Abstract. A small neural network forecasting the next-day range loses out of sample, and loses significantly, to the four-parameter GJR-GARCH model of Glosten, Jagannathan, and Runkle (1993). The same architecture family, set instead to approximate the Heston (1993) pricing map from self-generated surfaces, clears every gate fixed before the run. Data density, not model capacity, governs the outcome.
1. The question
An earlier note found GJR-GARCH beating a regime-switching model of my own construction. Nonlinear interaction is what a linear recursive formula cannot represent, so a network is the obvious next candidate. Two experiments follow under one out-of-sample discipline, with opposite verdicts.
2. Method
The network forecasts the range one day ahead from the predictor set given to the linear models, augmented with state inputs describing dealer positioning. Training was walk-forward, so each refit sees only data available at that date, and every forecaster faced the same window and thresholds. Table A carries the specification.
| Item | Value |
|---|---|
| Network size | ~2,000 parameters |
| Out-of-sample window | 1,794 daily observations |
| Refits, ensemble, stopping | monthly; 3 random seeds; early stopping |
| Daily volatility data available | ~250 observations per year |
| Newey–West t, error differential | −4.37 |
| Heston training sample | 9,000 self-generated vol surfaces |
| Degenerate case, vol-of-vol → 0 | recovers Black–Scholes to 4 decimals |
3. Results
A model of roughly two thousand parameters was beaten by one of four (Table 1), by a margin that is not luck. State inputs helped, consistent with earlier positioning work here, but no adjustment rescued the architecture at this scale. Daily volatility yields a few hundred observations a year, starvation for a network and ample for a parsimonious recursive model.
| Forecaster | MAE, range one day ahead (bps) |
|---|---|
| GJR-GARCH alone (4 parameters, 1993) | 41.6 |
| Full linear ladder (HAR + VIX + GJR) | 41.2 |
| Neural net, no state inputs | 63.4 |
| Neural net, with state inputs | 58.9 |
On the Heston (1993) pricing map, the function taking model parameters to a surface of implied volatility, the same family passed on accuracy, speed and skew (Table 2), following Horvath, Muguruza, and Tomas (2021). That training sample is dense and effectively unlimited, because the model generates it.
| Gate | Result | Outcome |
|---|---|---|
| Surface accuracy (gate < 0.5 vol pts) | 0.23 vol pts MAE | Pass |
| Speedup vs direct integration (gate ≥ 100x) | ~45,000x | Pass |
| Skew sanity (rho < 0 ⇒ downward skew) | put wing 24.3%, calls 14.3% | Pass |
4. What the split means
One architecture family, opposite verdicts. Neural capacity is not a modular upgrade to a forecast. It adds information where data is dense and structured, an option surface being thousands of simultaneous quotes tied together by arbitrage, and it loses to econometrics from forty years ago where data is scarce and noisy. Fitted to a single daily series, a network reproduces the error in Table 1, not the accuracy in Table 2. As a volatility forecaster it is rejected.
References
- Glosten, L. R., Jagannathan, R., and Runkle, D. E. (1993). On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks. Journal of Finance, 48(5), 1779–1801.
- Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies, 6(2), 327–343.
- Horvath, B., Muguruza, A., and Tomas, M. (2021). Deep Learning Volatility: A Deep Neural Network Perspective on Pricing and Calibration in (Rough) Volatility Models. Quantitative Finance, 21(1), 11–27.