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The Black‐Scholes Equation Revisited: Asymptotic Expansions And Singular Perturbations

Author

Listed:
  • Martin Widdicks
  • Peter W. Duck
  • Ari D. Andricopoulos
  • David P. Newton

Abstract

In this paper, novel singular perturbation techniques are applied to price European, American, and barrier options. Employment of these methods leads to a significant simplification of the problem in all cases, by reducing the number of parameters. For American options, the valuation problem is reduced to a procedure that may be performed on a rudimentary handheld calculator. The method also sheds light on the evolution of option prices for all of the cases considered, the results being particularly illuminating for American and barrier options.

Suggested Citation

  • Martin Widdicks & Peter W. Duck & Ari D. Andricopoulos & David P. Newton, 2005. "The Black‐Scholes Equation Revisited: Asymptotic Expansions And Singular Perturbations," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 373-391, April.
    • Handle: RePEc:bla:mathfi:v:15:y:2005:i:2:p:373-391
    DOI: 10.1111/j.0960-1627.2005.00224.x
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    Citations

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    Cited by:

    1. Nicholas Sharp & David Newton & Peter Duck, 2008. "An Improved Fixed-Rate Mortgage Valuation Methodology with Interacting Prepayment and Default Options," The Journal of Real Estate Finance and Economics, Springer, vol. 36(3), pages 307-342, April.
    2. Gao, Jianwei, 2010. "An extended CEV model and the Legendre transform-dual-asymptotic solutions for annuity contracts," Insurance: Mathematics and Economics, Elsevier, vol. 46(3), pages 511-530, June.
    3. Alev{s} v{C}ern'y & Stephan Denkl & Jan Kallsen, 2013. "Hedging in L\'evy Models and the Time Step Equivalent of Jumps," Papers 1309.7833, arXiv.org, revised Jul 2017.
    4. Peter W. Duck & Geoffrey W. Evatt & Paul V. Johnson, 2014. "Perpetual Options on Multiple Underlyings," Applied Mathematical Finance, Taylor & Francis Journals, vol. 21(2), pages 174-200, April.
    5. Yang, Nian & Chen, Nan & Liu, Yanchu & Wan, Xiangwei, 2017. "Approximate arbitrage-free option pricing under the SABR model," Journal of Economic Dynamics and Control, Elsevier, vol. 83(C), pages 198-214.
    6. Peter W. Duck & Chao Yang & David P. Newton & Martin Widdicks, 2009. "Singular Perturbation Techniques Applied To Multiasset Option Pricing," Mathematical Finance, Wiley Blackwell, vol. 19(3), pages 457-486, July.
    7. Pagliarani, Stefano & Pascucci, Andrea, 2011. "Analytical approximation of the transition density in a local volatility model," MPRA Paper 31107, University Library of Munich, Germany.
    8. Kristoffer Glover & Peter W Duck & David P Newton, 2010. "On nonlinear models of markets with finite liquidity: Some cautionary notes," Published Paper Series 2010-5, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    9. Seda Gulen & Catalin Popescu & Murat Sari, 2019. "A New Approach for the Black–Scholes Model with Linear and Nonlinear Volatilities," Mathematics, MDPI, vol. 7(8), pages 1-14, August.
    10. Stefano, Pagliarani & Pascucci, Andrea & Candia, Riga, 2011. "Expansion formulae for local Lévy models," MPRA Paper 34571, University Library of Munich, Germany.
    11. Adi Ben-Meir & Jeremy Schiff, 2012. "The Variance of Standard Option Returns," Papers 1204.3452, arXiv.org.

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