Applications of Eigenfunction Expansions in Continuous‐Time Finance

Abstract

We provide exact solutions for two closely related valuation problems in continuous‐time finance. The first problem is to value generalized European‐style options on stocks that pay dividends at a constant dollar rate. The second problem is to find the yield curve associated with the economy of R. C. Merton's “An Asymptotic Theory of Growth Under Uncertainty.” In Merton's economic growth model, the interest rate process has a volatility linear in the rate level and a linear/quadratic drift. Both problems are solved by an eigenfunction expansion technique. The main technical difficulty is handling the problem of payoff functions that are not square‐integrable with respect to the natural weight function of the models.