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Higher-Order Finite Element Solutions of Option Prices

Author

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  • Raahauge, Peter

    (Department of Finance, Copenhagen Business School)

Abstract

Kinks and jumps in the payoff function of option contracts prevent an effective implementation of higher-order numerical approximation methods. Moreover, the derivatives (the greeks) are not easily determined around such singularities, even with standard lower-order methods. This paper suggests a transformation to turn the original ill-conditioned pricing problem into a well-behaved numerical problem. For a standard test case, both vanilla- and binary call price functions are approximated with (tensor) B-splines of up to 10’th order. Polynomial convergence rates of orders up to approximately 10 are obtained for prices as well as for first and second order derivatives(delta and gamma). Unlike similar studies, numerical approximation errors are measured both as weighted averages and in the supnorm over a state space including time-to-maturities down to a split second.

Suggested Citation

  • Raahauge, Peter, 2004. "Higher-Order Finite Element Solutions of Option Prices," Working Papers 2004-5, Copenhagen Business School, Department of Finance.
  • Handle: RePEc:hhs:cbsfin:2004_005
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    File URL: http://openarchive.cbs.dk/cbsweb/handle/10398/7157
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    References listed on IDEAS

    as
    1. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, April.
    2. Rust, J., 1991. "Estimation of dynamic Structural Models: Problems and Prospects Part I : Discrete Decision Processes," Working papers 9106, Wisconsin Madison - Social Systems.
    3. Barone-Adesi, Giovanni & Bermudez, Ana & Hatgioannides, John, 2003. "Two-factor convertible bonds valuation using the method of characteristics/finite elements," Journal of Economic Dynamics and Control, Elsevier, vol. 27(10), pages 1801-1831, August.
    4. D. M. Pooley & P. A. Forsyth & K. R. Vetzal & R. B. Simpson, 2000. "Unstructured meshing for two asset barrier options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 7(1), pages 33-60.
    5. Steve Heston & Guofu Zhou, 2000. "On the Rate of Convergence of Discrete‐Time Contingent Claims," Mathematical Finance, Wiley Blackwell, vol. 10(1), pages 53-75, January.
    6. Bechmann, Ken L. & Raaballe, Johannes, 2000. "A Regulation of Bids for Dual Class Shares. Implication: Two Shares { One Price," Working Papers 2000-5, Copenhagen Business School, Department of Finance.
    7. P. A. Forsyth & K. R. Vetzal & R. Zvan, 1999. "A finite element approach to the pricing of discrete lookbacks with stochastic volatility," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(2), pages 87-106.
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    More about this item

    Keywords

    Numerical option pricing; Transformed state spaces; Higher-order;
    All these keywords.

    JEL classification:

    • G00 - Financial Economics - - General - - - General

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