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BOUNDS ON PRICES FOR ASIAN OPTIONS VIAFOURIER METHODS

Part of:Stochastic processesHarmonic analysis in several variablesMathematical finance

Published online by Cambridge University Press:  19 February 2016

SCOTT ALEXANDEROpen the ORCID record for SCOTT ALEXANDER[Opens in a new window],
ALEXANDER NOVIKOV and
NINO KORDZAKHIA
SCOTT ALEXANDER*
Affiliation:
University of Technology, Sydney, NSW, Australia email scott.alexander@student.uts.edu.au, alex.novikov@uts.edu.au
ALEXANDER NOVIKOV
Affiliation:
University of Technology, Sydney, NSW, Australia email scott.alexander@student.uts.edu.au, alex.novikov@uts.edu.au
NINO KORDZAKHIA
Affiliation:
Macquarie University, Sydney, NSW, Australia email nino.kordzakhia@mq.edu.au
*
✉scott.alexander@student.uts.edu.au
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Abstract

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The problem of pricing arithmetic Asian options is nontrivial, and has attracted much interest over the last two decades. This paper provides a method for calculating bounds on option prices and approximations to option deltas in a market where the underlying asset follows a geometric Lévy process. The core idea is to find a highly correlated, yet more tractable proxy to the event that the option finishes in-the-money. The paper provides a means for calculating the joint characteristic function of the underlying asset and proxy processes, and relies on Fourier methods to compute prices and deltas. Numerical studies show that the lower bound provides accurate approximations to prices and deltas, while the upper bound provides good though less accurate results.

Keywords

Asian optionlower boundcharacteristic functionFourier transforminverse Fourier transformexponential damping

MSC classification

Primary:60G51: Processes with independent increments; L'evy processes
Secondary:91G20: Derivative securities42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 57 , Issue 3 , January 2016, pp. 299 - 318
Copyright
© 2016 Australian Mathematical Society 

References

Alexander, S., Novikov, A. and Kordzakhia, N., “Bounds on prices for Asian-type options via Fourier methods”, 2014, working paper, University of Technology, Sydney.Google Scholar
Alziary, B., Décamps, J.-P. and Koehl, P.-F., “A P.D.E. approach to Asian options: analytical and numerical evidence”, J. Bank. Finance 21 (1997) 613–640; doi:10.1016/S0378-4266(96)00057-X.CrossRefGoogle Scholar
Borovkov, K. and Novikov, A., “On a new approach to calculating expectations for option pricing”, J. Appl. Probab. 39 (2002) 889–895http://www.researchgate.net/publication/38350538_On_a_new_approach_to_calculating_expectations_for_option_pricing.CrossRefGoogle Scholar
Boyle, P., “Options: a Monte Carlo approach”, J. Financ. Econ. 4 (1977) 323–338 doi:10.1016/0304-405X(77)90005-8.CrossRefGoogle Scholar
Boyle, P. and Potapchik, A., “Prices and sensitivities of Asian options: a survey”, Insurance Math. Econom. 42 (2008) 189–211; doi:10.1016/j.insmatheco.2007.02.003.CrossRefGoogle Scholar
Cai, N. and Kou, S., “Pricing Asian options under a hyper-exponential jump diffusion model”, J. Oper. Res. 60 (2012) 64–77; doi:10.1287/opre.1110.1006.Google Scholar
Carr, P. and Madan, D., “Option valuation using the fast Fourier transform”, J. Comput. Finance 2 (1999) 61–73; http://www.math.nyu.edu/research/carrp/papers/pdf/jcfpub.pdf.CrossRefGoogle Scholar
Cont, R. and Tankov, P., Financial modelling with jump processes (Chapman and Hall/CRC, Boca Raton, FL, 2004) https://www.crcpress.com/Financial-Modelling-with-Jump-Processes/Tankov-Cont/9781584884132.Google Scholar
Curran, M., “Valuing Asian and portfolio options by conditioning on the geometric mean price”, Manag. Sci. 40 (1994) 1705–1711; doi:10.1287/mnsc.40.12.1705.CrossRefGoogle Scholar
Dassios, A. and Nagaradjasarma, J., “The square-root process and Asian options”, Quant. Finance 6 (2006) 337–347; doi:10.1080/14697680600724775.CrossRefGoogle Scholar
Dufresne, D., “Laguerre series for Asian and other options”, Math. Finance 10 (2000) 407–428; doi:10.1111/1467-9965.00101.CrossRefGoogle Scholar
Forde, M. and Jacquier, A., “Robust approximations for pricing Asian options and volatility swaps under stochastic volatility”, Appl. Math. Finance 17 (2010) 241–259 doi:10.1080/13504860903335348.CrossRefGoogle Scholar
Foufas, G. and Larson, M., “Valuing Asian options using the finite element method and duality techniques”, J. Comput. Appl. Math. 222 (2008) 144–158; doi:10.1016/j.cam.2007.10.031.CrossRefGoogle Scholar
Fouque, J.-P. and Han, C.-H., “Pricing Asian options with stochastic volatility”, Quant. Finance 3 (2003) 353–362; http://www.pstat.ucsb.edu/faculty/fouque/PubliFM/asiansv3.pdf.CrossRefGoogle Scholar
Funahashi, H. and Kijima, M., “An extension of the chaos expansion approximation for the pricing of exotic basket options”, Appl. Math. Finance 21 (2013) 109–139 doi:10.1080/1350486X.2013.812855.CrossRefGoogle Scholar
Fusai, G. and Meucci, A., “Pricing discretely monitored Asian options under Lévy processes”, J. Bank. Finance 32 (2008) 2076–2088; doi:10.1016/j.jbankfin.2007.12.027.CrossRefGoogle Scholar
Geman, H. and Yor, M., “Bessel processes, Asian options, and perpetuities”, Math. Finance 3 (1993) 349–375; doi:10.1111/j.1467-9965.1993.tb00092.x.CrossRefGoogle Scholar
Kemna, A. and Vorst, A., “A pricing method for options based on average asset values”, J. Bank. Finance 14 (1990) 113–129; doi:10.1016/0378-4266(90)90039-5.CrossRefGoogle Scholar
Lemmens, D., Liang, L., Tempere, J. and De Schepper, A., “Pricing bounds for discrete arithmetic Asian options under Lévy models”, Physica A 389 (2010) 5193–5207 doi:10.1016/j.physa.2010.07.026.CrossRefGoogle Scholar
Lord, R., “Partially exact and bounded approximations for arithmetic Asian options”, J. Comput. Finance 10 (2006) 1–46; http://papers.ssrn.com/abstract=678041.CrossRefGoogle Scholar
Lukacs, E., “A characterization of stable processes”, J. Appl. Probab. 6 (1969) 409–418 doi:10.2307/3212010.CrossRefGoogle Scholar
March, C., “Methods for pricing average rate options in exponential Lévy models”, Ph. D. Thesis, 2008, http://www.researchgate.net/publication/256007174_Methods_for_Pricing_Average_Rate_ Options_in_Exponential_Lvy_Models.Google Scholar
Merton, R., “Option pricing when underlying stock returns are discontinuous”, J. Financ. Econ. 3 (1976) 125–144; doi:10.1016/0304-405X(76)90022-2.CrossRefGoogle Scholar
Milevsky, M. and Posner, S., “Asian options, the sum of lognormals, and the reciprocal gamma distribution”, J. Financ. Quant. Anal. 33 (1998) 409–421; http://papers.ssrn.com/abstract=103770.CrossRefGoogle Scholar
Novikov, A. and Kordzakhia, N., “On lower and upper bounds for Asian-type options: a unified approach”, Proc. Steklov Math. Inst. 287 (2014) 225–231; http://arxiv.org/abs/1309.2383.CrossRefGoogle Scholar
Novikov, A., Ling, T. and Kordzakhia, N., “Pricing of volume-weighted average options: analytical approximations and numerical results”, in: Inspired by finance (Springer International Publishing, Cham, Switzerland, 2014) 461–474; doi:10.1007/978-3-319-02069-3_21.CrossRefGoogle Scholar
Pascucci, A., PDE and martingale methods in option pricing (Springer, Milan, 2011) http://www.springer.com/us/book/9788847017801.CrossRefGoogle Scholar
Rogers, L. and Shi, Z., “The value of an Asian option”, J. Appl. Probab. 32 (1995) 1077–1088; doi:10.2307/3215221.CrossRefGoogle Scholar
Stace, A., “A moment matching approach to the valuation of a volume weighted average price option”, Int. J. Theor. Appl. Finance 10 (2007) 95–110http://www.researchgate.net/publication/23551900_A_moment_matching_approach_to_the_valuation_of_a_volume_weighted_average_price_option.CrossRefGoogle Scholar
Thompson, G., “Fast narrow bounds on the value of Asian options”, 1999, working paper, Centre for Financial Research, Judge Institute of Management, University of Cambridge, 1–12;https://www.jbs.cam.ac.uk/fileadmin/user_upload/research/workingpapers/wp0209.pdf.Google Scholar
Večeř, J., “Unified Asian pricing”, Risk 15 (2002) 113–116http://www.risk.net/data/Pay_per_view/risk/technical/2002/0602_vecer.pdf.Google Scholar