Measuring and Monitoring Volatility

Welcome to Volatility: The Heartbeat of Risk

Welcome! If you’ve ever wondered why stock prices go on a "rollercoaster" one day and stay calm the next, you’re looking at volatility. In this chapter, we are going to learn how to measure that "wildness." This is a crucial part of the Valuation and Risk Models section because we can’t calculate things like Value-at-Risk (VaR) or price an option accurately if we don’t know how much prices are likely to swing.

Don't worry if the math looks a bit intimidating at first. We’ll break it down into simple steps, using everyday analogies to help the concepts stick!

1. Understanding Daily Returns and Volatility

Before we measure volatility, we need to know what we are measuring. Usually, we look at daily returns. In the FRM world, we often use continuously compounded returns (log returns) because they are easier to work with mathematically.

The return for day \(n\), denoted as \(u_n\), is typically calculated as:
\(u_n = \ln(S_n / S_{n-1})\)

Quick Review: Volatility (\(\sigma\)) is essentially the standard deviation of these returns. It tells us the range within which we expect the price to move. Variance (\(\sigma^2\)) is just volatility squared.

Common Mistake: Students often forget the difference between variance and volatility. Just remember: Variance is the "squared" version; Volatility is the "original units" version. If a question asks for volatility but gives you variance, take the square root!

2. The Simple Moving Average (The "Old School" Way)

The simplest way to estimate variance is to take the average of the squared returns over the last \(m\) days:
\(\sigma_n^2 = \frac{1}{m} \sum_{i=1}^{m} u_{n-i}^2\)

The Problem: This method treats yesterday’s market crash the same as a tiny price move from 30 days ago. In reality, what happened yesterday is usually much more relevant to what will happen tomorrow.

Summary: Simple moving averages are easy but "slow" to react because they give equal weight to all data points in the window.

3. EWMA: The "Fading Memory" Model

The Exponentially Weighted Moving Average (EWMA) solves the equal-weighting problem. It gives more weight to the most recent data and less weight to older data.

The formula for the new variance (\(\sigma_n^2\)) is:
\(\sigma_n^2 = \lambda \sigma_{n-1}^2 + (1 - \lambda) u_{n-1}^2\)

The Components:
1. \(\lambda\) (Lambda): This is the decay factor. It’s usually between 0.90 and 0.99 (RiskMetrics uses 0.94).
2. \(\sigma_{n-1}^2\): Yesterday’s variance estimate.
3. \(u_{n-1}^2\): Yesterday’s actual squared return.

Analogy: Think of EWMA like your memory of a restaurant. Yesterday’s meal (\(u_{n-1}^2\)) has a big impact on your opinion, but your long-term overall experience (\(\sigma_{n-1}^2\)) still matters. As days pass, that one bad meal "decays" and matters less.

Key Takeaway: A higher \(\lambda\) means the model has a "long memory" (it changes slowly). A lower \(\lambda\) means it reacts very quickly to new information.

4. The GARCH (1,1) Model: Adding Mean Reversion

GARCH stands for Generalized Autoregressive Conditional Heteroskedasticity. (Don’t let the name scare you—it’s just a fancy way of saying "volatility that changes over time and cares about the long-run average.")

While EWMA assumes volatility can wander anywhere, GARCH (1,1) assumes volatility eventually wants to return to a long-run average. This is called mean reversion.

The formula for GARCH (1,1) is:
\(\sigma_n^2 = \omega + \alpha u_{n-1}^2 + \beta \sigma_{n-1}^2\)

The Three Ingredients:
1. \(\omega\) (Omega): A weighted version of the long-run variance (\(\gamma V_L\)).
2. \(\alpha\) (Alpha): The weight given to the recent news (yesterday's return).
3. \(\beta\) (Beta): The weight given to the previous variance.

Did you know? If you set \(\omega = 0\) and \(\alpha + \beta = 1\), the GARCH model actually turns back into the EWMA model! EWMA is just a special case of GARCH without mean reversion.

Calculating the Long-Run Average Variance (\(V_L\)):

You can find the long-run variance using this formula:
\(V_L = \frac{\omega}{1 - \alpha - \beta}\)

Condition for Stability: For the model to be stable and "mean-reverting," the sum of \(\alpha + \beta\) must be less than 1. If \(\alpha + \beta > 1\), the volatility is "explosive" and won't return to an average.

Key Takeaway: GARCH (1,1) is generally considered superior to EWMA for long-term forecasting because it accounts for the fact that volatility usually settles down eventually.

5. Maximum Likelihood Estimation (MLE)

You might wonder: How do we pick the best values for \(\lambda\), \(\alpha\), or \(\beta\)?

We use a technique called Maximum Likelihood Estimation (MLE). You don't need to do the complex calculus for the FRM exam, but you do need to understand the concept.

The Step-by-Step Logic of MLE:
1. Start with trial values for your parameters (\(\alpha, \beta\)).
2. Calculate the probability (likelihood) that your actual historical data would have occurred if those parameters were true.
3. Use a computer to keep changing the parameters until you find the ones that maximize that probability.
4. This "best fit" becomes your model.

6. Monitoring Correlations and Covariances

Volatility isn't just about one asset; we also care about how two assets move together. This is covariance.

The good news is that the math for covariance is almost identical to volatility! You can use EWMA or GARCH to update covariance:
EWMA for Covariance: \(cov_n = \lambda cov_{n-1} + (1 - \lambda) x_{n-1} y_{n-1}\)

Once you have the covariance and the individual volatilities, you can find the Correlation (\(\rho\)):
\(\rho_{xy} = \frac{cov_{xy}}{\sigma_x \sigma_y}\)

Mnemonic for Correlation: "Covariance divided by the product of the deviations." Just remember that correlation must always be between -1 and +1. If your calculation gives you 1.2, stop and check your math!

7. Final Quick Review & Summary

Let's recap the most important bits for your study session:

• Simple Moving Average: Equal weights, "ghosting effect" (old data drops off suddenly).
• EWMA: Exponential weights, reacts fast, uses \(\lambda\). No long-run mean.
• GARCH (1,1): Includes mean reversion. Best for seeing the "big picture."
• Persistence: In GARCH, this is \(\alpha + \beta\). It tells you how long a "volatility shock" will last.
• Weights: In any valid model, the total weights must sum to 1.

Encouraging Note: This chapter is very formula-heavy, but the exam often asks about the behavior of these models. Focus on understanding how changing \(\lambda\) or \(\alpha\) changes the way the model looks at the world. You’ve got this!