On the Hedging Strategy of a European Option with Discrete Stochastic Dividends

In this note we value and hedge European options on assets paying discrete stochastic dividends. Our hedging strategy is based only on the underlying asset, risk-free bonds and dividend strips. Our simple strategy is easily seen to be compatible with early results based, among other things, on the existence of a dividend forward contract. Such contracts, however, are not traded in the market place and, therefore, our simpler strategy gains in clarity and practical application. In this note we analyze the problem of valuing a European call option written on an asset paying discrete random dividends. Dividends are an important feature when characterizing the process of the underlying asset for the purpose of pricing derivatives, as stressed in the early papers of Merton (1973) and Black (1975). In Black (1975) the dividends were considered to be stochastic, although with a continuous deterministic yield, whereas in Merton (1973) dividends are discrete, paying a known amount at a given payment date. A paper by Chance, Kumar and Rich (2002) [Chance et al. hereafter] relaxes these assumptions, allowing for stochastic, discrete dividends. As usual, the valuation of a European option is given by the value of a self-financing portfolio that replicates the final payoff of the option at maturity. The application of this principle to random dividends is an important ∗Corresponding author’s contact: tel. +351.213.822.723, fax +351.213.873.973, email: [email protected].