Learning volatility-adaptive forgetting via GARCH-modulated LSTM for stock price forecasting

Xin-Rui Lu; Da-Qing Zhang

cs.CEOct 17, 2026

Learning volatility-adaptive forgetting via GARCH-modulated LSTM for stock price forecasting

Authors: Xin-Rui Lu, Da-Qing Zhang

Organizations: School of Science, University of Science and Technology Liaoning, Anshan, 114051, Liaoning, PR China

Abstract

Stock price forecasting is hindered by the heteroscedastic nature of financial time series, in which turbulent and tranquil regimes alternate. Although the Long Short-Term Memory (LSTM) network is widely used for this task, its gates are oblivious to such volatility variation, so memory retention remains fixed regardless of market conditions. A common remedy is to supply volatility estimates as an additional input feature, yet this leaves the internal memory dynamics untouched. Instead, volatility can serve as a direct regulatory signal over the gate activation itself. This paper presents the Gsigmoid-LSTM model, which embeds Generalized Autoregressive Conditional Heteroscedasticity (GARCH(1,1))-derived volatility estimates into the LSTM forgetting gate through the Gsigmoid activation function, a volatility-adaptive variant of the sigmoid that combines a volatility adjustment factor with slope regulation via a learnable parameter. The model thereby automatically prioritizes recent information during turbulent periods while preserving long-term dependencies in stable markets. In the engineering application of daily stock price forecasting, extensive experiments, including activation function comparisons, baseline evaluations, and ablation studies, demonstrate that this structural gating modification effectively improves forecasting accuracy and robustness. The proposed model significantly outperforms a feature-level fusion baseline carrying the same volatility signal on all five datasets, as confirmed by Diebold-Mariano tests with all p<0.01, attains the best average rank of 1.00 in the Friedman test across nine models, and achieves the lowest MSE and MAE on all five datasets.

Figures & tables

Study	Integration level	Role of volatility	Learnable	Signal pathway
Kim and Won (2018)	Feature	GARCH-type estimates as network inputs	$\times$	Input layer
Kayim and Yilmaz (2022)	Activation	Standard deviation adjusts neuron response	$\times$	Hidden-unit activations (not gate-specific)
Amirshahi and Lahmiri (2023)	Feature	GARCH-extracted features as DL inputs	$\times$	Input layer
Mutinda and Langat (2024)	Decision	GARCH and AI predictions combined	$\times$	Decision stage
Liu (2024)	Decision	Optimal weighting across predictors	Partial	Decision stage
Chen (2025)	Feature	Indicators concatenated with DL inputs	$\times$	Input layer

Table 1: Comparison of representative volatility-aware forecasting studies by integration level, ordered chronologically.

Figure 1: Schematic diagram of LSTM structure

Figure 2: Gsigmoid activation function under different fluctuation conditions (reference stock sample: IXIC)

Activation	Formula	Range	Monotonic	Deriv. bound	Saturation
Sigmoid	$S(z)$	$(0,1)$	Yes	$\leq 1/4$	Two-sided
Tanh	$\tanh(z)$	$(-1,1)$	Yes	$\leq 1$	Two-sided
Swish	$z\cdot S(z)$	$[-0.28,\infty)$	No	$\approx 1.10$	Left weak, right $\to\infty$
ReLU	$\max(0,z)$	$[0,\infty)$	Non-strict	0/1	Dead zone, $z<0$
Gsigmoid	$\rho_{t}\cdot S(\gamma z)$	$(0,\rho_{t}]$	Yes	$\leq 0.25\gamma$	By $\gamma$ , $\rho_{t}$

Table 2: Mathematical Properties of Activation Functions.

Figure 3: Gsigmoid-LSTM framework. (Left) Detailed architecture components: (a) volatility-adaptive gating construction; (b) Gsigmoid activation learning; (c) dual-channel forget gate and cell state update modules; (d) volatility-modulated gating and prediction. (Right) Overall model process flow.

Dataset	LM Stat	LM p-value	F Stat	F p-value	Heteroskedasticity
JNJ	26.67	2.42E-07 ∗∗∗	26.87	2.31E-07 ∗∗∗	Yes
MSFT	5.85	1.56E-02 ∗	5.86	1.56E-02 ∗	Yes
AMZN	4.00	4.56E-02 ∗	4.00	4.56E-02 ∗	Yes
IXIC	99.32	2.14E-23 ∗∗∗	102.37	1.02E-23 ∗∗∗	Yes
GOOG	23.31	1.38E-06 ∗∗∗	23.46	1.33E-06 ∗∗∗	Yes

Table 3: Heteroskedasticity Test Results

Hyperparameter	Explanation	Search Range
DeepESN (Deep Echo State Network)
spectral_radius	Spectral radius of reservoir weight matrix	[0.7, 0.8, 0.9, 1.0]
input_scaling	Scaling factor for input weights	[0.2, 0.3, 0.4, 0.5]
leaking_rate	Leaking rate of reservoir neurons	[0.05, 0.1, 0.15, 0.2]
n_layers	Number of stacked reservoir layers	[1, 2, 3]
n_units	Number of reservoir units per layer	[32, 64, 96, 128, 192]

Table 4: Hyperparameter Search Space for Baseline Models.

Training Parameters		Model Architecture		Training Control
learn_rate	0.001	dropout	0.3	epochs	80
batch_size	16	lstm_units	128	patience	15
optimizer	Adam	time_step	24	time_step( $w_{\mu}$ )	20

Table 5: Model Parameters Configuration

Figure 4: Different activation function curves

Dataset	Activation Function
	LSTM-Gsigmoid	LSTM-Swish	LSTM-Tanh	LSTM-ReLU	LSTM-Sigmoid
JNJ	3.26	3.55	3.66	3.59	3.59
MSFT	37.24	214.45	201.77	336.49	97.20
AMZN	12.85	16.63	17.55	17.51	45.80
IXIC	43685.02	214416.64	72794.55	255754.45	131913.75
GOOG	5.96	29.74	47.62	30.74	20.71

Table 6: Comparison of MSE (

\downarrow

) Performance of Different Activation Functions in LSTM Models across Various Datasets.

Dataset	Activation Function
	LSTM-Gsigmoid	LSTM-Swish	LSTM-Tanh	LSTM-ReLU	LSTM-Sigmoid
JNJ	1.37	1.46	1.49	1.47	1.46
MSFT	4.64	12.15	11.60	15.50	7.93
AMZN	2.68	3.19	3.27	3.27	5.79
IXIC	162.03	390.75	224.60	421.61	303.95
GOOG	1.82	4.34	5.40	4.34	3.61

Table 7: Comparison of MAE (

\downarrow

) Performance of Different Activation Functions in LSTM Models across Various Datasets.

Figure 5: MAPE (

\downarrow

) Performance of LSTM Variants with Different Activation Functions. Datasets split chronologically: 70% training, 10% validation, 20% testing. All comparisons are based on the same datasets under a unified experimental setup, and all results were obtained by the authors.

Figure 6:

R^{2}

(

\uparrow

Dataset	Gsigmoid-LSTM	LSTM	RNN	GRU	ANN	RVFL	DeepESN	BiLSTM	BiGRU
JNJ	3.26	3.59	6.75	4.04	4.41	11.66	5.99	7.95	5.82
MSFT	37.24	97.20	583.04	103.09	50.42	96.99	91.64	526.19	105.18
AMZN	12.85	45.80	24.45	19.41	19.48	69.95	27.73	14.65	24.46
IXIC	43685.02	131913.75	454395.50	116705.37	72992.23	84211.76	101325.21	85080.88	61690.30
GOOG	5.96	20.71	66.18	17.82	7.58	10.57	14.57	34.25	29.67

Table 8: Comparison of MSE (

\downarrow

) Performance of Different Models across Various Datasets.

Dataset	Gsigmoid-LSTM	LSTM	RNN	GRU	ANN	RVFL	DeepESN	BiLSTM	BiGRU
JNJ	1.37	1.46	2.16	1.57	1.66	2.80	1.97	2.37	1.95
MSFT	4.64	7.93	21.37	8.31	5.41	8.22	7.65	19.07	8.31
AMZN	2.68	5.79	4.04	3.55	3.44	8.36	4.19	2.91	4.05
IXIC	162.03	303.95	577.19	286.88	221.55	238.42	318.32	245.56	195.94
GOOG	1.82	3.61	6.78	3.34	2.12	2.64	3.08	4.69	4.38

Table 9: Comparison of MAE (

\downarrow

) Performance of Different Models across Various Datasets.

Figure 7: Forecasting Performance: MAPE (

\downarrow

) Comparison Across Models. Datasets split chronologically: 70% training, 10% validation, 20% testing. All comparisons are based on the same datasets under a unified experimental setup, and all results were obtained by the authors.

Figure 8: Forecasting Performance:

R^{2}

(

\uparrow

Panel of Figure 9 — Figure 9: Prediction comparisons on different datasets

Figure 10: Prediction comparison on dataset IXIC

Asset	Step	MSE ( $\downarrow$ )	MAE ( $\downarrow$ )	R 2 ( $\uparrow$ )	MAPE(%) ( $\downarrow$ )
JNJ	1day	3.26	1.37	0.97	0.83
	3days	8.47	2.32	0.92	1.40
	5days	12.79	2.82	0.88	1.70
	7days	19.03	3.51	0.82	2.11
MSFT	1day	37.24	4.64	0.97	1.73
	3days	176.45	10.66	0.87	3.86

Table 10: Multi-step Prediction Performance for Different Assets.

Window Size	MSE ( $\downarrow$ )	MAE ( $\downarrow$ )	R 2 ( $\uparrow$ )	MAPE(%) ( $\downarrow$ )
12	46835.32	172.14	0.9827	1.35
24	43685.02	162.03	0.9831	1.28
48	46851.14	166.34	0.9809	1.31
64	55683.83	185.42	0.9770	1.45
96	45169.04	163.83	0.9807	1.27

Table 11: Performance Comparison of Different Window Sizes.

Volatility Estimator	MSE ( $\downarrow$ )	MAE ( $\downarrow$ )	R 2 ( $\uparrow$ )	MAPE(%) ( $\downarrow$ )
GARCH(1,1)	43685.02	162.03	0.9831	1.28
GJR-GARCH(1,1)	83502.94	240.12	0.9677	1.82
EGARCH(1,1)	181471.67	359.37	0.9299	2.67

Table 12: Sensitivity of Gsigmoid-LSTM to the Volatility Estimator on IXIC.

Dataset	Gsigmoid	Fixed- $\gamma$	No-Vol	Concat	LSTM
JNJ	3.26	3.78	4.59	4.29	3.59
MSFT	37.24	81.62	407.61	248.48	97.20
AMZN	12.85	14.99	37.85	20.93	45.80
IXIC	43685.02	62084.48	119772.96	82419.70	131913.75
GOOG	5.96	20.41	27.22	20.69	20.71

Table 13: Comparison of MSE (

\downarrow

) Performance in Ablation Study.

Dataset	Gsigmoid	Fixed- $\gamma$	No-Vol	Concat	LSTM
JNJ	1.37	1.50	1.70	1.64	1.46
MSFT	4.64	6.95	17.45	13.23	7.93
AMZN	2.68	2.95	5.19	3.70	5.79
IXIC	162.03	200.55	287.86	240.62	303.95
GOOG	1.82	3.59	4.20	3.58	3.61

Table 14: Comparison of MAE (

\downarrow

) Performance in Ablation Study.

Figure 11: Comparison of MAPE (

\downarrow

) Performance in Ablation Study. Datasets split chronologically: 70% training, 10% validation, 20% testing. All comparisons are based on the same datasets under a unified experimental setup, and all results were obtained by the authors.

Figure 12: Comparison of

R^{2}

(

\uparrow

Figure 13: Forget-gate behavior of Gsigmoid-LSTM and the standard LSTM on the MSFT test window. (a) Daily log returns and the GARCH(1,1) conditional volatility

\hat{\sigma}_{t}

. Shaded bands mark high-volatility episodes, defined as test-window days on which

\hat{\sigma}_{t}

lies in its top quartile. (b) The volatility signal

\rho_{t}

consumed by the gating activation. (c) Unit-averaged forget-gate output

\hat{f}_{t}

at the last timestep of each prediction window. The traced models correspond to the ablation runs under the fixed random seed reported in Table 13 , with test MSE of 37.24 for Gsigmoid-LSTM and 97.20 for the LSTM baseline.

Comparison Model	JNJ		MSFT		AMZN		IXIC		GOOG
	DM	p	DM	p	DM	p	DM	p	DM	p
LSTM	-3.63 ∗∗	$<$ 0.01	-8.22 ∗∗	$<$ 0.01	-10.46 ∗∗	$<$ 0.01	-8.02 ∗∗	$<$ 0.01	-7.74 ∗∗	$<$ 0.01
RNN	-12.46 ∗∗	$<$ 0.01	-10.26 ∗∗	$<$ 0.01	-9.20 ∗∗	$<$ 0.01	-9.72 ∗∗	$<$ 0.01	-9.15 ∗∗	$<$ 0.01
GRU	-5.69 ∗∗	$<$ 0.01	-9.08 ∗∗	$<$ 0.01	-7.45 ∗∗	$<$ 0.01	-8.53 ∗∗	$<$ 0.01	-7.63 ∗∗	$<$ 0.01
ANN	-4.96 ∗∗	$<$ 0.01	-2.21 ∗	0.03	-6.33 ∗∗	$<$ 0.01	-8.15 ∗∗	$<$ 0.01	-4.12 ∗∗	$<$ 0.01
BiLSTM	-10.65 ∗∗	$<$ 0.01	-8.17 ∗∗	$<$ 0.01	-3.74 ∗∗	$<$ 0.01	-7.57 ∗∗	$<$ 0.01	-8.30 ∗∗	$<$ 0.01

Table 15: Diebold-Mariano Test Results for Model Comparison

Comparison Model	JNJ		MSFT		AMZN		IXIC		GOOG
	High	Low	High	Low	High	Low	High	Low	High	Low
LSTM	-2.98 ∗∗	-2.74 ∗∗	-5.60 ∗∗	-6.40 ∗∗	-4.79 ∗∗	-14.02 ∗∗	-3.83 ∗∗	-8.47 ∗∗	-4.40 ∗∗	-6.93 ∗∗
RNN	-8.70 ∗∗	-9.56 ∗∗	-7.36 ∗∗	-7.59 ∗∗	-3.73 ∗∗	-11.47 ∗∗	-5.50 ∗∗	-9.82 ∗∗	-5.69 ∗∗	-7.81 ∗∗
GRU	-4.66 ∗∗	-4.12 ∗∗	-6.26 ∗∗	-6.93 ∗∗	-1.88 ns	-10.38 ∗∗	-4.08 ∗∗	-9.06 ∗∗	-4.25 ∗∗	-6.89 ∗∗
ANN	-4.49 ∗∗	-3.15 ∗∗	-3.30 ∗∗	-0.24 ns	-4.82 ∗∗	-4.86 ∗∗	-5.87 ∗∗	-6.52 ∗∗	-3.45 ∗∗	-2.28 ∗
BiLSTM	-7.25 ∗∗	-9.08 ∗∗	-5.91 ∗∗	-6.49 ∗∗	-0.84 ns	-5.25 ∗∗	-4.48 ∗∗	-6.74 ∗∗	-5.00 ∗∗	-7.32 ∗∗

Table 16: Diebold-Mariano Statistics of Gsigmoid-LSTM Against Comparison Models within High- and Low-Volatility Regimes.

Model	Average rank	Rank
Gsigmoid-LSTM	1.00	1
ANN	3.00	2
GRU	4.80	3
DeepESN	5.20	4
BiGRU	5.40	5
RVFL	5.80	6

Table 17: Average Ranks of Nine Models in the Friedman Test.

Dataset	Metric	Comparison Model
		Fixed- $\gamma$	No-Vol	Concat	LSTM
JNJ	DM Statistic	-6.10	-7.38	-6.85	-3.63
	p-value	$<$ 0.01 ∗∗	$<$ 0.01 ∗∗	$<$ 0.01 ∗∗	$<$ 0.01 ∗∗
MSFT	DM Statistic	-6.37	-9.64	-8.88	-8.22
	p-value	$<$ 0.01 ∗∗	$<$ 0.01 ∗∗	$<$ 0.01 ∗∗	$<$ 0.01 ∗∗
AMZN	DM Statistic	-6.30	-9.49	-7.64	-10.46

Table 18: Diebold-Mariano Test Results for Ablation Study

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A decomposition-integration framework coupling singular spectrum analysis with a volatility-adaptive LSTM for stock price forecasting

Xin-Rui Lu, Da-Qing Zhang

School of Science, University of Science and Technology Liaoning, Anshan, 114051, Liaoning, PR China

Time Series Forecasting Financial Time Series Recurrent Neural Networks Time Series Decomposition

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Financial time series exhibit multi-scale, high-noise, and non-stationary volatility dynamics. Decomposition--integration frameworks built on complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) have advanced this field, yet existing CEEMDAN hybrids remain unstable: high-frequency mode mixing contaminates downstream predictors, and the heterogeneous volatility of decomposed components is not matched by any volatility-adaptive mechanism. To address these gaps, this paper proposes CEEMDAN-SE-SSA-VMGL. CEEMDAN adaptively decomposes the original series, and sample-entropy-driven K-means clustering groups the intrinsic mode functions into high-, mid-, and low-frequency and residual components. Singular spectrum analysis (SSA) is then applied to the high-frequency group, whose singular-value decomposition of the trajectory matrix and principal-component reconstruction decouple intertwined pseudo-oscillatory modes and alleviate mode mixing. We further design a plug-and-play robust LSTM variant, the volatility-modulated GARCH-LSTM (VMGL), which embeds a differentiable GARCH(1,1) layer inside the LSTM cell as a risk gate; the estimated conditional volatility multiplicatively scales the forget-gate activation, letting memory decay adapt to the instantaneous volatility while remaining fully end-to-end trainable. Component predictions are linearly combined. On five US and Chinese financial series (CSI 300, CSI 500, AAPL, GOOGL, S&P 500), the proposed model is compared with eleven benchmark architectures, lowering the forecasting error by 9% to 24% relative to the best benchmark and by 60% to 70% relative to a standard LSTM. Diebold--Mariano and model-confidence-set (MCS) tests confirm the statistical significance and stability of the gains, spectral analyses verify the SSA denoising efficacy, and backtesting demonstrates the framework's economic utility.

Abstract

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