Modeling of Tradeable Securities with Dividends

We propose a generalized framework for the modeling of tradeable securities with dividends which are not necessarily cash dividends at fixed times or continuously paid dividends. In our setup the dividend processes are only required to be semi-martingales. We give a definition of self-financing replication which incorporates dividend processes, and we show how this allows us to translate standard results for the pricing and hedging of derivatives on assets without dividends to the case of assets with dividends. We then apply this framework to analyze and compare the different assumptions that have been made in earlier dividend models. We also study the case where we have uncertain dividend dates, and we look at securities which are not equity-based such as futures and credit default swaps, since our weaker assumptions on the dividend process allow us to consider these other applications as well.