Analytic Approximation for Bachelier Option Prices and Applications

Alòs, Elisa; Burés, Òscar

doi:10.3390/e28060642

Open AccessArticle

Analytic Approximation for Bachelier Option Prices and Applications

by

Elisa Alòs

^1,*

and

Òscar Burés

²

¹

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

²

Departament de Matemàtica Econòmica, Financera i Actuarial, Universitat de Barcelona, Diagonal 690–696, 08034 Barcelona, Spain

^*

Author to whom correspondence should be addressed.

Entropy 2026, 28(6), 642; https://doi.org/10.3390/e28060642

Submission received: 7 May 2026 / Revised: 29 May 2026 / Accepted: 3 June 2026 / Published: 6 June 2026

(This article belongs to the Special Issue Stochastic Processes in Pricing Financial Derivatives)

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Abstract

It is well-known that, in the Bachelier model, when asset prices and volatilities are uncorrelated, the at-the-money implied volatility coincides with the fair value of the volatility swap. Using this identity as a starting point and applying classical Itô 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.

Keywords: Bachelier-type model; option price expansion; Monte Carlo prices

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MDPI and ACS Style

Alòs, E.; Burés, Ò. Analytic Approximation for Bachelier Option Prices and Applications. Entropy 2026, 28, 642. https://doi.org/10.3390/e28060642

AMA Style

Alòs E, Burés Ò. Analytic Approximation for Bachelier Option Prices and Applications. Entropy. 2026; 28(6):642. https://doi.org/10.3390/e28060642

Chicago/Turabian Style

Alòs, Elisa, and Òscar Burés. 2026. "Analytic Approximation for Bachelier Option Prices and Applications" Entropy 28, no. 6: 642. https://doi.org/10.3390/e28060642

APA Style

Alòs, E., & Burés, Ò. (2026). Analytic Approximation for Bachelier Option Prices and Applications. Entropy, 28(6), 642. https://doi.org/10.3390/e28060642

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Analytic Approximation for Bachelier Option Prices and Applications

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