A closed-form approximation formula for pricing European options under a three-factor model

Hye Mee Kil, Jeong Hoon Kim

Research output: Contribution to journalArticlepeer-review


The double-mean-reverting model, introduced by Gatheral [(2008). Consistent modeling of SPX and VIX options. In The Fifth World Congress of the Bachelier Finance Society London, July 18], is known to be a successful three-factor model that can be calibrated to both CBOE Volatility Index (VIX) and S&P 500 Index (SPX) options. However, the calibration of this model may be slow because there is no closed-form solution formula for European options. In this paper, we use a rescaled version of the model developed by Huh et al. [(2018). A scaled version of the double-mean-reverting model for VIX derivatives. Mathematics and Financial Economics 12: 495-515] and obtain explicitly a closed-form pricing formula for European option prices. Our formulas for the first and second-order approximations do not require any complicated calculation of integral. We demonstrate that a faster calibration result of the double-mean revering model is available and yet the practical implied volatility surface of SPX options can be produced. In particular, not only the usual convex behavior of the implied volatility surface but also the unusual concave down behavior as shown in the COVID-19 market can be captured by our formula.

Original languageEnglish
Pages (from-to)1214-1240
Number of pages27
JournalProbability in the Engineering and Informational Sciences
Issue number4
Publication statusPublished - 2022 Oct 1

Bibliographical note

Publisher Copyright:
Copyright © The Author(s), 2021. Published by Cambridge University Press.

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Management Science and Operations Research
  • Industrial and Manufacturing Engineering


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