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Short-time implied volatility in exponential Lévy models
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2015 (English)In: International Journal of Theoretical and Applied Finance, ISSN 0219-0249, Vol. 18, no 4, 1550025Article in journal (Other academic) Published
Abstract [en]

We show that a necessary and sufficient condition for the explosion of implied volatility near expiry in exponential Levy models is the existence of jumps towards the strike price in the underlying process. When such jumps do not exist, the implied volatility converges to the volatility of the Gaussian component of the underlying Levy process as the time to maturity tends to zero.Those results are proved by comparing  the short-time asymptotics of the Black-Scholes price to the explicit formulas for upper or lower bounds of option prices in exponential Levy models.

Place, publisher, year, edition, pages
2015. Vol. 18, no 4, 1550025
Keyword [en]
implied volatility; exponential Levy models; short-time asymptotic behavior.
National Category
Mathematics Other Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-209221DOI: 10.1142/S0219024915500259ISI: 000365773000004OAI: oai:DiVA.org:uu-209221DiVA: diva2:656385
Available from: 2013-10-15 Created: 2013-10-15 Last updated: 2017-12-06Bibliographically approved
In thesis
1. Calibration, Optimality and Financial Mathematics
Open this publication in new window or tab >>Calibration, Optimality and Financial Mathematics
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of a summary and five papers, dealing with financial applications of optimal stopping, optimal control and volatility.

In Paper I, we present a method to recover a time-independent piecewise constant volatility from a finite set of perpetual American put option prices.

In Paper II, we study the optimal liquidation problem under the assumption that the asset price follows a geometric Brownian motion with unknown drift, which takes one of two given values. The optimal strategy is to liquidate the first time the asset price falls below a monotonically increasing, continuous time-dependent boundary.

In Paper III, we investigate the optimal liquidation problem under the assumption that the asset price follows a jump-diffusion with unknown intensity, which takes one of two given values. The best liquidation strategy is to sell the asset the first time the jump process falls below or goes above a monotone time-dependent boundary.

Paper IV treats the optimal dividend problem in a model allowing for positive jumps of the underlying firm value. The optimal dividend strategy is of barrier type, i.e. to pay out all surplus above a certain level as dividends, and then pay nothing as long as the firm value is below this level.

Finally, in Paper V it is shown that a necessary and sufficient condition for the explosion of implied volatility near expiry in exponential Lévy models is the existence of jumps towards the strike price in the underlying process.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2013. 25 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 83
Keyword
perpetual put option, calibration of models, piecewise constant volatility, optimal liquidation of an asset, incomplete information, optimal stopping, jump-diffusion model, optimal distribution of dividends, singular stochastic control, implied volatility, exponential Lévy models, short-time asymptotic behavior.
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-209235 (URN)978-91-506-2377-2 (ISBN)
Public defence
2013-11-29, Ångström, rum 2001, Ångströmslaboratoriet , Lägerhyddsvägen 1, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2013-11-07 Created: 2013-10-15 Last updated: 2013-11-07

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