{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T11:22:05Z","timestamp":1785410525476,"version":"3.56.0"},"reference-count":17,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2026,6,6]],"date-time":"2026-06-06T00:00:00Z","timestamp":1780704000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003030","name":"AGAUR","doi-asserted-by":"publisher","award":["2025 FI-1 00580"],"award-info":[{"award-number":["2025 FI-1 00580"]}],"id":[{"id":"10.13039\/501100003030","id-type":"DOI","asserted-by":"publisher"}]},{"award":["2025 FI-1 00580"],"award-info":[{"award-number":["2025 FI-1 00580"]}],"id":[{"id":"https:\/\/ror.org\/01n4pqe45","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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 It\u00f4 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.<\/jats:p>","DOI":"10.3390\/e28060642","type":"journal-article","created":{"date-parts":[[2026,6,8]],"date-time":"2026-06-08T07:43:02Z","timestamp":1780904582000},"page":"642","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Analytic Approximation for Bachelier Option Prices and Applications"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4234-0671","authenticated-orcid":false,"given":"Elisa","family":"Al\u00f2s","sequence":"first","affiliation":[{"name":"Department of Economics and Business, Barcelona School of Economics, Universitat Pompeu Fabra, Ram\u00f3n Trias Fargas 25-27, 08005 Barcelona, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-1741-1308","authenticated-orcid":false,"given":"\u00d2scar","family":"Bur\u00e9s","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tica Econ\u00f2mica, Financera i Actuarial, Universitat de Barcelona, Diagonal 690\u2013696, 08034 Barcelona, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,6,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"21","DOI":"10.24033\/asens.476","article-title":"Th\u00e9orie de la sp\u00e9culation","volume":"17","author":"Bachelier","year":"1900","journal-title":"Ann. Sci. L\u2019\u00c9cole Norm. Sup\u00e9rieure"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"959","DOI":"10.1002\/fut.22315","article-title":"A Black\u2013Scholes user\u2019s guide to the Bachelier model","volume":"42","author":"Choi","year":"2022","journal-title":"J. Futur. Mark."},{"key":"ref_3","unstructured":"Lewis, A.L. (2016). Option Valuation Under Stochastic Volatility, II, Finance Press."},{"key":"ref_4","first-page":"249","article-title":"Managing smile risk","volume":"1","author":"Hagan","year":"2002","journal-title":"Best Wilmott"},{"key":"ref_5","unstructured":"Fouque, J.P., Papanicolaou, G., and Sircar, K.R. (2000). Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1648","DOI":"10.1137\/S0036139902401550","article-title":"Singular Perturbations in Option Pricing","volume":"63","author":"Fouque","year":"2003","journal-title":"SIAM J. Appl. 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Stoch."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1080\/14697688.2019.1642506","article-title":"Exponentiation of conditional expectations under stochastic volatility","volume":"20","author":"Gatheral","year":"2020","journal-title":"Quant. Financ."},{"key":"ref_12","unstructured":"Baviera, R., and Massaria, M.D. (2025). Smile asymptotic for Bachelier Implied Volatility. arXiv."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Floc\u2019h, F.L. (2022). On the Bachelier implied volatility at extreme strikes. arXiv.","DOI":"10.54946\/wilm.11076"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Al\u00f2s, E., Bur\u00e9s, \u00d2., and Vives, J. (2025). Short-time behavior of the At-The-Money implied volatility for the jump-diffusion stochastic volatility Bachelier model. arXiv.","DOI":"10.1137\/25M1776615"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1747","DOI":"10.1080\/14697688.2022.2071759","article-title":"Proof of non-convergence of the short-maturity expansion for the SABR model","volume":"22","author":"Lewis","year":"2022","journal-title":"Quant. Financ."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Antonov, A., Konikov, M., and Spector, M. (2019). Modern SABR Analytics, Springer. SpringerBriefs in Quantitative Finance.","DOI":"10.1007\/978-3-030-10656-0"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1537","DOI":"10.1007\/s11009-020-09827-5","article-title":"Small-t expansion for the Hartman-Watson distribution","volume":"23","author":"Pirjol","year":"2021","journal-title":"Methodol. Comput. Appl. Probab."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/6\/642\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T04:31:01Z","timestamp":1781152261000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/6\/642"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,6]]},"references-count":17,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2026,6]]}},"alternative-id":["e28060642"],"URL":"https:\/\/doi.org\/10.3390\/e28060642","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,6]]}}}