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I really liked a comparison generated by ChatGPT: Financial markets are rarely calm seas — they surge, crash, and ripple with waves of uncertainty. While this may be overly poetic, it conveys an important message. One thing that will always be present in the markets is volatility.
And although asset prices themselves may seem unpredictable, something that might be easier to forecast is volatility— or more precisely, volatility clusters. In simpler terms, these clusters refer to periods of high (or low) volatility that are often followed by more of the same. Capturing this behavior is crucial for traders, risk managers, and policymakers.
In this article, we will explore one of the approaches to forecasting time-varying volatility: the popular ARCH model.
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Volatility 101
Before we talk about forecasting volatility, it makes sense to first understand what volatility actually is. Volatility refers to the degree of variation in asset prices over time and is usually measured as the standard deviation or variance of returns.
Essentially, it tells us how much risk or uncertainty is associated with the future price of an asset:
High volatility implies that returns fluctuate significantly, indicating a riskier asset.
Low volatility implies that returns are more stable, suggesting a less risky asset.
That’s almost it for the basic introduction. One last thing to keep in mind is that volatility is not static — it changes over time in patterns that we can try to model and forecast.
Image generated with ChatGPT
Why Forecast Volatility?
Now that we know what volatility is, it should be quite clear why we’d want to forecast it. Some of the key use cases for volatility forecasting include:
Risk management — Financial institutions need to quantify and control risk. For example, a bank holding options should know by how much their value might change.
Asset allocation and portfolio management — As investors aim to balance returns and risk, they can use forecasts to manage their portfolios. For instance, if volatility is rising and they’re risk-averse, they might shift to safer assets.
Derivative valuation — The prices of derivatives, such as options, are highly sensitive to expected volatility.
The ARCH Model
The Autoregressive Conditional Heteroskedasticity (ARCH) model was introduced by Robert Engle in 1982 and was designed to forecast time-varying volatility. It is based on the concept of volatility clustering mentioned earlier — that is, periods of high volatility are typically followed by more high volatility, and low by low.
Many time series models (such as ARMA) assume that the variance of the error term is constant over time. However, this assumption is often violated in financial time series. To capture this behavior, the ARCH(q) model expresses the variance of the error term as a function of past squared errors. To be a bit more precise, it assumes that the variance of the errors follows an autoregressive model.
where:
ε_t is the error term at time t,
σ_t^2 is the conditional variance at time t,
α_0 > 0, and α_i ≥ 0 for I > 0 to ensure positive variance.
Let’s try to get an intuitive understanding of the formula. In the simplest case, we are dealing with the ARCH(1) model with the following formula:
The formula essentially states that today’s volatility depends on a constant plus the squared return shock from yesterday. So if there’s a large unexpected movement today, tomorrow’s forecasted volatility will be higher.
The biggest strength of the ARCH model is that it produces volatility estimates that exhibit excess kurtosis (fat tails as compared to the Normal distribution), which is in line with the empirical observations about stock returns.
Lastly, it is also important to be aware of the limitations of the ARCH model:
Financial time series often exhibit long memory in volatility, which ARCH cannot effectively model without including many lags.
ARCH ignores the leverage effect, assuming that positive and negative shocks of the same magnitude have identical impacts on volatility.
ARCH does not explain variations in volatility, so the model is likely to over-forecast volatility. That is because it is slow to respond to large, isolated shocks in the returns series.
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💻 Hands-On Example in Python
It is time to get our hands a bit dirty and create volatility forecasts using the arch library.
As always, we start by importing the required libraries: requests to download data, pandas for some data wrangling, matplotlib for plotting, and, arch for volatility forecasting.
The api_key file contains our personal API key to Financial Modelling Prep’s API, from where we will download the historical stock prices.
import requests
import pandas as pd
from arch import arch_model
from datetime import datetime
# plotting
import matplotlib.pyplot as plt
# settings
plt.style.use("seaborn-v0_8-colorblind")
plt.rcParams["figure.figsize"] = (16, 8)
# api key
from api_keys import FMP_API_KEYThen, we download the data. For this example, we use Apple’s stock prices from 2015 to April 2025. ARCH models estimate time-varying variance, which is generally more data-hungry than conditional mean models. With too little data, we risk unstable variance estimates and overfitting. Having enough data also helps capture different volatility regimes (for example, crises and quieter periods).
TICKER = "AAPL"
START_DATE = "2015-01-01"
END_DATE = "2025-04-30"
SPLIT_DATE = datetime(2025, 1, 1)
def get_adj_close_price(symbol: str, start_date: str, end_date: str) -> pd.DataFrame:
"""
Fetches adjusted close prices for a given ticker and date range from FMP.
"""
url = f"https://financialmodelingprep.com/api/v3/historical-price-full/{symbol}"
params = {
"from": start_date,
"to": end_date,
"apikey": FMP_API_KEY
}
response = requests.get(url, params=params)
r_json = response.json()
if "historical" not in r_json:
raise ValueError(f"No historical data found for {symbol}")
df = pd.DataFrame(r_json["historical"]).set_index("date").sort_index()
df.index = pd.to_datetime(df.index)
return df[["adjClose"]].rename(columns={"adjClose": symbol})
price_df = get_adj_close_price(TICKER, START_DATE, END_DATE)
price_df.plot(title=f"{TICKER} Adjusted Close Price", ylabel="Price ($)", xlabel="Date")Below we can see the downloaded stock prices of Apple.
Then, we need to convert the prices to returns.
return_df = 100 * price_df[TICKER].pct_change().dropna()
return_df.name = "asset_returns"
return_df.plot(title=f"{TICKER} returns - {START_DATE}:{END_DATE}");
Our dataset contains 2596 observations.
Thanks to the arch library, estimating ARCH models is as simple as instantiating the model and using the .fit() method. We will train two models, using 1 and 5 autoregressive components.
model_1 = arch_model(return_df, mean="Zero", vol="ARCH", p=1, q=0)
model_2 = arch_model(return_df, mean="Zero", vol="ARCH", p=5, q=0)
fitted_model_1 = model_1.fit(disp="off")
fitted_model_2 = model_2.fit(disp="off")For the mean model, we selected the zero-mean approach, which is suitable for many liquid financial assets. Another possible choice here could be a constant mean.
We can inspect the fitted models by using the summary method.
print(fitted_model_1.summary()) Zero Mean - ARCH Model Results
==============================================================================
Dep. Variable: asset_returns R-squared: 0.000
Mean Model: Zero Mean Adj. R-squared: 0.000
Vol Model: ARCH Log-Likelihood: -5158.24
Distribution: Normal AIC: 10320.5
Method: Maximum Likelihood BIC: 10332.2
No. Observations: 2596
Date: Mon, May 05 2025 Df Residuals: 2596
Time: 13:35:15 Df Model: 0
Volatility Model
========================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------
omega 2.5171 0.166 15.203 3.374e-52 [ 2.193, 2.842]
alpha[1] 0.2586 5.807e-02 4.453 8.455e-06 [ 0.145, 0.372]
========================================================================
Covariance estimator: robustWe will do this again for the ARCH(5) model to compare the parameter estimates.
Zero Mean - ARCH Model Results
==============================================================================
Dep. Variable: asset_returns R-squared: 0.000
Mean Model: Zero Mean Adj. R-squared: 0.000
Vol Model: ARCH Log-Likelihood: -5030.98
Distribution: Normal AIC: 10074.0
Method: Maximum Likelihood BIC: 10109.1
No. Observations: 2596
Date: Mon, May 05 2025 Df Residuals: 2596
Time: 13:35:15 Df Model: 0
Volatility Model
==========================================================================
coef std err t P>|t| 95.0% Conf. Int.
--------------------------------------------------------------------------
omega 1.4362 0.143 10.033 1.089e-23 [ 1.156, 1.717]
alpha[1] 0.1496 3.811e-02 3.925 8.660e-05 [7.490e-02, 0.224]
alpha[2] 0.0902 3.561e-02 2.532 1.133e-02 [2.038e-02, 0.160]
alpha[3] 0.1232 3.976e-02 3.100 1.938e-03 [4.531e-02, 0.201]
alpha[4] 0.1064 4.501e-02 2.364 1.809e-02 [1.818e-02, 0.195]
alpha[5] 0.1105 3.792e-02 2.915 3.556e-03 [3.622e-02, 0.185]
==========================================================================
Covariance estimator: robustNow we can inspect the standardized residuals and the conditional volatility series by plotting them. The standardized residuals were computed by dividing the residuals by the conditional volatility.
We use the annualize="D" argument of the plot method in order to annualize the conditional volatility series from daily data.
fitted_model_1.plot(annualize="D");
Having estimated the models, it is now time to generate the volatility forecasts. Using the ARCH model family, there are three main approaches to doing so:
Analytical: One-step forecasts are always available due to the model structure. Multi-step forecasts require forward recursion and are only feasible for models that are linear in squared residuals. This method is not suitable for pure ARCH models, but it can be used with their extensions (such as GARCH, HARCH, etc.).
Simulation: This approach simulates future volatility paths using random draws from a specified residual distribution. Averaging these paths yields the forecast. It works for any forecast horizon and converges to the analytical forecast as the number of simulations increases.
Bootstrap (aka, Filtered Historical Simulation): Similar to simulation, but draws standardized residuals from historical data using the estimated model parameters. This method requires only a small amount of in-sample data.
Due to the specification of ARCH class models, the first out-of-sample forecast will always be the same, regardless of which approach we use.
As you can see in the snippet below, we use the analytic method for generating the forecasts, but you could also pass in “simulation” or “bootstrap”.
fitted_model_1 = model_1.fit(last_obs=SPLIT_DATE, disp="off")
fitted_model_2 = model_2.fit(last_obs=SPLIT_DATE, disp="off")
forecasts_analytical_1 = fitted_model_1.forecast(
horizon=1,
start=SPLIT_DATE,
method="analytic",
reindex=False
)
forecasts_analytical_2 = fitted_model_2.forecast(
horizon=1,
start=SPLIT_DATE,
method="analytic",
reindex=False
)
a = forecasts_analytical_1.variance.rename(columns={"h.1": "arch(1)"})
b = forecasts_analytical_2.variance.rename(columns={"h.1": "arch(5)"})
a.join(b).plot(title="Analytical forecasts for horizon = 1");
As you can see in the generated plot, the volatility forecast varies quite a lot depending on the number of autoregressive terms used in estimating the ARCH model.
In the next article on volatility forecasting, we will explore the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model, an extension of the ARCH model.
Wrapping up
In this article, we have explored volatility forecasting with ARCH models. The main takeaways are:
Volatility forecasting aims to predict the variability of asset returns over time, which is crucial for risk management, derivative pricing, and portfolio allocation.
The ARCH model captures time-varying volatility by modeling the current period’s variance as a function of past squared returns.
It accounts for volatility clustering — periods of high or low volatility persisting over time.
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