Accurately forecasting volatility is a strategic necessity for modern risk management, enabling informed decisions on portfolio construction, hedging, and capital allocation. Simple historical volatility measures, which assume that future volatility will resemble the past, are often inadequate because they fail to capture the dynamic, time-varying nature of market risk. Our methodology is built on the recognition that markets move between periods of high and low turbulence, and we explicitly reject the assumption of constant volatility.

Our methodology is designed to capture the phenomenon known as conditional heteroscedasticity, where the magnitude of unpredictable price fluctuations varies over time. It is most visibly expressed as volatility clustering, where periods of high volatility tend to be followed by more high volatility, and calm periods tend to be followed by more calm.

3.1 The ARCH Model

The Autoregressive Conditional Heteroscedasticity (ARCH) model, developed by Robert Engle, was the first formal econometric model to capture this behavior. The simplest version, an ARCH(1) model, defines the return process and its conditional variance as follows:

• Rt = σt εt
• σt² = c + a1 R²t−1

In this model, the variance of an asset’s return at time t (σt²) is not constant. Instead, it depends on a constant term (c) plus a proportion of the squared return from the previous period (R²t−1). This elegantly captures the observation that large price movements (positive or negative) in one period tend to be followed by larger price movements in the next.

3.2 The GARCH Model

The Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model is a powerful and more parsimonious extension of the ARCH framework. It enhances the forecast by incorporating not only past returns but also past volatility forecasts. The general GARCH(p, q) model is expressed as:

σt² = c + a1 R²t−1 + … + ap R²t−p + b1 σ²t−1 + … + bq σ²t−q

The core insight of the GARCH model is that future volatility is a function of two key components:

1. Past Squared Returns (The ARCH term): Recent market shocks, as captured by R²t−p.
2. Past Volatility Forecasts (The GARCH term): The model’s own prior variance forecasts, as captured by σ²t−q.

In essence, the ARCH term represents the immediate impact of market shocks or “news,” while the GARCH term models the persistence of that shock, akin to how fear or complacency can linger in the market long after an event has passed. Our ability to model this persistence is critical for forward-looking risk management. In practice, a simple GARCH(1,1) model is often sufficient to capture the dynamics of volatility in financial time series.

GARCH models provide a robust, forward-looking estimate of volatility, which is a critical input for our risk management processes, including Value-at-Risk (VaR) calculations and the pricing of options. However, just as volatility is not constant, we recognize that the relationships between financial variables are not uniform across all market conditions, necessitating even more advanced techniques.