkursen und Aktienkursen befasst. Die Kurse von Rentenwerten oder Terminkontrakten auf Rentenwerte haben hingegen bisher kaum Beach- tung erfahren. Das vorliegende Arbeitspapier versucht diese Lücke zu schließen. Unser Augenmerk gilt dabei insbesondere drei Fragen: (i) Haben die Ren- diteverteilungen von Terminkontrakten auf Bundeswertpapiere "fat tails"? (ii) Ist die Wahrscheinlichkeit extremer Kursbewegungen im Zeitablauf kon- stant? (iii) Kann ein Tail-Index Informationen Äuber den Grad von Marktun- sicherheit liefern, die klassische Indikatoren wie die VolatilitÄat nicht liefern können? Die Antworten zu diesen drei Fragen sind (i) ja, (ii) nein und (iii) ja. Wir finden ein signifikantes "fat tails" Phänomen in der Renditeverteilung von BUND Future Kontrakten. Ein Tail-Index von circa 3 impliziert, dass das vierte und alle höheren Momente der Verteilung nicht existieren. Mit Hilfe kürzlich entwickelter Tests finden wir Brüche der Tail-Stärke der Ren- diteverteilungen. Interessanterweise bewegt sich der Tail-Index nicht immer in die gleiche Richtung wie die Volatilität. Dies lässt vermuten, dass die Betrachtung der Tails dem Risikomanagement Informationen liefert, die mit herkömmlichen Verfahren nicht gewonnen werden kÄonnen.

Time variation in the tail behaviour of bunds futures returns

The present paper focuses on three questions: (i) Are heavy tails a relevant feature of the distribution of BUND futures returns? (ii) Is the tail behaviour constant over time? (iii) If it is not, can we use the tail index as an indicator for financial market risk and does it add value in addition to classical indicators? The answers to these questions are (i) yes, (ii) no, and (iii) yes. The tail index is on average around 3, implying the nonexistence of the fourth moments. A recently developed test for changes in the tail behaviour indicated several breaks in the degree of heaviness of the return tails. Interestingly, the tails of the return distribution do not move in parallel to realised volatility. This suggests that the tails of futures returns contain information for risk management that complements that gained from more standard statistical measures. JEL Classification: C14, G13extreme value theory, futures returns, risk management, Tail index

A dynamic copula approach to recovering the index implied volatility skew

Equity index implied volatility functions are known to be excessively skewed in comparison with implied volatility at the single stock level. We study this stylized fact for the case of a major German stock index, the DAX, by recovering index implied volatility from simulating the 30 dimensional return system of all DAX constituents. Option prices are computed after risk neutralization of the multivariate process which is estimated under the physical probability measure. The multivariate models belong to the class of copula asymmetric dynamic conditional correlation models. We show that moderate tail-dependence coupled with asymmetric correlation response to negative news is essential to explain the index implied volatility skew. Standard dynamic correlation models with zero tail-dependence fail to generate a sufficiently steep implied volatility skew.Copula Dynamic Conditional Correlation, Basket Options, Multivariate GARCH Models, Change of Measure, Esscher Transform