On Valuing Stochastic Perpetuities Using New Long Horizon Stock Price Models Distinguishing Booms, Busts and Balanced Markets

Abstract

For longer horizons, assuming no dividend distributions, equilibrium models for discounted stock prices are formulated as conditional expectations of nontrivial terminal random variables defined at infinity. Observing that extant models fail to have this property, new models are proposed. The new equilibrium concept proposed here permits a distinction between unduly optimistic or pessimistic disequilibria. A tractable example is provided by the discounted variance gamma model. Calibrations to market data provide empirical support. Additionally, procedures are presented for the valuation of path dependent stochastic perpetuities. For these new discounted models, implied volatility curves do not flatten out at the larger maturities. Evidence is provided for long dated claims, paying coupon for the time spent by the stock price above a lower barrier, being underpriced by extant models relative to the new discounted ones. Given that such claims are now issued quite regularly, the resulting mispricing could possibly take some corrections.