Stochastic volatility, jumps and hidden time changes
AbstractStochastic volatility and jumps are viewed as arising from Brownian subordination given here by an independent purely discontinuous process and we inquire into the relation between the realized variance or quadratic variation of the process and the time change. The class of models considered encompasses a wide range of models employed in practical financial modeling. It is shown that in general the time change cannot be recovered from the composite process and we obtain its conditional distribution in a variety of cases. The implications of our results for working with stochastic volatility models in general is also described. We solve the recovery problem, i.e. the identification the conditional law for a variety of cases, the simplest solution being for the gamma time change when this conditional law is that of the first hitting time process of Brownian motion with drift attaining the level of the variation of the time changed process. We also introduce and solve in certain cases the problem of stochastic scaling. A stochastic scalar is a subordinator that recovers the law of a given subordinator when evaluated at an independent and time scaled copy of the given subordinator. These results are of importance in comparing price quality delivered by alternate exchanges.
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Bibliographic InfoArticle provided by Springer in its journal Finance and Stochastics.
Volume (Year): 6 (2002)
Issue (Month): 1 ()
Note: received: May 2000; final version received: March 2001
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Web page: http://www.springerlink.com/content/101164/
Find related papers by JEL classification:
- C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
- G12 - Financial Economics - - General Financial Markets - - - Asset Pricing
- G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
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- Rubenthaler, Sylvain & Wiktorsson, Magnus, 2003. "Improved convergence rate for the simulation of stochastic differential equations driven by subordinated Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 108(1), pages 1-26, November.
- Neil Shephard & Torben G. Andersen, 2008.
"Stochastic Volatility: Origins and Overview,"
OFRC Working Papers Series
2008fe23, Oxford Financial Research Centre.
- Neil Shephard & Torben Andersen, 2008. "Stochastic Volatility: Origins and Overview," Economics Papers 2008-W04, Economics Group, Nuffield College, University of Oxford.
- Neil Shephard & Torben G. Andersen, 2008. "Stochastic Volatility: Origins and Overview," Economics Series Working Papers 389, University of Oxford, Department of Economics.
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