Applications of time-delayed backward stochastic differential equations to pricing, hedging and portfolio management in insurance and finance

We investigate novel applications of a new class of equations which we call time-delayed backward stochastic differential equations. Time-delayed BSDEs may arise in insurance and finance when we want to find an investment strategy and an investment portfolio which should replicate a liability or meet a target depending on the applied strategy or the past values of the portfolio. In this setting, a managed investment portfolio serves simultaneously as the underlying security on which the liability/target is contingent and as a replicating portfolio for that liability/target. This is usually the case for capital-protected investments and performance-linked pay-offs. We give examples of pricing, hedging and portfolio management problems (asset-liability management problems) which couldbe investigated in the framework of time-delayed BSDEs. Our motivation comes from life insurance and we focus on participating contracts and variable annuities. We derive the corresponding time-delayed BSDEs and solve them explicitly or at least provide hints how to solve them numerically. We argue that time-delayed BSDEs could offer a tool for studying investment life insurance contracts from a new and desirable perspective.

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