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Pricing options with Green's functions when volatility, interest rate and barriers depend on time

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  • Gregor Dorfleitner
  • Paul Schneider
  • Kurt Hawlitschek
  • Arne Buch

Abstract

We derive the Green's function for the Black-Scholes partial differential equation with time-varying coefficients and time-dependent boundary conditions. We provide a thorough discussion of its implementation within a pricing algorithm that also accommodates American style options. Greeks can be computed as derivatives of the Green's function. Generic handling of arbitrary time-dependent boundary conditions suggests our approach to be used with the pricing of (American) barrier options, although options without barriers can be priced equally well. Numerical results indicate that knowledge of the structure of the Green's function together with the well-developed tools of numerical integration make our approach fast and numerically stable.

Suggested Citation

  • Gregor Dorfleitner & Paul Schneider & Kurt Hawlitschek & Arne Buch, 2008. "Pricing options with Green's functions when volatility, interest rate and barriers depend on time," Quantitative Finance, Taylor & Francis Journals, vol. 8(2), pages 119-133.
    • Handle: RePEc:taf:quantf:v:8:y:2008:i:2:p:119-133
    DOI: 10.1080/14697680601161480
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    References listed on IDEAS

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    1. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
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    Cited by:

    1. R'uben Sousa & Ana Bela Cruzeiro & Manuel Guerra, 2016. "Barrier Option Pricing under the 2-Hypergeometric Stochastic Volatility Model," Papers 1610.03230, arXiv.org, revised Aug 2017.
    2. Gregor Dorfleitner & Paul Schneider & Tanja Veža, 2011. "Flexing the default barrier," Quantitative Finance, Taylor & Francis Journals, vol. 11(12), pages 1729-1743.
    3. Amirhossein Sobhani & Mariyan Milev, 2017. "A Numerical Method for Pricing Discrete Double Barrier Option by Lagrange Interpolation on Jacobi Node," Papers 1712.01060, arXiv.org, revised Feb 2018.
    4. Amirhossein Sobhani & Mariyan Milev, 2017. "A Numerical Method for Pricing Discrete Double Barrier Option by Legendre Multiwavelet," Papers 1703.09129, arXiv.org, revised Mar 2017.
    5. Aleksandar Mijatović, 2010. "Local time and the pricing of time-dependent barrier options," Finance and Stochastics, Springer, vol. 14(1), pages 13-48, January.
    6. Marianito R. Rodrigo, 2020. "Pricing of Barrier Options on Underlying Assets with Jump-Diffusion Dynamics: A Mellin Transform Approach," Mathematics, MDPI, vol. 8(8), pages 1-20, August.

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