Dividends in the Theory of Derivative Securities Pricing

Abstract

This paper develops the fundamental aspects of the theory of martingale pricing of derivative securities in a setting where the cumulative gains processes are Ito processes while the cumulative dividend processes of both the underliers and the derivative securities are general enough to cover all cases encountered in practical applications. A key ingredient is a general formula for how to change the unit of account of a cumulative dividend process. The formula is inconsistent with parts of the earlier literature. It obeys a unit-invariance rule for trading strategies, satisfies a consistency property when the unit is changed twice in a row, gives the correct results in well-known and uncontroversial special cases, and fits perfectly into a generalization of the martingale valuation theory. Using that generalized theory, we show that the value of a dividend process equals the value of a claim to the nominal amount of dividends yet to be accumulated plus the value of a flow of interest on the cumulative dividends at each point in time.