  The issue of developing simple Black-Scholes type approximations for pricing
European options with large discrete dividends was popular since early 2000's
with a few different approaches reported during the last 10 years. Moreover, it
has been claimed that at least some of the resulting expressions represent
high-quality approximations which closely match results obtained by the use of
numerics.
  In this paper we review, on the one hand, these previously suggested
Black-Scholes type approximations and, on the other hand, different versions of
the corresponding Crank-Nicolson numerical schemes with a primary focus on
their boundary condition variations. Unexpectedly we often observe substantial
deviations between the analytical and numerical results which may be especially
pronounced for European Puts. Moreover, our analysis demonstrates that any
Black-Scholes type approximation which adjusts Put parameters identically to
Call parameters has an inherent problem of failing to detect a little known
Put-Call Parity violation phenomenon. To address this issue we derive a new
analytic approximation which is in a better agreement with the corresponding
numerical results in comparison with any of the previously known analytic
approaches for European Calls and Puts with large discrete dividends.
