这是使用GARCH预测股价波动的正确方法吗

Question

我正试图对未来某个时间（例如90天）的股票波动做出预测。看来GARCH是为此使用的传统模型。

我已经在下面使用Python的arch库实现了这一点。我所做的一切都在注释中进行了解释，运行代码唯一需要更改的是提供您自己的每日价格，而不是从我自己的API检索价格的地方。

import utils
import numpy as np
import pandas as pd
import arch
import matplotlib.pyplot as plt

ticker = 'AAPL'         # Ticker to retrieve data for
forecast_horizon = 90   # Number of days to forecast

# Retrive prices from IEX API
prices = utils.dw.get(filename=ticker, source='iex', iex_range='5y')
df = prices[['date', 'close']]

df['daily_returns'] = np.log(df['close']).diff()            # Daily log returns
df['monthly_std'] = df['daily_returns'].rolling(21).std()   # Standard deviation across trading month
df['annual_vol'] = df['monthly_std'] * np.sqrt(252)         # Annualize monthly standard devation
df = df.dropna().reset_index(drop=True)

# Convert decimal returns to %
returns = df['daily_returns'] * 100

# Fit GARCH model
am = arch.arch_model(returns[:-forecast_horizon])
res = am.fit(disp='off')

# Calculate fitted variance values from model parameters
# Convert variance to standard deviation (volatility)
# Revert previous multiplication by 100
fitted = 0.1 * np.sqrt(
    res.params['omega'] +
    res.params['alpha[1]'] *
    res.resid**2 +
    res.conditional_volatility**2 *
    res.params['beta[1]']
)

# Make forecast
# Convert variance to standard deviation (volatility)
# Revert previous multiplication by 100
forecast = 0.1 * np.sqrt(res.forecast(horizon=forecast_horizon).variance.values[-1])

# Store actual, fitted, and forecasted results
vol = pd.DataFrame({
    'actual': df['annual_vol'],
    'model': np.append(fitted, forecast)
})

# Plot Actual vs Fitted/Forecasted
plt.plot(vol['actual'][:-forecast_horizon], label='Train')
plt.plot(vol['actual'][-forecast_horizon - 1:], label='Test')
plt.plot(vol['model'][:-forecast_horizon], label='Fitted')
plt.plot(vol['model'][-forecast_horizon - 1:], label='Forecast')
plt.legend()
plt.show()

对于Apple，这将产生以下情节：

很明显，拟合值一直远远低于实际值，这也导致该预测也被大大低估了（鉴于苹​​果在此测试期间的波动性异常高，因此这是一个不佳的例子，我尝试过的公司，该模型总是低估了拟合值。

我所做的一切都正确吗，GARCH模型不是很强大，还是很难建模？还是我犯了一些错误？