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Smoothness of scale functions for spectrally negative Lévy processes
, 2006
"... Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that stan ..."
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Cited by 86 (19 self)
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Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.
The Russian option: Reduced regret
 Ann. Appl. Probab
, 1993
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Cited by 76 (3 self)
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Some calculations for Israeli options
 Finance and Stoch
"... Recently Kifer (2000) introduced the concept of an Israeli (or Game) option. That is a general Americantype option with the added possibility that the writer may terminate the contract early inducing a payment exceeding the holder's claim had they exercised at that moment. Kifer shows that pri ..."
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Cited by 27 (2 self)
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Recently Kifer (2000) introduced the concept of an Israeli (or Game) option. That is a general Americantype option with the added possibility that the writer may terminate the contract early inducing a payment exceeding the holder's claim had they exercised at that moment. Kifer shows that pricing and hedging of these options reduces to evaluating an optimal stopping problem assocaited with Dynkin games. In this short text we give two examples of perpetual Israeli options where the solutions are explicit.
Symmetry and duality in Lévy markets
 Quantitative Finance
"... The aim of this paper is to introduce the notion of symmetry in a Lévy market. This notion appears as a particular case of a general known relation between prices of put and call options, of both the European and the American type, that is also reviewed in the paper, and that we call putcall duali ..."
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Cited by 9 (0 self)
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The aim of this paper is to introduce the notion of symmetry in a Lévy market. This notion appears as a particular case of a general known relation between prices of put and call options, of both the European and the American type, that is also reviewed in the paper, and that we call putcall duality. Symmetric Lévy markets have the distinctive feature of producing symmetric smile curves, in the log of strike/futures prices. PutCall Duality is obtained as a consequence of a change of the risk neutral probability measure through Girsanov’s Theorem, when considering the discounted and reinvested stock price as the numeraire. Symmetry is defined when a certain law before and after the change of measure through Girsanov’s Theorem coincides. A parameter characterizing the departure from symmetry is introduced, and a necessary and sufficient condition for symmetry to hold is obtained, in terms of the jump measure of the Lévy process, answering a question raised by Carr and Chesney (1996). Some empirical evidence is shown, supporting that in general markets are not symmetric.