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Short-time behavior of the At-The-Money implied volatility for the jump-diffusion stochastic volatility Bachelier model

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  • Elisa Al`os
  • `Oscar Bur'es
  • Josep Vives

Abstract

In this paper we use Malliavin Calculus techniques in order to obtain expressions for the short-time behavior of the at-the-money implied volatility (ATM-IV) level and skew for a jump-diffusion stock price. The diffusion part is assumed to be the stochastic volatility Bachelier model and the jumps are modeled by a pure-jump L\'evy process with drift so that the stock price is a martingale. Regarding the level, we show that the short-time behavior of the ATM-IV level is the same for all pure-jump L\'evy processes and, regarding the skew, we give conditions on the law of the jumps for the skew to exist. We also give several numerical examples of stochastic volatilities and L\'evy processes that confirm the theoretical results found in the paper.

Suggested Citation

  • Elisa Al`os & `Oscar Bur'es & Josep Vives, 2025. "Short-time behavior of the At-The-Money implied volatility for the jump-diffusion stochastic volatility Bachelier model," Papers 2503.22282, arXiv.org.
    • Handle: RePEc:arx:papers:2503.22282
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    References listed on IDEAS

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    1. Jean-Philippe Aguilar, 2021. "Explicit option valuation in the exponential NIG model," Quantitative Finance, Taylor & Francis Journals, vol. 21(8), pages 1281-1299, August.
    2. Elisa Alòs & Jorge León & Josep Vives, 2007. "On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility," Finance and Stochastics, Springer, vol. 11(4), pages 571-589, October.
    3. Elisa Alòs & Jorge A. León & Monique Pontier & Josep Vives, 2008. "A Hull and White formula for a general stochastic volatility jump-diffusion model with applications to the study of the short-time behavior of the implied volatility," Economics Working Papers 1081, Department of Economics and Business, Universitat Pompeu Fabra.
    4. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    5. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," The Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    6. Elisa Alòs & Kenichiro Shiraya, 2019. "Estimating the Hurst parameter from short term volatility swaps: a Malliavin calculus approach," Finance and Stochastics, Springer, vol. 23(2), pages 423-447, April.
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