New Analytic Approach to Address Put--Call Parity Violation due to Discrete Dividends

The issue of developing simple Black--Scholes (BS)-type approximations for pricing European options with large discrete dividends was popular since the early 2000s 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 the results obtained by the use of numerics. In this article we review, on the one hand, these previously suggested BS-type approximations and, on the other hand, different versions of the corresponding Crank--Nicolson (CN) 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 BS-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 pricing approximation which is in 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.