A pricing formula for delayed claims: Appreciating the past to value the future

TL;DR

This paper derives a pricing formula for contingent claims with delayed dynamics in a Black-Scholes market, emphasizing the importance of past information in valuing future cashflows, with applications to human capital valuation.

Contribution

It introduces a novel pricing formula that decomposes the value into past and present components, applicable to infinite-dimensional stochastic control problems.

Findings

Explicit formula for human capital valuation with wage rigidity

Decomposition of claim value into past and present market values

Successful application to infinite-dimensional stochastic control problems

Abstract

We consider the valuation of contingent claims with delayed dynamics in a Black&Scholes complete market model. We find a pricing formula that can be decomposed into terms reflecting the market values of the past and the present, showing how the valuation of future 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 different settings.