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On The Implied Volatility Of European And Asian Call Options Under The Stochastic Volatility Bachelier Model

Author

Listed:
  • ELISA ALÃ’S

    (Department of Economics and Business, Universitat Pompeu Fabra and Barcelona School of Economics, Ramón Trias Fargas 25-27, 08005 Barcelona, Catalonia, Spain)

  • EULALIA NUALART

    (Department of Economics and Business, Universitat Pompeu Fabra and Barcelona School of Economics, Ramón Trias Fargas 25-27, 08005 Barcelona, Catalonia, Spain)

  • MAKAR PRAVOSUD

    (Department of Economics and Business, Universitat Pompeu Fabra and Barcelona School of Economics, Ramón Trias Fargas 25-27, 08005 Barcelona, Catalonia, Spain)

Abstract

In this paper, we study the short-time behavior of the at-the-money implied volatility for European and arithmetic Asian call options with fixed strike price. The asset price is assumed to follow the Bachelier model with a general stochastic volatility process. Using techniques of the Malliavin calculus such as the anticipating Itô’s formula, we first compute the level of the implied volatility when the maturity converges to zero. Then, we find a short-maturity asymptotic formula for the skew of the implied volatility that depends on the roughness of the volatility model. We apply our general results to the stochastic alpha–beta–rho (SABR), fractional Bergomi and local volatility models, and provide some numerical simulations that confirm the accurateness of the asymptotic formula for the skew.

Suggested Citation

  • Elisa Alã’S & Eulalia Nualart & Makar Pravosud, 2024. "On The Implied Volatility Of European And Asian Call Options Under The Stochastic Volatility Bachelier Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 27(07n08), pages 1-28, December.
    • Handle: RePEc:wsi:ijtafx:v:27:y:2024:i:07n08:n:s0219024925500037
    DOI: 10.1142/S0219024925500037
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