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Topics in importance sampling and derivatives pricing
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics. (Tillämpad matematik och beräkningsmatematik)
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers, presented in Chapters 2-5, on the topics of derivatives pricing and importance sampling for stochastic processes.

In the first paper a model for the evolution of the forward density of the future value of an asset is proposed. The model is constructed with the aim of being both simple and realistic, and avoid the need for frequent re-calibration. The model is calibrated to liquid options on the S\&P 500 index and an empirical study illustrates that the model provides a good fit to option price data.

In the last three papers of this thesis efficient importance sampling algorithms are designed for computing rare-event probabilities in the setting of stochastic processes. The algorithms are based on subsolutions of partial differential equations of Hamilton-Jacobi type and the construction of appropriate subsolutions is facilitated by a minmax representation involving the \mane potential.

In the second paper, a general framework is provided for the case of one-dimensional diffusions driven by Brownian motion. An analytical formula for the \mane potential is provided and the performance of the algorithm is analyzed in detail for geometric Brownian motion and for the Cox-Ingersoll-Ross process. Depending on the choice of the parameters of the models, the importance sampling algorithm is either proven to be asymptotically optimal or its good performance is demonstrated in numerical investigations.

The third paper extends the results from the previous paper to the setting of high-dimensional stochastic processes. Using the method of characteristics, the partial differential equation for the \mane potential is rewritten as a system of ordinary differential equations which can be efficiently solved. The methodology is used to estimate loss probabilities of large portfolios in the Black-Scholes model and in the stochastic volatility model proposed by Heston. Numerical experiments indicate that the algorithm yields significant variance reduction when compared with standard Monte-Carlo simulation.

In the final paper, an importance sampling algorithm is proposed for computing the probability of voltage collapse in a power system. The power load is modeled by a high-dimensional stochastic process and the sought probability is formulated as an exit problem for the diffusion. A particular challenge is that the boundary of the domain cannot be characterized explicitly. Simulations for two power systems shows that the algorithm can be effectively implemented and provides a viable alternative to existing system risk indices.

The thesis begins with a historical review of mathematical finance, followed by an introduction to importance sampling for stochastic processes.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2015. , viii, 28 p.
Series
TRITA-MAT-A, 2015:2
National Category
Probability Theory and Statistics
Research subject
Applied and Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-159640ISBN: 978-91-7595-445-5 (print)OAI: oai:DiVA.org:kth-159640DiVA: diva2:786454
Public defence
2015-03-03, F3, Lindstedtsvägen 26, KTH, Stockholm, 14:00 (English)
Opponent
Supervisors
Note

QC 20150206

Available from: 2015-02-06 Created: 2015-02-05 Last updated: 2015-02-06Bibliographically approved
List of papers
1. A simple time-consistent model for the forward density process
Open this publication in new window or tab >>A simple time-consistent model for the forward density process
2013 (English)In: International Journal of Theoretical and Applied Finance, ISSN 0219-0249, Vol. 16, no 8, 13500489- p.Article in journal (Refereed) Published
Abstract [en]

In this paper, a simple model for the evolution of the forward density of the future value of an asset is proposed. The model allows for a straightforward initial calibration to option prices and has dynamics that are consistent with empirical findings from option price data. The model is constructed with the aim of being both simple and realistic, and avoid the need for frequent re-calibration. The model prices of n options and a forward contract are expressed as time-varying functions of an (n + 1)-dimensional Brownian motion and it is investigated how the Brownian trajectory can be determined from the trajectories of the price processes. An approach based on particle filtering is presented for determining the location of the driving Brownian motion from option prices observed in discrete time. A simulation study and an empirical study of call options on the S&P 500 index illustrate that the model provides a good fit to option price data.

Keyword
mixture models, Option pricing
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-136972 (URN)10.1142/S0219024913500489 (DOI)2-s2.0-84892834759 (Scopus ID)
Note

QC 20140319. Updated from accepted to published.

Available from: 2013-12-10 Created: 2013-12-10 Last updated: 2017-12-06Bibliographically approved
2. A note on efficient importance sampling for one-dimensional diffusions
Open this publication in new window or tab >>A note on efficient importance sampling for one-dimensional diffusions
(English)Manuscript (preprint) (Other academic)
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-159644 (URN)
Note

QS 2015

Available from: 2015-02-06 Created: 2015-02-06 Last updated: 2015-02-06Bibliographically approved
3. Efficient importance sampling to compute loss probabilities in financial portfolios
Open this publication in new window or tab >>Efficient importance sampling to compute loss probabilities in financial portfolios
(English)Manuscript (preprint) (Other academic)
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-159645 (URN)
Note

QS 2015

Available from: 2015-02-06 Created: 2015-02-06 Last updated: 2015-02-06Bibliographically approved
4. Efficient importance sampling to assess the risk of voltage collapse in power systems
Open this publication in new window or tab >>Efficient importance sampling to assess the risk of voltage collapse in power systems
(English)Manuscript (preprint) (Other academic)
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-159646 (URN)
Note

QS 2015

Available from: 2015-02-06 Created: 2015-02-06 Last updated: 2015-02-06Bibliographically approved

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