Neural Volatility Forecasting Against GARCH

GARCH forecasts tomorrow's volatility from a tight, three-parameter recipe: yesterday's shock, yesterday's variance, and a long-run average, combined in a fixed formula. It's been the industry default for decades because it's simple, stable, and hard to overfit with only three parameters to estimate. A neural network can, in principle, learn a far richer relationship — but "in principle" is doing a lot of work, because a network with thousands of parameters trained on a few thousand days of returns is exactly the setup where flexibility becomes a liability rather than an asset.

An analogy: a thermostat versus a weather model

GARCH is like a thermostat: yesterday's temperature and a simple rule tell you roughly what to expect today, and it rarely misfires because the rule is so constrained. A neural volatility model is like a full weather model that ingests pressure fronts, humidity, and dozens of other signals to make a genuinely better forecast — when it has enough reliable data to calibrate against. Point it at a city with ten years of dense weather records and it can beat the thermostat easily; point it at a city with two years of patchy records and it will confidently predict nonsense, because it has too many moving parts and not enough grounding.

What the network gets that GARCH doesn't

A GARCH(1,1) forecast is:

$$\sigma_t^2 = \omega + \alpha \, \epsilon_{t-1}^2 + \beta \, \sigma_{t-1}^2$$

In plain English: today's variance is a long-run baseline $\omega$, plus a piece proportional to yesterday's surprise, plus a piece carried over from yesterday's own variance — a fixed, linear rule with three numbers to fit. A neural model (typically an LSTM over a window of past returns and other features — volume, implied vol, macro surprises) replaces that fixed rule with a learned, nonlinear function, which can in principle capture asymmetries and cross-asset effects GARCH's fixed formula cannot.

Worked example: where each one wins

On a single liquid equity index with fifteen years of daily history, a GARCH(1,1) and an LSTM trained on the same data are compared by one-day-ahead volatility forecast error (using realized volatility as the target). GARCH achieves a mean absolute error of roughly 0.85 (in daily variance units); the LSTM, trained with care and validated out of sample, edges it out slightly at 0.80 — a real but modest improvement, consistent with published results showing neural models beat GARCH by single-digit percentages on deep, liquid single-asset histories. Now repeat the comparison on a two-year-old small-cap stock with three hundred usable trading days: GARCH still produces a reasonable forecast because it only needs to fit three parameters, while the LSTM's error is worse than GARCH's — it has enough parameters to fit noise in such a short history, and its extra flexibility works against it. The lesson is not "neural networks are better" or "GARCH is better," but that the answer depends heavily on how much reliable history is available.

What this means in practice

Neural volatility models are worth the added complexity where there's deep, clean history and a real need to capture nonlinear effects GARCH's fixed formula misses — often multi-asset or cross-sectional volatility problems rather than single-series forecasting, where GARCH already does a solid job cheaply. Any neural volatility model needs to be benchmarked against a GARCH baseline on genuinely held-out data before it earns a place in production; "more flexible" is not the same as "more accurate," and the gap between the two is exactly where overfitting lives.