IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1805.03172.html
   My bibliography  Save this paper

Sum of all Black-Scholes-Merton models: An efficient pricing method for spread, basket, and Asian options

Author

Listed:
  • Jaehyuk Choi

Abstract

Contrary to the common view that exact pricing is prohibitive owing to the curse of dimensionality, this study proposes an efficient and unified method for pricing options under multivariate Black-Scholes-Merton (BSM) models, such as the basket, spread, and Asian options. The option price is expressed as a quadrature integration of analytic multi-asset BSM prices under a single Brownian motion. Then the state space is rotated in such a way that the quadrature requires much coarser nodes than it would otherwise or low varying dimensions are reduced. The accuracy and efficiency of the method is illustrated through various numerical experiments.

Suggested Citation

  • Jaehyuk Choi, 2018. "Sum of all Black-Scholes-Merton models: An efficient pricing method for spread, basket, and Asian options," Papers 1805.03172, arXiv.org.
    • Handle: RePEc:arx:papers:1805.03172
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1805.03172
    File Function: Latest version
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Hélyette Geman & Marc Yor, 1993. "Bessel Processes, Asian Options, And Perpetuities," Mathematical Finance, Wiley Blackwell, vol. 3(4), pages 349-375, October.
    2. Kemna, A. G. Z. & Vorst, A. C. F., 1990. "A pricing method for options based on average asset values," Journal of Banking & Finance, Elsevier, vol. 14(1), pages 113-129, March.
    3. Caldana, Ruggero & Fusai, Gianluca, 2013. "A general closed-form spread option pricing formula," Journal of Banking & Finance, Elsevier, vol. 37(12), pages 4893-4906.
    4. Deelstra, G. & Liinev, J. & Vanmaele, M., 2004. "Pricing of arithmetic basket options by conditioning," Insurance: Mathematics and Economics, Elsevier, vol. 34(1), pages 55-77, February.
    5. Ruggero Caldana & Gianluca Fusai & Alessandro Gnoatto & Martino Grasselli, 2016. "General closed-form basket option pricing bounds," Quantitative Finance, Taylor & Francis Journals, vol. 16(4), pages 535-554, April.
    6. Turnbull, Stuart M. & Wakeman, Lee Macdonald, 1991. "A Quick Algorithm for Pricing European Average Options," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 26(3), pages 377-389, September.
    7. Griselda Deelstra & Jan Liinev & Michèle Vanmaele, 2004. "Pricing of arithmetic basket options by conditioning," ULB Institutional Repository 2013/7600, ULB -- Universite Libre de Bruxelles.
    8. Moshe Arye Milevsky & Steven E. Posner, 1999. "Asian Options, The Sum Of Lognormals, And The Reciprocal Gamma Distribution," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar, chapter 7, pages 203-218, World Scientific Publishing Co. Pte. Ltd..
    9. Fusai, Gianluca & Meucci, Attilio, 2008. "Pricing discretely monitored Asian options under Levy processes," Journal of Banking & Finance, Elsevier, vol. 32(10), pages 2076-2088, October.
    10. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.
    11. Mark Broadie & Özgür Kaya, 2006. "Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes," Operations Research, INFORMS, vol. 54(2), pages 217-231, April.
    12. Li, Minqiang, 2008. "Closed-Form Approximations for Spread Option Prices and Greeks," MPRA Paper 6994, University Library of Munich, Germany.
    13. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    14. Levy, Edmond, 1992. "Pricing European average rate currency options," Journal of International Money and Finance, Elsevier, vol. 11(5), pages 474-491, October.
    15. Petter Bjerksund & Gunnar Stensland, 2014. "Closed form spread option valuation," Quantitative Finance, Taylor & Francis Journals, vol. 14(10), pages 1785-1794, October.
    16. Margrabe, William, 1978. "The Value of an Option to Exchange One Asset for Another," Journal of Finance, American Finance Association, vol. 33(1), pages 177-186, March.
    17. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    18. Vadim Linetsky, 2004. "Spectral Expansions for Asian (Average Price) Options," Operations Research, INFORMS, vol. 52(6), pages 856-867, December.
    19. E. Benhamou, 2001. "Fast Fourier Transform for discrete Asian Options," Computing in Economics and Finance 2001 6, Society for Computational Economics.
    20. Michael Curran, 1994. "Valuing Asian and Portfolio Options by Conditioning on the Geometric Mean Price," Management Science, INFORMS, vol. 40(12), pages 1705-1711, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Jaehyuk Choi & Minsuk Kwak & Chyng Wen Tee & Yumeng Wang, 2021. "A Black-Scholes user's guide to the Bachelier model," Papers 2104.08686, arXiv.org, revised Feb 2022.
    2. Matteo Gardini & Piergiacomo Sabino, 2022. "Exchange option pricing under variance gamma-like models," Papers 2207.00453, arXiv.org.
    3. Olkhov, Victor, 2020. "Classical Option Pricing and Some Steps Further," MPRA Paper 105431, University Library of Munich, Germany, revised 28 Dec 2020.
    4. Michael Samet & Christian Bayer & Chiheb Ben Hammouda & Antonis Papapantoleon & Ra'ul Tempone, 2022. "Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in L\'evy Models," Papers 2203.08196, arXiv.org, revised Oct 2023.
    5. Jaehyuk Choi & Minsuk Kwak & Chyng Wen Tee & Yumeng Wang, 2022. "A Black–Scholes user's guide to the Bachelier model," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(5), pages 959-980, May.
    6. Anatoliy A. Pogorui & Anatoliy Swishchuk & Ramón M. Rodríguez-Dagnino, 2022. "Asymptotic Estimation of Two Telegraph Particle Collisions and Spread Options Valuations," Mathematics, MDPI, vol. 10(13), pages 1-14, June.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jinke Zhou & Xiaolu Wang, 2008. "Accurate closed‐form approximation for pricing Asian and basket options," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 24(4), pages 343-358, July.
    2. Yijuan Liang & Xiuchuan Xu, 2019. "Variance and Dimension Reduction Monte Carlo Method for Pricing European Multi-Asset Options with Stochastic Volatilities," Sustainability, MDPI, vol. 11(3), pages 1-21, February.
    3. Chueh-Yung Tsao & Chao-Ching Liu, 2012. "Asian Options with Credit Risks: Pricing and Sensitivity Analysis," Emerging Markets Finance and Trade, Taylor & Francis Journals, vol. 48(S3), pages 96-115, September.
    4. Gianluca Fusai & Ioannis Kyriakou, 2016. "General Optimized Lower and Upper Bounds for Discrete and Continuous Arithmetic Asian Options," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 531-559, May.
    5. Dan Pirjol & Lingjiong Zhu, 2017. "Asymptotics for the Discrete-Time Average of the Geometric Brownian Motion and Asian Options," Papers 1706.09659, arXiv.org.
    6. Pingping Zeng & Yue Kuen Kwok, 2016. "Pricing bounds and approximations for discrete arithmetic Asian options under time-changed Lévy processes," Quantitative Finance, Taylor & Francis Journals, vol. 16(9), pages 1375-1391, September.
    7. Kwangil Bae, 2019. "Valuation and applications of compound basket options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 39(6), pages 704-720, June.
    8. Ping Wu & Robert J. Elliott, 2017. "A simple efficient approximation to price basket stock options with volatility smile," Annals of Finance, Springer, vol. 13(1), pages 1-29, February.
    9. Dai, Min & Li, Peifan & Zhang, Jin E., 2010. "A lattice algorithm for pricing moving average barrier options," Journal of Economic Dynamics and Control, Elsevier, vol. 34(3), pages 542-554, March.
    10. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.
    11. Boyle, Phelim & Potapchik, Alexander, 2008. "Prices and sensitivities of Asian options: A survey," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 189-211, February.
    12. Sander Willems, 2018. "Asian Option Pricing with Orthogonal Polynomials," Papers 1802.01307, arXiv.org, revised Sep 2018.
    13. Xueping Wu & Jin Zhang, 1999. "Options on the minimum or the maximum of two average prices," Review of Derivatives Research, Springer, vol. 3(2), pages 183-204, May.
    14. Lemmens, D. & Liang, L.Z.J. & Tempere, J. & De Schepper, A., 2010. "Pricing bounds for discrete arithmetic Asian options under Lévy models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(22), pages 5193-5207.
    15. Manuel Moreno & Javier F. Navas, 2008. "Australian Options," Australian Journal of Management, Australian School of Business, vol. 33(1), pages 69-93, June.
    16. Leccadito, Arturo & Paletta, Tommaso & Tunaru, Radu, 2016. "Pricing and hedging basket options with exact moment matching," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 59-69.
    17. Chih-Chen Hsu & Chung-Gee Lin & Tsung-Jung Kuo, 2020. "Pricing of Arithmetic Asian Options under Stochastic Volatility Dynamics: Overcoming the Risks of High-Frequency Trading," Mathematics, MDPI, vol. 8(12), pages 1-16, December.
    18. Keng‐Hsin Lo & Kehluh Wang & Ming‐Feng Hsu, 2008. "Pricing European Asian options with skewness and kurtosis in the underlying distribution," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 28(6), pages 598-616, June.
    19. Georges Dionne & Genevieve Gauthier & Nadia Ouertani & Nabil Tahani, 2011. "Heterogeneous Basket Options Pricing Using Analytical Approximations," Multinational Finance Journal, Multinational Finance Journal, vol. 15(1-2), pages 47-85, March - J.
    20. Geon Ho Choe & Minseok Kim, 2021. "Closed‐form lower bounds for the price of arithmetic average Asian options by multiple conditioning," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 41(12), pages 1916-1932, December.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1805.03172. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.