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Pricing Derivatives under Self-Exciting Dynamics: A Finite-Difference and Transform Approach

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.

Pricing Derivatives under Self-Exciting Dynamics: A Finite-Difference and Transform Approach

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 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.
Paper Structure (38 sections, 25 theorems, 119 equations, 5 figures, 1 table)

This paper contains 38 sections, 25 theorems, 119 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and Setting
  3. Clamped extension (off-grid evaluation).
  4. Framework and partial integro-differential equations
  5. Pricing measure
  6. Interpretation and calibration of $\theta$.
  7. Gamma mixture-of-experts for marks.
  8. Estimation by Expectation-Maximization (EM).
  9. Calibration of $\theta$.
  10. Spectral reduction in the accumulation variable $u$
  11. Numerical plan (Bromwich--IMEX--Gauss--Laguerre).
  12. Choice of $\delta$, $Y_{\max}$ and grids.
  13. IMEX discretization for the $(t,\lambda)$ PIDE at fixed frequency
  14. Grids and discrete operators
  15. Boundary conditions in $\lambda$.
  16. ...and 23 more sections

Key Result

Proposition 1

If for any $t > 0$ and a given measurable function $f:\mathbb{R}^4 \to \mathbb{R}$, it holds that $f$ has continuous partial derivatives with respect to $t$, $\lambda$, $u$ the map is measurable, with $\mathbb E_{\eta}$ the expectation under $\mathbb P$ given $X_0=\eta$ then $f$ is in the domain of the extended generator of the process $t \mapsto (t,\lambda_t,N_t,U_t)$, i.e. $f \in \mathcal{D}(\m

Figures (5)

  • Figure 1: Option price as a function of the self-excitation strength $\beta$ for $T=150$ days and $(K,C)=(1.2,3.0)$. Dashed line with markers: IMEX--Gauss--Laguerre--Bromwich price. Squares with $95\%$ error bars: Monte Carlo benchmark (exact PDMP simulation). All remaining parameters are fixed.
  • Figure 2: Convergence diagnostics against a finer internal numerical reference $V_{\mathrm{ref}}$. Left: time refinement. Right: spatial refinement along a coupled path where $\Delta t$ decreases as $N_\lambda$ increases.
  • Figure 3: Convergence of the Bromwich inversion with respect to the frequency grid size $N_y$ for a fixed truncation level $Y_{\max}$. The error is measured against an internal numerical reference $V_{\mathrm{ref}}$ computed with a larger $(Y_{\max},N_y)$.
  • Figure 4: Option price as a function of the initial jump intensity $\lambda_0$. The monotonic increase reflects the higher probability of accumulating large marks prior to maturity.
  • Figure 5: The "intensity Greek" (Delta) $\partial_{\lambda_0}V$ estimated via centred finite differences. The smooth, bell-shaped profile highlights the diminishing marginal impact of extreme initial intensities (saturation effect).

Theorems & Definitions (56)

  • Proposition 1
  • proof
  • Proposition 2: Discounted value function and Backward PIDE
  • proof
  • Proposition 3: Backward PIDE under Esscher pricing
  • proof : Proof sketch
  • Remark 1: Stability under OU drift
  • Definition 1: Gamma mixture-of-experts for the mark law
  • Lemma 1: Esscher transform of a Gamma MoE
  • Lemma 2: Domain of $\theta$
  • ...and 46 more