Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-14T17:24:30.550Z Has data issue: false hasContentIssue false
  • English
  • Français

Efficient Numerical Valuation of Continuous Installment Options

Published online by Cambridge University Press:  03 June 2015

Anton Mezentsev ,
Anton Pomelnikov  and
Matthias Ehrhardt
Anton Mezentsev*
Affiliation:
IDE, Halmstad University, Box 823, 301 18, Halmstad, Sweden
Anton Pomelnikov*
Affiliation:
IDE, Halmstad University, Box 823, 301 18, Halmstad, Sweden
Matthias Ehrhardt*
Affiliation:
Lehrstuhl für Angewandte Mathematik und Numerische Analysis, Fachbereich C–Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany
*
Corresponding author. URL: http://www-num.math.uni-wuppertal.de/~ehrhardt/ Email: [email protected]
Email: [email protected]
Email: [email protected]
Get access

Abstract

In this work we investigate the novel Kryzhnyi method for the numerical inverse Laplace transformation and apply it to the pricing problem of continuous installment options. We compare the results with the one obtained using other classical methods for the inverse Laplace transformation, like the Euler summation method or the Gaver-Stehfest method.

Keywords

65M1078A48Installment optionsBlack-Scholes equationnumerical inverse Laplace transformGaver-Stehfest methodEuler summation methodoptimal stopping problemnon-monotonic stopping boundary
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 3 , Issue 2 , April 2011 , pp. 141 - 164
Copyright
Copyright © Global-Science Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Abate, J. and Valkó, P. P., Multi-precision Laplace transform inversion, Int. J. Numer. Meth. Engng., 60 (2004), pp. 979–993.Google Scholar
[2] Alobaidi, G., Mallier, R. and Deakin, A. S., Laplace transforms and installment options, Math. Models. Meth. Appl. Sci., 8 (2004), pp. 1167–1189.Google Scholar
[3] Ballester, C., Company, R. and Jódar, L., An efficient method for option pricing with discrete dividend payment, Comput. Math. Appl., 56 (2008), pp. 822–835.Google Scholar
[4] Bateman, H., Two systems of polynomials for the solution of Laplace’s integral equation, Duke. Math. J., 2 (1936), pp. 569–577.Google Scholar
[5] Ben-Ameur, H., Breton, M. and Francois, P., A dynamic programming approach to price installment options, Euro. J. Oper. Res., 169 (2006), pp. 667–676.Google Scholar
[6] Bryan, K., Elementary Inversion of the Laplace Transform, 2006, http://www.rose-hulman.edu/∼bryan/invlap.pdf.Google Scholar
[7] Ciurlia, P. and Roko, I., Valuation of American continuous-installment options, Comput. Econ., 25 (2005), pp. 143–165.Google Scholar
[8] Cohen, A., Numerical Methods for Laplace Transform Inversion, Springer, New York, 2007.Google Scholar
[9] Crump, K. S., Numerical inversion of Laplace transforms using a Fourier series approximation, J. ACM., 23 (1976), pp. 89–96.Google Scholar
[10] Davies, B. and Martin, B., Numerical inversion of the Laplace transform: a survey and comparison of methods, J. Comput. Phys., 33 (1979), pp. 1–32.Google Scholar
[11] Davis, M., Schachermayer, W. and Tompkins, R., Pricing, no-arbitrage bounds and robust hedging of installment options, Quant. Finance., 1 (2001), pp. 597–610.Google Scholar
[12] Davis, M., Schachermayer, W. and Tompkins, R., Installment options and static hedging, J. Risk. Finance., 3 (2002), pp. 46–52.Google Scholar
[13] Dixit, A. R. and Pindyck, R. S., Investment Under Uncertainty, Princeton University Press, 1994.Google Scholar
[14] Dong, C. W., A regularization method for numerical inversion of the Laplace transform, SIAM J. Numer. Anal., 30 (1993), pp. 759–773.Google Scholar
[15] Duffy, D. G., On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications, ACM Trans. Math. Soft., 19 (1993), pp. 333–359.Google Scholar
[16] Ehrhardt, M., Discrete Artificial Boundary Conditions, Ph.D. Thesis, Technische Uni-versität Berlin, 2001.Google Scholar
[17] Gaver, D. P., Observing stochastic processes and approximate transform inversion, Op. Res., 14 (1966), pp. 444–459.Google Scholar
[18] Griebsch, S., kühn, C. and Wystup, U., Instalment Options: a Closed-Form Solution and the Limiting Case, Mathematical Control Theory and Finance, Springer, Heidelberg, 2008.Google Scholar
[19] De Hoog, F.R., Knight, J. H. and Stokes, A. N., An improved method for numerical inversion of Laplace transforms, SIAM J. Sci. Stat. Comp, 3 (1982), pp. 357–366.Google Scholar
[20] Karsenty, F. and Sikorav, J., Installment plan, over the rainbow, Risk., (1993), pp. 36–40.Google Scholar
[21] Kimura, T., Valuing continuous-installment options, Euro. J. Oper. Res., 201 (2010), pp. 222–230.Google Scholar
[22] Kryzhnyi, V., Numerical inversion of the Laplace transform: analysis via regularized analytic continuation, Inverse. Problems., 22 (2006), pp. 579–597.Google Scholar
[23] Kryzhnyi, V., >Freeware DLL for Matlab at http://www.laplacetransform.org/.Freeware+DLL+for+Matlab+at+http://www.laplacetransform.org/.>Google Scholar
[24] Macrae, C. D., The Employee Stock Option: An Installment Option, 2008, Available at SSRN: http://ssrn.com/abstract=1286928.Google Scholar
[25] Migneron, R. and Narayanan, K. S. S., Numerical inversion of Mellin moments and Laplace transforms, Comput. Phys. Commun., 49 (1988), pp. 457–463.Google Scholar
[26] O’cinneide, C., Euler summation for Fourier series and Laplace transform inversion, Stochastic. Models., 13 (1997), pp. 315–337.Google Scholar
[27] Post, E. L., Generalized differentiation, Trans. Am. Math. Soc., 32 (1930), pp. 723–781.Google Scholar
[28] Sneddon, I. H., The Use of Integral Transforms, McGraw-Hill Book Company, 1972.Google Scholar
[29] Stehfest, H., Algorithm 368: numerical inversion of Laplace transform, Commun. ACM., 13 (1970), pp. 47–49.Google Scholar
[30] Thomassen, L. and Van Wouwe, M., Statistics for Industry and Technology, Springer-Verlag, Berlin, 2004.Google Scholar
[31] Valkoó, P. P. and Abate, J., Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion, Comput. Math. Appl., 48 (2004), pp. 629–636.Google Scholar
[32] Weideman, J. A. C., Algorithms for parameter selection in the Weeks method for inverting the Laplace transform, SIAM J. Sci. Comput., 21 (1999), pp. 111–128.Google Scholar
[33] Widder, D. V., The inversion of the Laplace integral and the related moment problem, Trans. Math. Soc., 36 (1934), pp. 107–200.Google Scholar