Drawdowns, drawups, and their applications
Item
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Title
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Drawdowns, drawups, and their applications
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Identifier
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d_2009_2013:a86dc38f3b37:10704
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identifier
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10769
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Creator
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Zhang, Hongzhong,
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Contributor
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Olympia Hadjiliadis
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Date
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2010
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics | Applied mathematics | CUSUM | Diffusion processes | Drawdowns and drawups | Quickest detection | Static replication | Stopping time
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Abstract
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This thesis studies the probability characteristics of drawdown and drawup processes of general linear diffusions. The drawdown process is defined as the current drop from the running maximum, while the drawup process is defined as the current rise over the running minimum. Attention is drawn to the first hitting times of the drawdown and the drawup processes, also known as the drawdown and the drawup respectively, and their applications in managing financial risks and detecting abrupt changes in random processes. The probabilities that the drawdown of a units precedes the drawup of equal size are derived in a biased simple random walk model and a drifted Brownian motion model. It is then shown that there exists an analytical formula for the Laplace transform of the drawdown of a units when it precedes the drawup of b units. The above problem can be related to the arbitrage-free pricing of a digital option related to the drawdowns and the drawups. Several static and semi-static replications are developed to hedge the risk exposure of these options. Finally, we study the properties of the drawups as a means of detecting abrupt changes in random processes with multi-source observations. In particular, we study extensions of the cumulative sum (CUSUM) stopping rule, which is the drawup of the log-likelihood ratio process. It is shown that the N-CUSUM stopping rule is at least second-order asymptotically optimal as the meantime to the first false alarm tends to infinity.
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Type
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dissertation
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Source
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2009_2013.csv
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degree
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Ph.D.
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Program
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Mathematics
