Paper: arXiv 2603.12602

Authors: Aqib Ahmed, Heiðar Eyjólfsson

Abstract

We consider the pricing of derivatives written on accumulated marks, such as weather derivatives or aggregate loss claims, using a self-exciting marked point process. The jump intensity mean-reverts between events and increases at jump times by an amount proportional to the mark. The resulting state process, where the variable $U_t$ accumulates jump magnitudes, is a piecewise deterministic Markov process (PDMP). We derive the discounted pricing equation as a backward partial integro-differential equation (PIDE) in two spatial dimensions. To overcome the dimensionality, we propose an exponential (Laplace/Fourier) transform in the accumulated mark variable, which diagonalizes the translation operator and reduces the pricing problem to a family of one-dimensional PIDEs in the intensity variable along a Bromwich contour. For Gamma-mixture mark laws (under actuarial or Esscher-tilted measures), the nonlocal jump term is efficiently approximated by generalized Gauss–Laguerre quadrature. We solve the reduced PIDEs backward in time using a monotone IMEX finite difference scheme (implicit upwind drift and discounting, explicit jump operator) and recover option prices via numerical inversion. We provide a rigorous, term-by-term global error bound covering time and space discretization, quadrature, interpolation, and boundary effects, supported by numerical experiments and Monte Carlo benchmarks.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a sophisticated theoretical framework for pricing derivatives under self-exciting dynamics using partial integro-differential equations (PIDEs), Fourier/Laplace transforms, and numerical scheme error bounds. While the mathematical formulations and numerical analysis are highly rigorous, empirical validation is limited to synthetic Monte Carlo benchmarks rather than empirical market data backtests.

Research Flowchart

  flowchart TD
    A[Goal: Price Weather/Insurance Derivatives under Self-Exciting Dynamics] --> B[Inputs: Contract Specs, Parameters & Gamma-Mixture Mark Laws]
    B --> C[Derive 2D Backward PIDE for PDMP State Process]
    C --> D[Apply Laplace/Fourier Transform to Reduce PIDE from 2D to Family of 1D PIDEs]
    D --> E[Compute Jump Term via Gauss-Laguerre Quadrature & Solve via Monotone IMEX Finite Difference]
    E --> F[Invert Transform along Bromwich Contour to Recover Derivative Prices]
    F --> G[Outcomes: Fast, Accurate Option Pricing with Proven Global Error Bounds]

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0 of 74 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.