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

TL;DR

This paper develops a novel finite-difference and transform method for pricing derivatives on accumulated marks within self-exciting marked point processes, effectively reducing dimensionality and providing rigorous error bounds.

Contribution

It introduces a transform-based approach to efficiently solve high-dimensional PIDEs for derivatives on self-exciting processes, with detailed error analysis and numerical validation.

Findings

Efficient numerical scheme for pricing derivatives with self-exciting dynamics.

Reduction of the pricing problem to one-dimensional PIDEs via exponential transform.

Validated accuracy through numerical experiments and Monte Carlo benchmarks.

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.