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Static super-replicating strategies for a class of exotic options

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  • Chen, Xinliang
  • Deelstra, G.
  • Dhaene, Jan
  • Vanmaele, M.

Abstract

In this paper, we investigate static super-replicating strategies for European-type call options written on a weighted sum of asset prices. This class of exotic options includes Asian options and basket options among others. We assume that there exists a market where the plain vanilla options on the different assets are traded and hence their prices can be observed in the market. Both the infinite market case (where prices of the plain vanilla options are available for all strikes) and the finite market case (where only a finite number of plain vanilla option prices are observed) are considered. We prove that the finite market case converges to the infinite market case when the number of observed plain vanilla option prices tends to infinity. We show how to construct a portfolio consisting of the plain vanilla options on the different assets, whose pay-off super-replicates the pay-off of the exotic option. As a consequence, the price of the super-replicating portfolio is an upper bound for the price of the exotic option. The super-hedging strategy is model-free in the sense that it is expressed in terms of the observed option prices on the individual assets, which can be e.g. dividend paying stocks with no explicit dividend process known. This paper is a generalization of the work of Simon et al. [Simon, S., Goovaerts, M., Dhaene, J., 2000. An easy computable upper bound for the price of an arithmetic Asian option. Insurance Math. Econom. 26 (2-3), 175-184] who considered this problem for Asian options in the infinite market case. Laurence and Wang [Laurence, P., Wang, T.H., 2004. What's a basket worth? Risk Mag. 17, 73-77] and Hobson et al. [Hobson, D., Laurence, P., Wang, T.H., 2005. Static-arbitrage upper bounds for the prices of basket options. Quant. Fin. 5 (4), 329-342] considered this problem for basket options, in the infinite as well as in the finite market case. As opposed to Hobson et al. [Hobson, D., Laurence, P., Wang, T.H., 2005. Static-arbit

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Bibliographic Info

Paper provided by Katholieke Universiteit Leuven in its series Open Access publications from Katholieke Universiteit Leuven with number urn:hdl:123456789/199873.

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Date of creation: Jun 2008
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Publication status: Published in Insurance: Mathematics & Economics (2008-06) v.42, p.1067-1085
Handle: RePEc:ner:leuven:urn:hdl:123456789/199873

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References

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  1. Albrecher, H & Dhaene, Jan & Goovaerts, Marc & Schoutens, Wim, 2003. "Static hedging of Asian options under Lévy models: the comonotonicity approach," Open Access publications from Katholieke Universiteit Leuven urn:hdl:123456789/118273, Katholieke Universiteit Leuven.
  2. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
  3. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-51, October.
  4. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(04), pages 627-627, November.
  5. Kaas, R & Dhaene, Jan & Goovaerts, Marc, 2000. "Upper and lower bounds for sums of random variables," Open Access publications from Katholieke Universiteit Leuven urn:hdl:123456789/223713, Katholieke Universiteit Leuven.
  6. Jamshidian, Farshid, 1989. " An Exact Bond Option Formula," Journal of Finance, American Finance Association, vol. 44(1), pages 205-09, March.
  7. Dhaene, Jan & Wang, Shaun & Young, Virginia & Goovaerts, Marc, 2000. "Comonotonicity and maximal stop-loss premiums," Open Access publications from Katholieke Universiteit Leuven urn:hdl:123456789/200165, Katholieke Universiteit Leuven.
  8. Dhaene, J. & Denuit, M. & Goovaerts, M. J. & Kaas, R. & Vyncke, D., 2002. "The concept of comonotonicity in actuarial science and finance: theory," Insurance: Mathematics and Economics, Elsevier, vol. 31(1), pages 3-33, August.
  9. L. Rüschendorf, 1983. "Solution of a statistical optimization problem by rearrangement methods," Metrika, Springer, vol. 30(1), pages 55-61, December.
  10. Simon, S & Goovaerts, Marc & Dhaene, Jan, 1999. "An easy computable upper bound for the price of an arithmetic Asian option," Open Access publications from Katholieke Universiteit Leuven urn:hdl:123456789/118653, Katholieke Universiteit Leuven.
  11. Dhaene, J. & Denuit, M. & Goovaerts, M. J. & Kaas, R. & Vyncke, D., 2002. "The concept of comonotonicity in actuarial science and finance: applications," Insurance: Mathematics and Economics, Elsevier, vol. 31(2), pages 133-161, October.
  12. David Hobson & Peter Laurence & Tai-Ho Wang, 2005. "Static-arbitrage upper bounds for the prices of basket options," Quantitative Finance, Taylor and Francis Journals, vol. 5(4), pages 329-342.
  13. Muller, Alfred, 1997. "Stop-loss order for portfolios of dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 21(3), pages 219-223, December.
  14. Nielsen, J. Aase & Sandmann, Klaus, 2003. "Pricing Bounds on Asian Options," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 38(02), pages 449-473, June.
  15. Reynaerts, Huguette & Vanmaele, Michele & Dhaene, Jan & Deelstra, Griselda, 2006. "Bounds for the price of a European-style Asian option in a binary tree model," European Journal of Operational Research, Elsevier, vol. 168(2), pages 322-332, January.
  16. Dhaene, Jan, 2002. "The concept of comonotonicity in actuarial science and finance: theory," Open Access publications from Université catholique de Louvain info:hdl:2078.1/105441, Université catholique de Louvain.
  17. J. Dhaene & S. Vanduffel & M. Goovaerts, 2007. "Comonotonicity," Review of Business and Economics, Katholieke Universiteit Leuven, Faculteit Economie en Bedrijfswetenschappen, vol. 0(2), pages 265-278.
  18. Dhaene, Jan, 2002. "The concept of comonotonicity in actuarial science and finance: applications," Open Access publications from Université catholique de Louvain info:hdl:2078.1/105440, Université catholique de Louvain.
  19. Kaas, R. & Dhaene, Jan & Vyncke, D. & Goovaerts, Marc & Denuit, M., 2002. "A simple geometric proof that comonotonic risks have the convex-largest sum," Open Access publications from Katholieke Universiteit Leuven urn:hdl:123456789/200098, Katholieke Universiteit Leuven.
  20. Goovaerts, Marc & Dhaene, Jan & De Schepper, A, 1999. "Stochastic upper bounds for present value functions," Open Access publications from Katholieke Universiteit Leuven urn:hdl:123456789/118656, Katholieke Universiteit Leuven.
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Citations

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Cited by:
  1. Laurence, Peter & Wang, Tai-Ho, 2009. "Sharp distribution free lower bounds for spread options and the corresponding optimal subreplicating portfolios," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 35-47, February.
  2. Dhaene, Jan & Dony, Julia & Forys, Monika & Linders, Daniël & Schoutens, Wim, 2011. "FIX - The fear index. Measuring market fear," Open Access publications from Katholieke Universiteit Leuven urn:hdl:123456789/313200, Katholieke Universiteit Leuven.
  3. Linders, Daniël & Dhaene, Jan & Hounnon, Hippolyte & Vanmaele, Michèle, 2012. "Index options: A model-free approach," Open Access publications from Katholieke Universiteit Leuven urn:hdl:123456789/339066, Katholieke Universiteit Leuven.
  4. Xu, Guoping & Zheng, Harry, 2009. "Approximate basket options valuation for a jump-diffusion model," Insurance: Mathematics and Economics, Elsevier, vol. 45(2), pages 188-194, October.
  5. Dhaene, Jan & Linders, Daniël & Schoutens, Wim & Vyncke, David, 2011. "The herd behavior index: A new measure for systemic risk in financial markets," Open Access publications from Katholieke Universiteit Leuven urn:hdl:123456789/320654, Katholieke Universiteit Leuven.
  6. Mathias Beiglb\"ock & Pierre Henry-Labord\`ere & Friedrich Penkner, 2011. "Model-independent Bounds for Option Prices: A Mass Transport Approach," Papers 1106.5929, arXiv.org, revised Feb 2013.
  7. Griselda Deelstra & Alexandre Petkovic & Michèle Vanmaele, 2008. "Pricing and Hedging Asian Basket Spread Options," Working Papers ECARES 2008_004, ULB -- Universite Libre de Bruxelles.

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