Combining Deep Learning and GARCH Models for Financial Volatility and Risk Forecasting

Abstract: In this paper, we develop a hybrid approach to forecasting the volatility and risk of financial instruments by combining common econometric GARCH time series models with deep learning neural networks. For the latter, we employ Gated Recurrent Unit (GRU) networks, whereas four different specifications are used as the GARCH component: standard GARCH, EGARCH, GJR-GARCH and APARCH. Models are tested using daily logarithmic returns on the S&P 500 index as well as gold price Bitcoin prices, with the three assets representing quite distinct volatility dynamics. As the main volatility estimator, also underlying the target function of our hybrid models, we use the price-range-based Garman-Klass estimator, modified to incorporate the opening and closing prices. Volatility forecasts resulting from the hybrid models are employed to evaluate the assets' risk using the Value-at-Risk (VaR) and Expected Shortfall (ES) at two different tolerance levels of 5% and 1%. Gains from combining the GARCH and GRU approaches are discussed in the contexts of both the volatility and risk forecasts. In general, it can be concluded that the hybrid solutions produce more accurate point volatility forecasts, although it does not necessarily translate into superior VaR and ES forecasts.

• The paper demonstrates that integrating GARCH models with GRU networks improves volatility forecasting accuracy using metrics like MSE and MAE.
• It leverages multiple GARCH variants alongside GRU layers to capture complex market dynamics across assets such as the S&P 500, Bitcoin, and gold.
• Empirical results indicate that model performance varies by asset and risk tolerance, highlighting the need for tailored approaches for VaR and ES predictions.

Hybrid GARCH and Deep Learning Models for Financial Forecasting

This paper explores a hybrid approach to financial volatility and risk forecasting, integrating traditional GARCH models with deep learning techniques, specifically GRU neural networks. The authors evaluate this hybrid methodology using daily logarithmic returns from the S&P 500 index, Bitcoin, and gold, and assess the performance of the models via Value-at-Risk (VaR) and Expected Shortfall (ES) metrics.

Methodology

The methodology combines GARCH models, including standard GARCH, EGARCH, GJR-GARCH, and APARCH, with GRU neural networks.

GARCH Models

The paper employs four GARCH specifications to model conditional variance:

• GARCH: Defined by the equation:

  $$h_t = a_0 + \sum_{i=1}^{q} \alpha_i \epsilon_{t-i}^2 + \sum_{j=1}^{p} \beta_j h_{t-j}$$

  where $h_t$ is the conditional variance, $\epsilon_t$ are error terms, and $\alpha_i$ and $\beta_j$ are parameters subject to constraints ensuring positive variance.

• GJR-GARCH: Which incorporates an indicator function to capture asymmetric responses to positive and negative shocks:

  $$h_t = a_0 + \sum_{i=1}^{q} \alpha_i \epsilon_{t-i}^2 + \sum_{i=1}^{q} \theta_i I_{t-i} \epsilon_{t-i}^2 + \sum_{j=1}^{p} \beta_j h_{t-j}$$

  where $I_{t-i}$ equals 1 if $\epsilon_{t-i} \leq 0$ and 0 otherwise, and $\theta_i$ are parameters capturing the asymmetric effect.

• EGARCH: Uses the logarithm of the conditional variance to allow for asymmetry:

  $$ln(h_t) = a_0 + \sum_{i=1}^{q} \alpha_i \left\{ \theta z_{t-i} + \gamma \left[ \left| z_{t-i} \right| - E(\left| z_{t-i} \right|) \right] \right\} + \sum_{j=1}^{p} \beta_j ln(h_{t-j})$$

  where $z_t$ are standardized residuals, and $\theta$ and $\gamma$ are parameters capturing the size and sign effects, respectively.

• APARCH: Which generalizes other ARCH variants by including a power term:

  $$h_t^{\delta} = a_0 + \sum_{i=1}^{q} \alpha_i \left( \left| \epsilon_{t-i} \right| - \gamma_i \epsilon_{t-i} \right)^{\delta} + \sum_{j=1}^{p} \beta_j h_{t-j}^{\delta}$$

  where $\delta$ is a power term, and $\gamma_i$ allows for asymmetry.

The error term $z_t$ is modeled using normal, Student's t-, and skewed Student's t-distributions. Estimation is performed via maximum likelihood using the rugarch package in R.

GRU and Hybrid Models

The paper uses GRU networks, a simplified version of LSTM networks, for the neural network component. The GRU network cells are defined by the following equations:

$z_t = \sigma(W^z x_t + U^z o_{t-1} + b^z)$r_t = \sigma(W^r x_t + U^r o_{t-1} + b^r)$$\tilde{o}_t = \tanh(W x_t + U (r_t \odot o_{t-1}) + b)$$o_t = (1 - z_t) \odot o_{t-1} + z_t \odot \tilde{o}_t$</p> <p>where$x_t$is the input vector,$o_t$is the output vector,$z_t$is the update gate vector, and$r_t$$is the reset gate vector. The hybrid GARCH-GRU model incorporates GARCH volatility forecasts as inputs to the GRU network. The network is trained to predict the Garman-Klass-Yang-Zhang (GKYZ) volatility estimator, modified to include overnight price gaps. The GKYZ volatility estimator is given by:</p> <p>$$\sigma_i^2 = \frac{1}{n} \sum_{i=1}^{n} \left\{ \frac{1}{2} (\ln(H_i) - \ln(L_i))^2 - (2\ln(2) - 1)(\ln(C_i) - \ln(O_i))^2 \right\}$</p> <p>where$O_i$,$H_i$,$L_i$, and$C_i$$are the open, high, low, and close prices at time$$i$.

The GRU network architecture consists of three layers with 512, 256, and 128 neurons, respectively, and a single-neuron dense output layer. ReLU activation, dropout regularization (0.3), and L2 kernel regularization (0.00001) are applied. The Adam optimizer is used with a learning rate of 0.0009, and the loss function is the mean squared error between GKYZ volatility estimates and network predictions.

Evaluation Metrics

The volatility forecasts are evaluated using MSE, MAE, and HMSE. The Diebold-Mariano test is used to assess the statistical significance of differences in MSEs between GARCH and GARCH-GRU models. Risk measure accuracy is assessed using VaR and ES, with backtesting performed via the Kupiec test and the Christoffersen test for VaR, and the McNeil and Frey test for ES.

Empirical Analysis

The empirical analysis uses daily logarithmic returns for the S&P 500, Bitcoin, and gold, spanning from 2009 to 2020. A rolling window scheme is employed, with a window size of 504 days for GARCH models and rolling training sets of 1008 observations for the neural network stage.

Results for S&P 500

The EGARCH-GRU model with a skewed Student's t-distribution yields the best MSE. Combining GARCH models with GRU networks generally enhances forecast accuracy, with statistically significant Diebold-Mariano test p-values. The GJR-GARCH-GRU models with normal and skewed Student's t-distributions achieve accurate VaR hit coverage for the 5% tolerance level, while the APARCH model with a skewed Student's t-distribution performs best for the 1% tolerance.

Results for Bitcoin

The APARCH-GRU structure with a normal distribution performs best in terms of point volatility forecasts. However, the improvement in volatility forecasts does not consistently translate to superior VaR prediction performance. "Sheer" APARCH models produce VaR estimates closest to the expected hit ratio.

Results for Gold

A hybrid EGARCH-GRU model with a Student's t-distribution produces the most accurate volatility forecasts. However, the gains from hybridization do not unanimously improve VaR prediction. Hybrid models perform best at the 5% tolerance level, while "sheer" GARCH models perform best for the 1% VaR prediction.

Conclusion

The paper concludes that hybrid GARCH-GRU models can improve financial volatility forecasting. However, the improved volatility forecasts do not always translate into better VaR and ES predictions. The choice of the best model depends on the specific asset and tolerance level. The authors suggest that future research should focus on redefining the target function underlying the GRU components in the hybrid models, specifically for VaR and ES prediction tasks.