Universal Bad News Principle and Pricing of Options on Dividend-paying Assets

We solve the pricing problem for perpetual American puts and calls on dividend-paying assets. The dependence of a dividend process on the underlying stochastic factor is fairly general: any non-decreasing function is admissible. The stochastic factor follows a Lévy process. This specification allows us to consider assets that pay no dividends at all when the level of the underlying factor (say, the assets of the firm) is too low, and assets that pay dividends at a fixed rate when the underlying stochastic process remains in some range. Certain dividend processes exhibiting mean-reverting features can be modelled as appropriate increasing functions of Lévy processes. The pay-offs of both the American put and call options can be represented as the expected present value (EPV) of a certain stream of dividends: g(Xt) = δ(Xt)− qK and g(Xt) = qK − δ(Xt), respectively, and we show that the option must be exercised the first time the EPV of the stream g(X t ), where X t = inf0≤s≤t Xs is the infimum process starting from the current level X0, becomes positive. Thus, the exercise threshold depends only on the record setting bad news. The results can be applied to the theory of real options as well; as one of possible applications, we consider the problem of incremental capital expansion.