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Veröffentlichungen:
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| [5] |
C. Jonen, Efficient Pricing of High-Dimensional
American-Style Derivatives: A Robust Regression
Monte Carlo Method, http://kups.ub.uni-koeln.de/4442/.
Pricing high-dimensional American-style derivatives is still a challenging task, as the complexity
of numerical methods for solving the underlying mathematical problem rapidly grows with the
number of uncertain factors. We tackle the problem of developing efficient algorithms for valuing
these complex financial products in two ways. In the first part of this thesis we extend the
important class of regression-based Monte Carlo methods by our Robust Regression Monte Carlo
(RRM) method. The key idea of our proposed approach is to fit the continuation value at every
exercise date by robust regression rather than by ordinary least squares; we are able to get a more
accurate approximation of the continuation value due to taking outliers in the cross-sectional
data into account. In order to guarantee an efficient implementation of our RRM method, we
suggest a new Newton-Raphson-based solver for robust regression with very good numerical
properties. We use techniques of the statistical learning theory to prove the convergence of our
RRM estimator. To test the numerical efficiency of our method, we price Bermudan options
on up to thirty assets. It turns out that our RRM approach shows a remarkable convergence
behavior; we get speed-up factors of up to over four compared with the state-of-the-art Least
Squares Monte Carlo (LSM) method proposed by Longstaff and Schwartz (2001). In the second
part of this thesis we focus our attention on variance reduction techniques. At first, we propose
a change of drift technique to drive paths in regions which are more important for variance and
discuss an efficient implementation of our approach. Regression-based Monte Carlo methods
might be combined with the Andersen-Broadie (AB) method (2004) for calculating lower and
upper bounds for the true option value; we extend our ideas to the AB approach and our
technique leads to speed-up factors of up to over twenty. Secondly, we research the effect of
using quasi-Monte Carlo techniques for producing lower and upper bounds by the AB approach
combined with the LSM method and our RRM method. In our study, efficiency has high priority
and we are able to accelerate the calculation of bounds by factors of up to twenty. Moreover, we
suggest some simple but yet powerful acceleration techniques; we research the effect of replacing
the double precision procedure for the exponential function and introduce a modified version
of the AB approach. We conclude this thesis by combining the most promising approaches
proposed in this thesis, and, compared with the state-of-the-art AB method combined with
the LSM method, it turns out that our ultimate algorithm shows a remarkable performance;
speed-up factors of up to over sixty are quite possible.
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| [4] |
C. Jonen, Valuing High-Dimensional American-Style Derivatives: A Robust Regression Monte Carlo Method, in review.
Pricing high-dimensional American-style derivatives is still a challenging task, as the complexity of numerical methods for solving the underlying mathematical problem rapidly grows with the number of uncertain factors. In this paper we extend the important class of regression-based Monte Carlo methods for valuing these complex financial products. The key idea of our proposed approach is to fit the continuation value at every exercise date by robust regression rather than by ordinary least squares. By using robust regression, we are able to get a more accurate approximation of the continuation value due to taking outliers in the cross-sectional data into account. In order to guarantee an efficient implementation of our Robust Regression Monte Carlo (RRM) method, we suggest a new Newton-Raphson-based solver for robust regression with very good numerical properties. We use techniques of the statistical learning theory to prove the convergence of our RRM estimator. In order to test the numerical efficiency of our proposed method, we price Bermudan options on up to thirty assets. It turns out that our RRM approach shows a remarkable convergence behavior; we get speed-up factors of up to over four compared with the state-of-the-art Least Squares Monte Carlo method proposed by Longstaff and Schwartz (2001).
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| [3] |
A. Laude and C. Jonen, Biomass and CCS:
The influence of learning effect, in review.
Combining bioethanol production and Carbon Capture and Storage technologies (BECCS) provides an opportunity to create negative emissions while producing biofuels. However, high capture costs reduce the profitability. This article tackles carbon price uncertainty and technological uncertainty through a real option approach. We first compare the case of an early versus a delayed CCS deployment. An early technological progress may arise from aggressive R&D and pilot project programs, but the expected reduction in costs is still uncertain. We show thus that the amount of avoided emissions is higher and that investments are carry out sooner but after 2030. In a second set of experiments, we apply an additional aid which consists in rewarding sequestered emissions rather than avoided emissions. In other words, CO2 emissions from the CCS implementation are not taken into account anymore. Investment level is then higher and the project may becomes attractive before 2030.
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| [2] |
C. Jonen, An efficient implementation of a least squares Monte Carlo method for valuing
American-style options, International Journal of Computer Mathematics, 86 (2009), pp. 1024-1039.
Several methods for valuing high-dimensional American-style options were proposed in the last years.
Longstaff and Schwartz (LS) have suggested a regression-based Monte Carlo approach, namely the least
squares Monte Carlo method. This article is devoted to an efficient implementation of this algorithm. First,
we suggest a code for faster runs. Regression-based Monte Carlo methods are sensitive to the choice of
basis functions for pricing high-dimensional American-style options and, like all Monte Carlo methods,
to the underlying random number generator. For this reason, we secondly propose an optimal selection of
basis functions and a random number generator to guarantee stable results. Our basis depends on the payoff
of the high-dimensional option and consists of only three functions. We give a guideline for an efficient
option price calculation of high-dimensional American-style options with the LS algorithm, and we test it
in examples with up to 10 dimensions.
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| [1] |
C. Jonen, Die Least-Squares-
Monte-Carlo-Methode
zur Bewertung von
amerikanischen
Finanzoptionen, Diplomarbeit (unveröffentlicht), 2008.
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Vorträge/Poster:
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| 03/2011 |
Pricing High-Dimensional American-Style Derivatives:
A Robust Regression Monte Carlo Method,
Amsterdam-Cologne Workshop on Computational Finance, Amsterdam, The Netherlands.
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| 11/2010 |
Pricing High-Dimensional American-Style Derivatives:
A Robust Regression Monte Carlo Method,
SIAM Conference on Financial Mathematics and Engineering,
San Francisco, California, USA.
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| 06/2010 |
A Robust Regression Monte Carlo Method for Pricing High-Dimensional American-Style Options,
6th World Congress of the Bachelier Finance Society,
Toronto, Canada.
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| 03/2010 |
A Robust Regression Monte Carlo Method for Pricing High-Dimensional American-Style Options,
10th MathFinance Conference, Derivatives and Risk
Management in Theory and Practice, Frankfurt, Germany.
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| 08/2009 |
A Robust Regression Monte Carlo Method for Pricing
High-Dimensional American-Style Options,
2nd SMAI European Summer School,
Financial Mathematics, Paris, France.
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| 07/2009 |
A Robust Regression Monte Carlo Method for Pricing
High-Dimensional American-Style Options,
23rd European Conference, Operational Research,
Bonn, Germany.
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English