The implied volatility surface in the presence of dividends

In South African markets, European call and put options trade liquidly at certain strikes and maturities on the TOPI40, an index of the top-40 shares based on market capitalisation. In general, it is assumed that dividends on indexes pay frequently, hence it is appropriate to make use of a dividend yield, and that implied volatilities for non-market options can be interpolated from liquid options. In this article, we show that these assumptions can introduce arbitrage around dividend dates. We also provide specific arbitrage formulae, which relate implied volatilities on
either side of dividend dates. It is reasonable to assume that different institutions will have different dividend forecasts. Furthermore, these forecasts may be
modelled as discrete dividends, either proportional to the stock price or an absolute cash amount. In this article we will show how absolute and proportional dividend payments give rise to different arbitrage constraints on the implied volatility surface.
Consequently, two institutions who agree on all the market option prices may imply quite different volatilities and prices for nonstandard options.