Tensor Networks Speed Up Eight-Dimensional Option Pricing

Pricing a complex basket option can mean wrestling with a huge grid of possible market conditions. This paper shows a way to compress that grid into a tensor-train, a compact representation built directly from black-box price evaluations. The method was tested on five-asset basket options across an eight-dimensional parameter space, including asset spot levels, strike, interest rate, and time to maturity. For European geometric basket puts, the tensor surrogate reached lower test error in less training time than standard Gaussian process regression, because it could scale to much larger effective training sets. For American arithmetic basket puts trained on least-squares Monte Carlo data, it scaled more favorably as the training set grew and returned prices in millisecond-level time per query. The authors also found that hyperparameter optimization consistently preferred a large kernel length-scale, and in that limit the Gaussian process predictor becomes multilinear interpolation for off-grid inputs. The practical message is simple: when option pricing lives in many dimensions, tensor networks can make the problem far easier to evaluate, with overall runtime often dominated by data generation rather than the surrogate itself.

Picture a risk desk before the market opens. It must reprice many basket options at once. Each one depends on five assets, plus strike, rate, and time left. That makes eight knobs to turn. A plain grid grows too fast. STN-GPR tackles that sprawl with a tensor train, a compact way to store many values. The surprise is not just speed. In one regime, the model's best guess becomes interpolation. That means a tool built from advanced math can fall back to a simple rule. For anyone who has watched a slow pricing run stall a workflow, that shift matters right away.

How a huge pricing grid gets squeezed

STN-GPR was tested on two basket option sets. One was a European geometric basket put. The other was an American arithmetic basket put trained on LSMC data. LSMC means least-squares Monte Carlo, a way to price with many simulated paths. On the European case, the tensor surrogate beat standard GPR on test error. It also trained faster. The gain came from scale. The tensor form could handle much larger effective training sets. Hyperparameter search kept favoring a large kernel length-scale, the knob that sets how wide the smoothing reaches. In that wide-smooth limit, the GPR predictor turns into multilinear interpolation for off-grid inputs. On the American case, the same approach scaled more gently as training data grew. It also returned each price in milliseconds. Overall runtime was mostly spent making the data, not asking the surrogate for an answer.

What makes the surrogate work

TT-cross approximation builds the tensor train from price calls alone. It never needs the full training grid. That matters because the grid would explode in eight dimensions. A tensor train is a compact chain of small tables. For inference, STN-GPR uses a Laplacian kernel, a distance rule that fades with absolute gap. The method also writes the kernel matrix in tensor-train form. A matrix is a square table of numbers. In the noise-free case, it also gives the inverse a closed-form tensor-train form. That lets Gaussian process regression, a smooth fit method, run without dense matrix factorization or repeated solve steps.

Why this matters for market risk

That setup matters for market risk management. VaR, or Value at Risk, is a one-number loss check. Expected Shortfall looks at the average of the worst losses. Both need large portfolio revaluation runs. STN-GPR gives a compact surrogate, which is a stand-in model for the price surface. That can cut the cost of asking for many prices across many market states. The American basket result adds a second lesson. When simulation creates the prices, data generation can dominate the clock. The surrogate then becomes cheap enough to query in milliseconds. In other words, the bottleneck moves upstream.

What to watch next

The sharpest twist stays the same. A wide kernel length-scale can make GPR act like interpolation. That means a fancy smoother can collapse into a plain rule. A useful check is whether the same pattern stays stable on the American arithmetic basket puts trained on LSMC data. If it does, the slow part will still be data generation. The fast part will still be the surrogate. For five-asset basket revaluation, that split is a practical win.