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Tools for Computational Finance pp 179–257Cite as

Standard Methods for Standard Options

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Abstract

This chapter elaborates the numerical solution of the Black-Scholes equation for European plain-vanilla options, and of the corresponding inequalities for the American-style case. Following the Black-Scholes model, this chapter is confined to constant coefficients. This allows to solve an equivalent partial-differential equation of the simplest parabolic type. Several finite-difference schemes are explained, as well as numerical stability. Boundary conditions are introduced, which lead to obstacle problems in the American-style case and to a formulation as linear complementarity problem. The solution of the free-boundary problem is tackled by the Brennan-Schwartz approach. Error control and accuracy are discussed. The final part of this chapter is devoted to analytic methods. This includes the interpolation method, the quadratic approximation, the analytic method of lines, and quadrature methods for an integral representation. Finally we discuss criteria for the comparison of different methods and for judging their efficiency.

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Notes

  1. 1.

    But the corresponding solutions V (S, t) and their early-exercise structure will be different. The Notes and Comments summarize how to correctly compensate for a discrete dividend payment.

  2. 2.

    Too large values of | a | or b can lead to underflow or overflow when evaluating the exponential function.

  3. 3.

    The zeros in the corner of the matrix G symbolize the triangular zero structure of (4.13).

  4. 4.

    For S = 0 the PDE is no longer parabolic.

  5. 5.

    Of course, the holder may wish to sell the option.

  6. 6.

    The final balance for a put after exercising is Ker(T−t). The reader is encouraged to show that holding is less profitable (Seδ(T−t) < Ker(T−t)), at least for small r(T − t). When a discrete dividend is paid, the stopping region is not necessarily connected ( → Exercise 4.1b).

  7. 7.

    Up to localization.

  8. 8.

    This is illustrated in Topic 9 of the Topics fCF.

  9. 9.

    Notation: In this Sect. 4.6.2, x does not have the meaning of transformation (4.3), and r not that of an interest rate, and y is no PDE solution. Here, \(x,y \in \mathbb{R}^{m-1}\).

  10. 10.

    For the KKT (Karush-Kuhn-Tucker or Kuhn-Tucker) theory we refer to [348, 350]. In our context, m − 1.

  11. 11.

    Successive overrelaxation, SOR. For an introduction to classic iterative methods for the solution of systems of linear equations Ax = b we refer to Appendix C.2.

  12. 12.

    The S-interval must be large enough, S 1 < S f.

  13. 13.

    With m = 20000, our best result was 1.8816935.

  14. 14.

    The latter situation might cause some uncertainty on the costs.

  15. 15.

    The files BENCHMARK00 for δ = 0 and BENCHMARK01 for δ = 0. 1 can be found on www.compfin.de.

  16. 16.

    All of the above methods were implemented in FORTRAN (F90 compiler) and run on a DS20 processor.

    Fig. 4.19
    figure 19

    Computing times and RMS errors of several methods, see the text. Points mark calculated RMS errors; corresponding points are connected by lines

    Full size image
  17. 17.

    For tree methods, dividends are discussed in Appendix D.2.

  18. 18.

    As already mentioned in Sect. 4.7, the risk of having chosen an inappropriate model is mostly larger than the risk of inaccurate digits.

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Seydel, R.U. (2017). Standard Methods for Standard Options. In: Tools for Computational Finance. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-7338-0_4

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