A Fourth-Order Solver Prices Bates-Model Options Up to 100 Times Faster

If an asset can jump suddenly instead of moving smoothly, pricing its options becomes a much harder math problem. The Bates model handles that reality by combining changing volatility with discontinuous price jumps, but the resulting equation is costly to solve. This paper proposes a fourth-order compact finite-difference scheme for the transformed Bates partial integro-differential equation, using an IMEX Crank–Nicolson split for the local terms and Simpson quadrature for the jump integral. In benchmarks, the new HOC-FD method reached near-fourth-order spatial accuracy, while quadratic finite elements reached near-second-order spatial accuracy. All the time integrators showed second-order temporal convergence. The striking result is speed: HOC-FD matched the accuracy of finite elements at up to two orders of magnitude lower runtime. That makes it a practical baseline for option pricing under stochastic volatility jump-diffusion models.

An option price can jump the way a stock does after bad news. That makes the math much harder than a smooth price path. The Bates model handles both moving volatility and sudden jumps. It is built for markets that do not move in neat lines. This paper asks a practical question. Can that messy model be solved fast enough to matter in real use? The answer is yes, and the surprise is sharp. A compact grid method gives near-fourth-order spatial accuracy. It also reaches similar accuracy to finite elements with far less time.

The speed-up hidden inside the grid

The headline result is simple. The new high-order compact finite-difference scheme, or HOC-FD, beats older baselines on speed. Benchmarks against second-order finite differences and quadratic finite elements show near-fourth-order spatial accuracy for HOC-FD. The finite-element method, or FEM, reaches near-second-order spatial accuracy. All time solvers show second-order time convergence, which means the error falls at the same rate when the time step gets smaller. The big practical gain is runtime. HOC-FD gets similar accuracy at up to two orders of magnitude lower runtime than FEM. That makes the compact grid method a strong baseline for this kind of pricing problem.

How the solver handles jumps

The solver starts with a transformed Bates partial integro-differential equation, or PIDE. A PIDE is a differential equation with an extra integral term. That integral term captures jumps. The local, smooth parts use an implicit-explicit Crank-Nicolson scheme. Crank-Nicolson is a time-stepping rule that averages the present and next step. Implicit-explicit, or IMEX, means some parts are treated in a stable implicit way, while others are handled directly. The jump term uses Simpson quadrature, a weighted sum that estimates the area under a curve. Together, these pieces let the solver keep sharp detail without paying the cost of a very heavy mesh.

Why this matters for option pricing

Option pricing models are only useful when they run in time. The Bates model is richer than a plain volatility model because it allows both drifting risk and sudden jumps. That extra realism usually brings extra cost. HOC-FD cuts that cost without giving up the detail the model needs. The result matters for anyone who wants a practical baseline, not just a neat equation. A method like this can make heavy jump-diffusion models easier to test, compare, and use as reference points in later work. It also shows that high accuracy does not always require the most expensive tool in the box.

What to watch next

The strongest test now is not the formula on paper. It is the solver under harder market cases. The next check is whether this same speed and accuracy hold across more transformed Bates setups and wider pricing tasks. The paper has already shown that a compact grid can beat a heavier finite-element build on runtime. The open question is where that edge stays intact as the jump term or the volatility path becomes more demanding. If it does, fast pricing for jumpy markets stops being a special case. It becomes the starting point.