A Transform Cuts Self-Exciting Derivative Pricing Down to One Dimension

Pricing a contract tied to rainfall, heating demand, or insurance losses is hard when events come in bursts. A new paper tackles that problem with a self-exciting marked point process, where each jump can make the next one more likely and the accumulated jump size drives the payoff. The authors derive the pricing problem as a backward partial integro-differential equation in two spatial dimensions, then shrink it by applying an exponential transform in the accumulated mark variable. That move turns the original problem into a family of one-dimensional PIDEs in the intensity variable, solved along a Bromwich contour. For Gamma-mixture mark laws, they approximate the jump term with generalized Gauss–Laguerre quadrature and march backward in time with a monotone IMEX finite difference scheme. They also prove a term-by-term global error bound covering time and space discretization, quadrature, interpolation, and boundary effects, and they test the method against numerical experiments and Monte Carlo benchmarks. The result is a computational route for pricing derivatives on clustered, self-exciting event streams.

Rain can arrive in streaks. So can claims after a big storm. Contracts tied to those bursts are hard to price. Each new event can change the odds of the next one. Each jump can also add to the final payout. That leaves two moving parts at once. One tracks the pace of new events. The other tracks the total piled up so far. The surprise is that this two-part maze can be flattened. A transform does the flattening. It rewrites the price in a new lens. Then the hard part becomes easier to march through. That matters for rainfall, heating demand, and loss claims. They jump, pause, and jump again.

Why bursty events turn pricing into a maze

The core move is an exponential transform. People also call it a Laplace or Fourier transform, a math trick that changes the view. It acts on the accumulated mark variable. That variable is the running total of jump sizes. After that step, the jump shift no longer ties all states together. The full two-dimensional pricing equation becomes a family of one-dimensional PIDEs, price equations with jump terms. A Bromwich contour undoes the transform. Gamma-mixture mark laws blend Gamma-shaped jump-size models. The jump term gets a fast quadrature rule. Esscher-tilted measures are pricing-weighted views of risk. The method also comes with a term-by-term global error bound. The result is a thinner problem that still keeps the jump physics.

How the reduction stays stable

Once the transform cuts the state down, the solver can move on a grid. It uses a monotone IMEX grid method. IMEX means it treats some parts implicitly and others explicitly. Monotone means it avoids wild price flips. It steps backward in time. It uses an implicit upwind rule for drift and discounting. That keeps flow moving one way. It treats the jump part explicitly. That keeps each step manageable. For Gamma-mixture marks, generalized Gauss-Laguerre quadrature replaces a hard integral with a weighted sum. The method then inverts the transform to recover the price. The error tally includes time steps, space steps, quadrature, interpolation, and boundary cuts.

Why the shortcut matters

This matters because these contracts live on bursty things. Rainfall, heating demand, and aggregate losses do not arrive like smooth stock prices. They come in clumps. A two-variable pricing problem can be costly to solve. That is especially true when you want a sharp error check. This method gives both. It cuts the state space and tracks each error source. That makes the price calculation more practical for weather derivatives and aggregate loss claims. It also gives a cleaner bridge between the jump model and the final quote. In other words, the method turns a messy stream into a grid-friendly one. Numerical tests compare it with Monte Carlo benchmarks, which use random draws.

What still has to hold up next

The next check is weather derivatives with Gamma-mixture marks. That is the kind of case this setup aims at. The open question is how well the error bound holds as bursts get sharper. The same test should also cover aggregate loss claims with strong clustering. If it passes, one two-variable price problem after another becomes a one-variable job. The surprise stays the same. A bursty stream does not have to stay a two-dimensional trap.