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Calibration, Optimality and Financial Mathematics
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of 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 [en]
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: urn:nbn:se:uu:diva-209235ISBN: 978-91-506-2377-2 (print)OAI: oai:DiVA.org:uu-209235DiVA: diva2:656421
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
List of papers
1. Recovering a piecewise constant volatility from perpetual put option prices
Open this publication in new window or tab >>Recovering a piecewise constant volatility from perpetual put option prices
2010 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 47, no 3, 680-692 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we present a method to recover a time-homogeneous piecewise constant volatility from a finite set of perpetual put option prices. The whole calculation process of the volatility is decomposed into easy computations in many fixed disjoint intervals. In each interval, the volatility is obtained by solving a system of nonlinear equations.

Keyword
Perpetual put option, calibration of models, piecewise constant volatility
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-134147 (URN)10.1239/jap/1285335403 (DOI)000282856000005 ()
Available from: 2010-11-24 Created: 2010-11-22 Last updated: 2017-12-12Bibliographically approved
2. Optimal selling of an asset under incomplete information
Open this publication in new window or tab >>Optimal selling of an asset under incomplete information
2011 (English)In: International Journal of Stochastic Analysis, ISSN 2090-3332, E-ISSN 2090-3340, Vol. 2011, 543590- p.Article in journal (Refereed) Published
Abstract [en]

We consider an agent who wants to liquidate an asset with unknown drift. The agent believes that the drift takes one of two given values and has initially an estimate for the probability of either of them. As time goes by, the agent observes the asset price and can thereforeupdate his beliefs about the probabilities for the drift distribution. We formulate an optimal stopping problem that describes the liquidation problem, and we demonstrate that the optimal strategy is to liquidate the first time the asset price falls below a certain time-dependent boundary. Moreover, this boundary is shown to be monotonically increasing, continuous and to satisfy a nonlinear integral equation.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-141329 (URN)10.1155/2011/543590 (DOI)
Available from: 2011-01-11 Created: 2011-01-11 Last updated: 2017-12-11Bibliographically approved
3. Optimal selling of an asset with jumps under incomplete information
Open this publication in new window or tab >>Optimal selling of an asset with jumps under incomplete information
2013 (English)In: Applied Mathematical Finance, ISSN 1350-486X, E-ISSN 1433-4313, Vol. 20, no 6, 599-610 p.Article in journal (Refereed) Published
Abstract [en]

We study the optimal liquidation strategy of an asset with price process satisfying a jump diffusion model with unknown jump intensity. It is assumed that the intensity takes one of two given values, and we have an initial estimate for the probability of both of them. As time goes by, by observing the price fluctuations, we can thus update our beliefs about the probabilities for the intensity distribution. We formulate an optimal stopping problem describing the optimal liquidation problem. It is shown that the optimal strategy is to liquidate the first time the point process falls below (goes above) a certain time-dependent boundary.

National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-164690 (URN)10.1080/1350486X.2013.810462 (DOI)
Available from: 2011-12-22 Created: 2011-12-22 Last updated: 2017-12-08Bibliographically approved
4. The optimal dividend problem in the dual model
Open this publication in new window or tab >>The optimal dividend problem in the dual model
2014 (English)In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 46, no 3, 746-765 p.Article in journal (Refereed) Published
Abstract [en]

We study de Finetti's optimal dividend problem, also known as the optimal harvesting problem, in the dual model. In this model, the firm value is affected both by continuous fluctuations and by upward directed jumps. We use a fixed point method to show that the solution of the optimal dividend problem with jumps can be obtained as the limit of a sequence of stochastic control problems for a diffusion. In each problem, the optimal dividend strategy is of barrier type, and the rate of convergence of the barrier and the corresponding value function is exponential.

Keyword
Optimal distribution of dividends, de Finetti’s dividend problem, optimal harvesting, singular stochastic control, jump-diffusion model.
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-209219 (URN)000341280200008 ()
Available from: 2013-10-15 Created: 2013-10-15 Last updated: 2017-12-06Bibliographically approved
5. Short-time implied volatility in exponential Lévy models
Open this publication in new window or tab >>Short-time implied volatility in exponential Lévy models
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.

Keyword
implied volatility; exponential Levy models; short-time asymptotic behavior.
National Category
Mathematics Other Mathematics
Identifiers
urn:nbn:se:uu:diva-209221 (URN)10.1142/S0219024915500259 (DOI)000365773000004 ()
Available from: 2013-10-15 Created: 2013-10-15 Last updated: 2017-12-06Bibliographically approved

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