Abstract
We study some properties of a continuous local martingale stopped at the first time when its range (the difference between the running maximum and minimum) reaches a certain threshold. The laws and the conditional laws of its value, maximum, and minimum at this time are simple and do not depend on the local martingale in question. As a consequence, the price and hedge of options which mature when the range reaches a given level are both model-free within the class of arbitrage-free models with continuous paths, which makes these products very convenient for hedging.
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© 2009 Springer-Verlag Berlin Heidelberg
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Cherny, A., Dupire, B. (2009). On Certain Distributions Associated with the Range of Martingales. In: Optimality and Risk - Modern Trends in Mathematical Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02608-9_2
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DOI: https://doi.org/10.1007/978-3-642-02608-9_2
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02607-2
Online ISBN: 978-3-642-02608-9
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