Paper: arXiv 2405.02115

Authors: Tiziano De Angelis, Alessandro Milazzo, Gabriele Stabile

Abstract

In this paper we provide a theoretical analysis of Variable Annuities with a focus on the holder’s right to an early termination of the contract. We obtain a rigorous pricing formula and the optimal exercise boundary for the surrender option. We also illustrate our theoretical results with extensive numerical experiments. The pricing problem is formulated as an optimal stopping problem with a time-dependent payoff which is discontinuous at the maturity of the contract and non-smooth. This structure leads to non-monotonic optimal stopping boundaries which we prove nevertheless to be continuous and regular in the sense of diffusions for the stopping set. The lack of monotonicity of the boundary makes it impossible to use classical methods from optimal stopping. Also more recent results about Lipschitz continuous boundaries are not applicable in our setup. Thus, we contribute a new methodology for non-monotone stopping boundaries.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly rigorous mathematical treatment of variable annuities, introducing novel optimal stopping techniques to address non-monotonic exercise boundaries. However, it relies primarily on theoretical proofs and numerical simulations rather than empirical market data or historical backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Price Variable Annuities & Determine Optimal Surrender Boundary] --> B[Inputs: Contract Parameters, Surrender Charges & Diffusion Assumptions]
    B --> C[Methodology: Formulate Non-Smooth, Discontinuous Optimal Stopping Problem]
    C --> D[Computational Process: Develop New Mathematical Framework for Non-Monotone Boundaries]
    D --> E[Numerical Experiments: Simulate Pricing & Exercise Boundaries]
    E --> F[Key Outcomes: Rigorous Pricing Formula & Proof of Boundary Continuity/Regularity]

Cited References (38)

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  8. Variable annuities in a Lévy-based hybrid model with surrender risk (2019) — link
and 30 more
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0 of 38 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.