AI Prices Caplets Up to 4.5 Million Times Faster Than Monte Carlo

When a pricing model has to follow an entire interest-rate curve, the computation can balloon fast. This paper tackles that problem by turning a stochastic pricing task into a deterministic partial differential equation (PDE). It then uses finance-informed neural networks, or FINNs, to solve that PDE directly, with automatic differentiation computing the exact derivatives needed along the way. On caplets, the method matches Monte Carlo benchmarks within 0.04 to 0.07 cents per dollar of contract value. Once trained, it prices caplets in a few microseconds, and the speedup grows from 300,000 to 4.5 million times as the forward-curve discretization expands from 10 to 150 nodes. The paper also shows that key sensitivities — theta and curve deltas — come at zero marginal cost during PDE evaluation. The same framework is described as extending to caps, swaptions, and callable bonds, with only boundary condition changes.

A caplet, one slice of an interest-rate cap, can be priced 4.5 million times faster than Monte Carlo at 150 curve nodes. That sounds like a spreadsheet trick until you see what it replaces. The old way uses Monte Carlo, which prices by repeated random simulation. Each run follows the whole interest-rate curve. This new method learns the price rule once. Then it reuses that rule. That matters if you care about speed, cost, or live risk checks. It also matters if you want the same answer, not a fresh roll of the dice. Interest-rate desks move fast. A few microseconds can turn a batch job into an instant quote. The surprise is not that the model is clever. The surprise is that the model can act like the pricing law itself. That is the shift this method makes.

Why the full curve makes pricing hard

The method matches Monte Carlo within 0.04 to 0.07 cents per dollar of contract value. That is a tiny gap in trading terms. Once trained, FINNs price caplets in a few microseconds. The speed stays fast even as the forward curve gets finer. The results show speedups from 300,000 to 4.5 million times. That range covers state spaces from 10 to 150 nodes. The model also gives theta and curve deltas at zero extra cost. Theta tracks time decay. Curve deltas track how the price shifts when rates move. Monte Carlo needs a full rerun for each of those checks. Traders do not just want one price. They want a live map of risk. FINNs give that map while they solve the price. So the same run answers both the quote and the sensitivity check.

How FINNs turn randomness into a rule

HJM stands for Heath-Jarrow-Morton. It tracks the whole forward curve at once. A forward curve is the set of future rates. Risk-neutral valuation prices the payoff by averaging future outcomes after discounting. Feynman-Kac then turns that random task into a deterministic PDE. A PDE is a rule written with rates of change. FINNs solve that rule directly. They shrink errors in the PDE and the boundary condition, the edge rule. Automatic differentiation computes the exact derivatives the PDE needs. That lets the model learn the price surface without random sampling. The derivatives come from the calculation itself.

Contracts that can reuse the same engine

Why speed changes the job

Speed changes who can use a pricing model. Microseconds make live quoting far less awkward. They also make repeated risk checks cheaper. The bigger gain is reuse. One core engine handles more than one contract type. The method only changes the boundary condition for new payoffs. That keeps the main pricing machinery in place. Caps fit that setup. Swaptions, or options to enter a swap later, fit it too. Callable bonds, which the issuer can buy back early, fit it as well. That cuts the need to rebuild each simulation from scratch. It also keeps the risk numbers ready during the same run.

What to test next

The real surprise is that the model can act like the pricing law itself. That makes caplets a test case, not the finish line. The next hard test is caps, swaptions, and callable bonds. Each one keeps the same forward-curve engine. Each one asks for a new boundary condition. If the microsecond speed and tiny pricing gap hold there, the old habit of rerunning huge simulations looks dated. That would move the method from a clever demo to a shared tool. It would also make sensitivity checks easier to repeat.