Review of Quantitative Finance and Accounting

, Volume 38, Issue 1, pp 109–129

Non-parametric method for European option bounds

Original Research


There is much research whose efforts have been devoted to discovering the distributional defects in the Black–Scholes model, which are known to cause severe biases. However, with a free specification for the distribution, one can only find upper and lower bounds for option prices. In this paper, we derive a new non-parametric lower bound and provide an alternative interpretation of Ritchken’s (J Finance 40:1219–1233, 1985) upper bound to the price of the European option. In a series of numerical examples, our new lower bound is substantially tighter than previous lower bounds. This is prevalent especially for out of the money options where the previous lower bounds perform badly. Moreover, we present how our bounds can be derived from histograms which are completely non-parametric in an empirical study. We discover violations in our lower bound and show that those violations present arbitrage profits. In particular, our empirical results show that out of the money calls are substantially overpriced (violate the lower bound).


Option bounds Non-parametric Black–Scholes model 

JEL Classification

C14 C68 G12 G13 

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Graduate Institute of Finance and BankingNational Cheng-Kung UniversityTainanTaiwan
  2. 2.School of BusinessFordham UniversityNew YorkUSA
  3. 3.Rutgers Business SchoolRutgers UniversityPiscatawayUSA

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