A Pricing Formula for Delayed Claims: Appreciating the Past to Value the Future

Abstract

We consider the valuation of contingent claims with delayed dynamics in a Samuelson complete
market model. We find a pricing formula that can be decomposed into terms reflecting the
current market values of the past and the future, showing how the valuation of prospective
cashflows cannot abstract away from the contribution of the past. As a practical application,
we provide an explicit expression for the market value of human capital in a setting with
wage rigidity. The formula we derive has successfully been used to explicitly solve the infinite
dimensional stochastic control problems addressed in [7], [6] and [16].

Keywords: Stochastic functional differential equations, delay equations, no-arbitrage pricing, human capital, sticky wages