Dynamic Asset Pricing Theory for Life Contingent Risks

TL;DR

This paper addresses the pricing of life-contingent risks within a non-arbitrage, complete market by leveraging the Fundamental Theorem of Asset Pricing (FTAP) and the stochastic discount factor (SDF). It starts with a one-step model to derive the SDF via Farkas' Lemma and then generalizes to a multi-step setting using information sets and a pricing kernel, deriving discounted-martingale pricing when dividends are absent. It applies these tools to life-contingent contracts by linking force of mortality with survival probabilities and deriving continuous-time valuation formulas for whole-life policies and life-contingent annuities, including continuous-time variants. The framework yields explicit valuation formulas, clarifies the relationship between mortality dynamics and financial pricing, and provides a coherent financial-intuition-based approach for pricing life-insurance products and annuities in a dynamic, complete-market setting, with potential applications in risk management and product design.

Abstract

Although the valuation of life contingent assets has been thoroughly investigated under the framework of mathematical statistics, little financial economics research pays attention to the pricing of these assets in a non-arbitrage, complete market. In this paper, we first revisit the Fundamental Theorem of Asset Pricing (FTAP) and the short proof of it. Then we point out that discounted asset price is a martingale only when dividends are zero under all random states of the world, using a simple proof based on pricing kernel. Next, we apply Fundamental Theorem of Asset Pricing (FTAP) to find valuation formula for life contingent assets including life insurance policies and life contingent annuities. Last but not least, we state the assumption of static portfolio in a dynamic economy, and clarify the FTAP that accommodates the valuation of a portfolio of life contingent policies.