Review of Derivatives Research

, Volume 22, Issue 1, pp 1–40 | Cite as

A general closed form option pricing formula

  • Ciprian NeculaEmail author
  • Gabriel Drimus
  • Walter Farkas


A new method to retrieve the risk-neutral probability measure from observed option prices is developed and a closed form pricing formula for European options is obtained by employing a modified Gram–Charlier series expansion, known as the Gauss–Hermite expansion. This expansion converges for fat-tailed distributions commonly encountered in the study of financial returns. The expansion coefficients can be calibrated from observed option prices and can also be computed, for example, in models with the probability density function or the characteristic function known in closed form. We investigate the properties of the new option pricing model by calibrating it to both real-world and simulated option prices and find that the resulting implied volatility curves provide an accurate approximation for a wide range of strike prices. Based on an extensive empirical study, we conclude that the new approximation method outperforms other methods both in-sample and out-of-sample.


European options Expansion based approximation of risk-neutral density Gauss–Hermite series expansion Calibration 

JEL Classification

C63 G13 



We thank an anonymous referee as well as the participants at the conferences Global Derivatives 2016, 9th World Congress of the Bachelier Finance Society 2016, Challenges in Derivatives Markets 2015, Stochastics and Computational Finance 2015, International Conference on Operations Research 2015, and Quantitative Methods in Finance 2015 for helpful comments.


The research leading to these results has received funding from the SCIEX Project 11.159 and from the European Union Seventh Framework Programme (FP7/2007-2013) under the MC-IEF Grant Agreement No. 627701.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Banking and FinanceUniversity of ZurichZurichSwitzerland
  2. 2.Department of Money and Banking, DOFINBucharest University of Economic StudiesBucharestRomania
  3. 3.Department of MathematicsETH ZurichZurichSwitzerland
  4. 4.Swiss Finance InstituteZurichSwitzerland

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