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Analytic approximation for Bachelier option prices and applications

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Listed:
  • Elisa Al`os
  • `Oscar Bur'es

Abstract

It is well-known that, in the Bachelier model, when asset prices and volatilities are uncorrelated, the implied volatility coincides with the fair value of the volatility swap. In this paper, via classical It\^o calculus and Taylor expansions, we write the price for out-of-the-money (OTM) and in-the-money (ITM) options as an expansion with respect to the moneyness, where the coefficients are related to the negative (non-integer) powers of the future mean volatility. As an a application, we use it as a control variate to reduce the variance of Monte Carlo option prices in the correlated case.

Suggested Citation

  • Elisa Al`os & `Oscar Bur'es, 2026. "Analytic approximation for Bachelier option prices and applications," Papers 2605.02040, arXiv.org, revised May 2026.
    • Handle: RePEc:arx:papers:2605.02040
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    References listed on IDEAS

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    1. Masaaki Fukasawa, 2011. "Asymptotic analysis for stochastic volatility: martingale expansion," Finance and Stochastics, Springer, vol. 15(4), pages 635-654, December.
    2. Fabio Antonelli & Sergio Scarlatti, 2009. "Pricing options under stochastic volatility: a power series approach," Finance and Stochastics, Springer, vol. 13(2), pages 269-303, April.
    3. Elisa Alòs & Jim Gatheral & Radoš Radoičić, 2020. "Exponentiation of conditional expectations under stochastic volatility," Quantitative Finance, Taylor & Francis Journals, vol. 20(1), pages 13-27, January.
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