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In this paper we investigate novel applications of a new class of equationswhich we call time-delayed backward stochastic differential equations.Time-delayed BSDEs may arise in finance when we want to find an investmentstrategy and an investment portfolio which should replicate a liability or meeta target depending on the applied strategy or the past values of the portfolio.In this setting, a managed investment portfolio serves simultaneously as theunderlying security on which the liability/target is contingent and as areplicating portfolio for that liability/target. This is usually the case forcapital-protected investments and performance-linked pay-offs. We give examplesof pricing, hedging and portfolio management problems (asset-liabilitymanagement problems) which could be investigated in the framework oftime-delayed BSDEs. Our motivation comes from life insurance and we focus onparticipating contracts and variable annuities. We derive the correspondingtime-delayed BSDEs and solve them explicitly or at least provide hints how tosolve them numerically. We give a financial interpretation of the theoreticalfact that a time-delayed BSDE may not have a solution or may have multiplesolutions.

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