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August 2003 Volatility time and properties of option prices
Svante Janson, Johan Tysk
Ann. Appl. Probab. 13(3): 890-913 (August 2003). DOI: 10.1214/aoap/1060202830


We use a notion of stochastic time, here called volatility time, to show convexity of option prices in the underlying asset if the contract function is convex as well as continuity and monotonicity of the option price in the volatility. The volatility time is obtained as the almost surely unique stopping time solution to a random ordinary differential equation related to volatility. This enables us to write price processes, or processes modeled by local martingales, as Brownian motions with respect to volatility time. The results are shown under very weak assumptions and are of independent interest in the study of stochastic differential equations. Options on several underlying assets are also studied and we prove that if the volatility matrix is independent of time, then the option prices decay with time if the contract function is convex. However, the option prices are no longer necessarily convex in the underlying assets and the option prices do not necessarily decay with time, if a time-dependent volatility is allowed.


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Svante Janson. Johan Tysk. "Volatility time and properties of option prices." Ann. Appl. Probab. 13 (3) 890 - 913, August 2003.


Published: August 2003
First available in Project Euclid: 6 August 2003

zbMATH: 1061.91028
MathSciNet: MR1994040
Digital Object Identifier: 10.1214/aoap/1060202830

Primary: 35K15 , 60H30 , 91B28

Keywords: Contingent claim , convexity , stochastic time , time decay , Volatility

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.13 • No. 3 • August 2003
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