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Interest, Coupons and Yields Read chapter Read first chapter

2013 | OriginalPaper | Chapter

10. Numerical Methods

Authors : Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

Published in: Introduction to Quantitative Methods for Financial Markets

Publisher: Springer Basel

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Abstract

Numerical techniques prove particularly useful when explicit solution formulas in a certain model cannot be derived even for simple derivatives (e.g. in the Black-Karasinski model) or when the to-be-priced financial instrument has a complex structure so that analytical methods fail (e.g. if multiple cancelation rights exist).

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previous chapter Interest Rate Models
next chapter Simulation Methods
Footnotes
1
Such a digital condition could be ‘the option value remains 0, once the underlying stock price exceeds a certain barrier’.
 
2
A detailed analysis of this aspect can be found in Zulehner [76].
 
3
Such non-structured grids could, for example, include triangular or tetrahedral grids, or grids that become finer in certain parts of the computational region.
 
4
It will be practical if W only takes non-zero values on a small interval (see Figure 10.7). In this case the first term on the right-hand side of the above equation disappears, while the second (integral) term has a correspondingly small integration domain.
 
5
In the Black-Scholes differential equation the term with the second derivative with respect to S is the diffusion term, which has smoothing properties. The term with the first derivative with respect to S is the so-called convection term. Heat transmission, for example, can occur by heat conduction (diffusion) or by the flow of fluids such as liquids or gases (convection). A central heating system in a house would be an example of convection. It is often challenging to treat convection numerically. The so-called upwind techniques or streamline diffusion techniques increase the stability of problems that show dominant convection. In the case of mean-reverting interest rate models, convection can be significant.
 
6
This is the case for most stock price models discussed here, in particular for the Heston and the Merton model.
 
7
This can be justified, as the integrand is absolutely integrable under the given assumption for α.
 
Literature
1.
M. Aichinger and A. Binder. A Workout in Computational Finance. Wiley (to appear), 2013.
6.
A. Binder and A. Schatz. Finite elements and streamline diffusion for the pricing of structured financial instruments, in: The Best of Wilmott 2, P. Wilmott, ed., pages 351–363, Wiley, Chichester, 2006.
15.
P. Carr and D. B. Madan. Option valuation using the Fast Fourier Transform. Journal of Computational Finance, 2(4):61–73, 1999.
36.
G. Fusai and A. Roncoroni. Implementing Models in Quantitative Finance: Methods and Cases. Springer Finance. Springer-Verlag, Berlin, 2008.MATH
42.
J. C. Hull and A. D. White. Numerical procedures for implementing term structure models II: two-factor models. Journal of Derivatives, 2(2):37–48, 1994.CrossRef
43.
J. C. Hull and A. D. White. Using Hull-White interest rate trees. Journal of Derivatives, 3(3):26–36, 1996.CrossRef
50.
S. Larsson, V. Thomée. Partial Differential Equations with Numerical Methods, Springer, Berlin, 2003MATH
51.
R. Lee. Option pricing by transform methods: extensions, unification and error control. Journal of Computational Finance, 7(3):51–86, 2004.
54.
R. Lord and C. Kahl. Optimal Fourier inversion in semi-analytical option pricing. Journal of Computational Finance, 10(4):1–30, 2007.
66.
H.-G. Roos, M. Stynes, and L. Tobiska. Numerical Mehods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems. Springer-Verlag, Berlin, 1996.
70.
R. U. Seydel. Tools for Computational Finance. 4th edition. Springer-Verlag, Berlin, 2009.MATH
73.
J. Topper. Financial Engineering with Finite Elements. Wiley, Chichester, 2005.
76.
W. Zulehner. Numerische Mathematik: eine Einführung anhand von Differentialgleichungsproblemen. Band 1. Stationäre Probleme. Mathematik Kompakt. Birkhäuser Verlag, Basel, 2008.
Metadata
Title
Numerical Methods
Authors
Hansjoerg Albrecher
Andreas Binder
Volkmar Lautscham
Philipp Mayer
Copyright Year
2013
Publisher
Springer Basel
Book
Introduction to Quantitative Methods for Financial Markets

Print ISBN: 978-3-0348-0518-6

Electronic ISBN: 978-3-0348-0519-3

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-3-0348-0519-3_10