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

Łukasz Delong|arXiv (Cornell University)|May 24, 2010

Stochastic processes and financial applications23 references17 citations

TL;DR

This paper introduces time-delayed backward stochastic differential equations (BSDEs) for pricing, hedging, and managing portfolios in financial and insurance contexts where liabilities depend on past portfolio performance or strategy. It demonstrates that such equations model capital-protected investments and performance-linked payoffs—like participating contracts and variable annuities—more accurately than classical BSDEs, and provides theoretical and numerical frameworks for solving them, including a heuristic algorithm based on Picard iterations and Monte Carlo simulation.

ABSTRACT

In this paper 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 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 could be 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 give a financial interpretation of the theoretical fact that a time-delayed BSDE may not have a solution or may have multiple solutions.

Motivation & Objective

• To address the gap in modeling financial instruments where liabilities depend on the past performance of the replicating portfolio or investment strategy.
• To extend classical backward stochastic differential equations by incorporating time delays in both the generator and terminal condition, enabling modeling of self-referential financial claims.
• To provide the first practical applications of time-delayed BSDEs in asset-liability management, particularly in life insurance products such as participating contracts and variable annuities.
• To offer a financial interpretation of the theoretical issue of non-existence or multiplicity of solutions in time-delayed BSDEs.
• To propose a heuristic numerical algorithm for solving time-delayed BSDEs using Picard iterations and Monte Carlo methods, despite the lack of existing solvers.

Proposed method

• Formulates time-delayed BSDEs where the generator and terminal condition depend on past values of the solution (Y, Z), capturing path-dependent liabilities.
• Applies the equations to model capital-protected investments and performance-linked payoffs in life insurance, such as variable annuities with guarantees.
• Uses a change of measure to the risk-neutral measure (ℚ) to derive the dynamics of the replicating portfolio and strategy.
• Employs a Picard iteration scheme to approximate solutions, starting from an initial guess and iteratively refining the process.
• Proposes a numerical approach based on Monte Carlo simulations to estimate conditional expectations in the iteration process.
• Considers Markovian structures and least-squares Monte Carlo to estimate the value function in forward-backward systems, though convergence remains an open issue.

Experimental results

Research questions

• RQ1 How can time-delayed BSDEs be used to model financial instruments where the liability depends on the past performance of the replicating portfolio?
• RQ2 What are the theoretical conditions under which a time-delayed BSDE admits a unique solution, and how does this relate to financial feasibility?
• RQ3 How can one numerically solve time-delayed BSDEs when no standard solvers exist?
• RQ4 What is the financial interpretation of non-existence or multiple solutions in time-delayed BSDEs?
• RQ5 Can time-delayed BSDEs improve the modeling of participating contracts and variable annuities compared to classical BSDEs?

Key findings

• The time-delayed BSDE (7.2) modeling a retirement product with performance-linked withdrawals has a unique square-integrable solution for small γ or short time horizons, under the condition that Y(0) ≥ 0.
• The solution Y is strictly positive, reflecting the economic reality that a liability with positive terminal payoff and positive cost stream cannot be zero.
• Explicit solutions are derived for specific cases, such as the constant interest rate and deterministic bonus structure, but general explicit solutions remain elusive.
• A heuristic numerical algorithm based on Picard iterations and Monte Carlo simulation shows promising performance on small time intervals, though it becomes computationally infeasible for longer horizons.
• The existence of multiple or no solutions in time-delayed BSDEs is interpreted financially as a sign of market incompleteness or risk mispricing in self-referential claims.
• The framework reveals that classical BSDEs fail to capture the feedback loop between the portfolio and its liability, which is critical in structured insurance products.