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USING MODEL-INDEPENDENT LOWER BOUNDS TO IMPROVE PRICING OF ASIAN STYLE OPTIONS IN LÉVY MARKETS

Published online by Cambridge University Press:  19 February 2014

Griselda Deelstra ,
Grégory Rayée ,
Steven Vanduffel  and
Jing Yao
Griselda Deelstra
Affiliation:
Department of Mathematics, Université Libre de Bruxelles, Bruxelles, Belgium E-Mail: griselda.deelstra@ulb.ac.be
Grégory Rayée
Affiliation:
Department of Mathematics, Université Libre de Bruxelles, Bruxelles, Belgium E-Mail: Gregory.Rayee@ulb.ac.be
Steven Vanduffel*
Affiliation:
Department of Economics, Vrije Universiteit Brussel, Bruxelles, Belgium
Jing Yao
Affiliation:
Department of Economics, Vrije Universiteit Brussel, Bruxelles, Belgium E-mail: jingyao@vub.ac.be
*
E-mail: steven.vanduffel@vub.ac.be
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Abstract

Albrecher et al. (Albrecher, H., Mayer Ph., Schoutens, W. (2008) General lower bounds for arithmetic Asian option prices. Applied Mathematical Finance, 15, 123–149) have proposed model-independent lower bounds for arithmetic Asian options. In this paper we provide an alternative and more elementary derivation of their results. We use the bounds as control variates to develop a simple Monte Carlo method for pricing contracts with Asian-style features. The conditioning idea that is inherent in our approach also inspires us to propose a new semi-analytic pricing approach. We compare both approaches and conclude that these both have their merits and are useful in practice. In particular, we point out that our newly proposed Monte Carlo method allows to deal with Asian-style products that appear in insurance (e.g., unit-linked contracts) in a very efficient way, and outperforms other known Monte Carlo methods that are based on control variates.

Keywords

Asian-style optionsconditional expectationcontrol variatesstochastic clock
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 44 , Issue 2 , May 2014 , pp. 237 - 276
Copyright
Copyright © ASTIN Bulletin 2014 

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References

Abramowitz, M. and Stegun, C.A. (1964) Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover.Google Scholar
Albrecher, H. and Predota, M. (2004) On asian option pricing for NIG lévy processes. Journal of Computational and Applied Mathemaics, 172, 153–168.CrossRefGoogle Scholar
Albrecher, H., Mayer, Ph. and Schoutens, W. (2008) General lower bounds for arithmetic Asian option prices. Applied Mathematical Finance, 15, 123–149.CrossRefGoogle Scholar
Bachelier, L. (1900) Théorie de la spéculation. PhD thesis, Gauthier-Villars, Paris, France.CrossRefGoogle Scholar
Ballotta, L. (2010) Efficient pricing of ratchet equity-indexed annuities in a variance-gamma economy. North American Actuarial Journal, 14, 355–368.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. (1998) Processes of normal inverse gaussian type. Finance and Stochastics, 2, 41–68.CrossRefGoogle Scholar
Boyle, P. and Potapchik, A. (2008) Prices and sensitivities of Asian options: A survey. Insurance: Mathematics and Economics, 42, 189–211.Google Scholar
Browne, S., Milevsky, M.A. and Salisbury, T.S. (2003) Asset allocation and the liquidity premium for illiquid annuities. Journal of Risk and Insurance, 70, 509–226.CrossRefGoogle Scholar
Carr, P. and Chou, A. (1997) Breaking barriers. Risk, 10, 139–145.Google Scholar
Carr, P. and Madan, D. (1998) Option valuation using the fast fourier transform. Journal of Computational Finance, 2, 61–73.CrossRefGoogle Scholar
Carr, P., Geman, H., Madan, D. and Yor, M. (2002) The fine structure of asset returns: An empirical investigation. Journal of Business, 75, 305–332.CrossRefGoogle Scholar
Cont, R. and Tankov, P. (2004) Financial Modeling with Jump Processes, CRC Financial Mathematics Series. London: Chapman & Hall.Google Scholar
Dingec, K.D. and Hormann, W. (2012) A general control variate method for option pricing under lévy process. European Journal of Operational Research, 221, 368–377.CrossRefGoogle Scholar
Eberlein, E. and Keller, U. (1995) Hyperbolic distributions in finance. Bernoulli, 1, 281–299.CrossRefGoogle Scholar
Eberlein, E., Keller, U. and Prause, K. (1998) New insights into smile, mis-pricing and value at risk: The hyperbolic model. Journal of Business, 71, 371–405.CrossRefGoogle Scholar
Fenton, L.F. (1960) The sum of log-normal probability distributions in scatter transmission systems. IEEE Transactions on Communication Systems, 8, 57–67.CrossRefGoogle Scholar
Fu, M., Madan, D. and Wang, T. (1999) Pricing continuous Asian options: A comparison of Monte Carlo and Laplace transform inversion methods. Journal of Computational Finance, 2, 49–74.CrossRefGoogle Scholar
Fusai, G. and Meucci, A. (2008) Pricing discretely monitored Asian option under lévy processes. Journal of Banking and Finance, 32, 2076–2088.CrossRefGoogle Scholar
Garcia, J., Goosens, S., Viktoriya, M. and Schoutens, W. (2007) Lévy Base Correlation. Technical report 07-04, Section of Statistics, K.U. Leuven, TU/e, the Netherlands.Google Scholar
Gerber, H. and Shiu, S.W. (1994) Option pricing by Esscher transforms. Transactions of the Society of Actuaries, 46, 99–191.Google Scholar
Glasserman, P. (2003) Monte Carlo Methods in Financial Engineering. New York, NY: Springer.CrossRefGoogle Scholar
Kaas, R., Dhaene, J. and Goovaerts, M. (2000) Upper and lower bounds for sums of random variables. Insurance: Mathematics and Economics, 27, 151–168.Google Scholar
Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2008) Modern Actuarial Risk Theory. New York, NY: Springer.CrossRefGoogle Scholar
Kemna, A. and Vorst, A. (1990) A pricing method for options based on average asset values. Journal of Banking and Finance, 14, 113–130.CrossRefGoogle Scholar
L'Ecuyer, P. (1994) Efficiency improvement and variance reduction. In Proceedings of the 1994 Winter Simulation Conference, Dec. 1994, 122–132.Google Scholar
Lin, X. and Tan, K. (2003) Valuation of equity-indexed annuities under stochastic interest rates. North American Actuarial Journal, 7, 72–93.Google Scholar
Madan, D. and Senata, E. (1990) The variance gamma model for share market returns. Journal of Business, 63, 511–524.CrossRefGoogle Scholar
Madan, D., Carr, P. and Chang, E. (1998) The variance gamma process and option pricing. European Finance Review, 2, 79–105.CrossRefGoogle Scholar
Madan, D.B., Pistorius, M.R. and Schoutens, W. (2011) The valuation of structured products using Markov chain models. Quantitative Finance, 13, 125–136.CrossRefGoogle Scholar
Mandelbrot, B. (1963) The variation of certain speculative prices. Journal of Business, 36, 394–419.CrossRefGoogle Scholar
Marrion, J. (2000) Advantage 2000 equity index report (November). Available online at http://www.indexannuity.org.Google Scholar
Marrion, J. (2001) Advantage 2001 equity index report (March). Available online at http://www.indexannuity.org.Google Scholar
Milevsky, M. and Posner, S. (2003) The titanic option: Valuation of the guaranteed minimum death benefit in variable annuities and mutual funds. Journal of Risk and Insurance, 68, 93–128.CrossRefGoogle Scholar
Monroe, I. (1978) Processes that can be embedded in Brownian motion. Annals of Probability, 6, 42–56.CrossRefGoogle Scholar
Müller, A. and Stoyan, D. (2002) Comparison Methods for Stochastic Models and Risks. Hoboken, NJ: Wiley.Google Scholar
Prause, K. (1999) The Generalized Hyperbolic Model: Estimation, Financial Derivatives, and Risk Measures. PhD thesis, University of Freiburg, Germany.Google Scholar
Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (1992) Numerical Recipes in C. Cambridge, UK: Cambridge University Press.Google Scholar
Romeo, M., Da Costa, V. and Bardou, F. (2003) Broad distribution effects in sums of lognormal random variables. European Physical Journal B, 32, 513–525.CrossRefGoogle Scholar
Sato, K. (1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge, UK: Cambridge University Press.Google Scholar
Schoutens, W. (2003) Lévy Processes in Finance: Pricing Financial Derivatives. Hoboken, NJ: Wiley.CrossRefGoogle Scholar
Schrager, D. and Pelsser, A. (2004) Pricing rate of return guarantees in regular premium unit linked insurance. Insurance: Mathematics and Economics, 35, 369–398.Google Scholar
Shaked, M. and Shanthikumar, J.G. (1994) Stochastic Orders and Their Applications. New York, NY: Academic Press.Google Scholar
Stroud, A.H. and Secrest, D. (1966) Gaussian Quadrature Formulas. Upper Saddle River, NJ: Prentice-Hall.Google Scholar
Valdez, E.A., Dhaene, J., Maj, M. and Vanduffel, S. (2009) Bounds and approximations for sums of dependent log-elliptical random variables. Insurance: Mathematics and Economics, 44 (3), 385–397.Google Scholar
Vanduffel, S., Chen, X., Dhaene, J., Goovaerts, M., Henrard, L. and Kaas, R. (2008a) Optimal approximations for risk measure of sums of log normals based on conditional expectations. Journal of Computational and Applied Mathematics, 221, 202–218.CrossRefGoogle Scholar
Vanduffel, S., Shang, Z., Henrard, L., Dhaene, J. and Valdez, E.A. (2008b) Analytic bounds and approximations for annuities and asian options. Insurance: Mathematics and Economics, 42 (3), 1109–1117.Google Scholar