New analytic approach to address Put - Call parity violation due to discrete dividends

TL;DR

This paper reviews existing Black-Scholes type approximations for European options with large discrete dividends, identifies their limitations especially for puts, and introduces a new analytic approximation that better captures Put-Call parity violations.

Contribution

It proposes a novel analytic approximation that accurately accounts for Put-Call parity violations caused by discrete dividends, improving upon previous methods.

Findings

Existing approximations often deviate significantly from numerical results.

Black-Scholes type adjustments fail to detect Put-Call parity violations.

The new approximation aligns more closely with numerical schemes for both calls and puts.

Abstract

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.