Boundary evolution equations for american options

Daniel Mitchell, Jonathan Goodman, Kumar Muthuraman

    Research output: Contribution to journalArticle

    5 Citations (Scopus)

    Abstract

    We consider the problem of finding optimal exercise policies for American options, both under constant and stochastic volatility settings. Rather than work with the usual equations that characterize the price exclusively, we derive and use boundary evolution equations that characterize the evolution of the optimal exercise boundary. Using these boundary evolution equations we show how one can construct very efficient computational methods for pricing American options that avoid common sources of error. First, we detail a methodology for standard static grids and then describe an improvement that defines a grid that evolves dynamically while solving the problem. When integral representations are available, as in the Black-Scholes setting, we also describe a modified integral method that leverages on the representation to solve the boundary evolution equations. Finally we compare runtime and accuracy to other popular numerical methods. The ideas and methodology presented herein can easily be extended to other optimal stopping problems.

    Original languageEnglish (US)
    Pages (from-to)505-532
    Number of pages28
    JournalMathematical Finance
    Volume24
    Issue number3
    DOIs
    StatePublished - 2014

    Fingerprint

    American Options
    Computational methods
    Evolution Equation
    Numerical methods
    Exercise
    Costs
    Grid
    Black-Scholes
    Optimal Stopping Problem
    Stochastic Volatility
    Methodology
    Integral Method
    methodology
    Leverage
    Integral Representation
    Computational Methods
    Pricing
    pricing
    Numerical Methods
    American options

    Keywords

    • American options
    • Dynamic grid
    • Early exercise boundary
    • Free-boundary problem
    • Optimal stopping
    • Stochastic volatility

    ASJC Scopus subject areas

    • Accounting
    • Social Sciences (miscellaneous)
    • Finance
    • Economics and Econometrics
    • Applied Mathematics

    Cite this

    Mitchell, D., Goodman, J., & Muthuraman, K. (2014). Boundary evolution equations for american options. Mathematical Finance, 24(3), 505-532. https://doi.org/10.1111/mafi.12002

    Boundary evolution equations for american options. / Mitchell, Daniel; Goodman, Jonathan; Muthuraman, Kumar.

    In: Mathematical Finance, Vol. 24, No. 3, 2014, p. 505-532.

    Research output: Contribution to journalArticle

    Mitchell, D, Goodman, J & Muthuraman, K 2014, 'Boundary evolution equations for american options', Mathematical Finance, vol. 24, no. 3, pp. 505-532. https://doi.org/10.1111/mafi.12002
    Mitchell, Daniel ; Goodman, Jonathan ; Muthuraman, Kumar. / Boundary evolution equations for american options. In: Mathematical Finance. 2014 ; Vol. 24, No. 3. pp. 505-532.
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