Everyone seems to agree that the option prices predicted by the Black-Merton-Scholes model are inconsistent with what is observed in reality. Still, many people rely on the model by using "the wrong number in the wrong formula to get the right price".
Question. What are some of the most important contradictions that one encounters in quantitative finance? Are there any model-independent inconsistencies? Are some of these apparent paradoxes born more equal than the others (i.e. lead to better models)?
I would like to limit the scope of the question to the contradictions arising in quantitative finance (so the well-documented paradoxes of economics and probability theory such as the St. Petersburg paradox or Allais paradox are deliberately excluded).
Edit (to adress Shane's comment). Hopefully, this question is different in focus and has a slightly more narrow scope than the previous question concerning the most dangerous concepts in quantitative finance work. For instance, using VaR "naively" does not lead to immediate contradictions the way naive application of the BS model does. VaR may be considered inadequate because it seriously underestimates tail risks but it is not self-contradictory per se (please correct me if I'm wrong). Similarly, the EMH in its weaker forms may not be inconsistent with the market reality (at least the opposite has not been demonstrated decisively yet).