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FRM Part I · FRM Exam Part I

Measuring and Monitoring Volatility for FRM Part I

Measuring and monitoring volatility means estimating how much an asset's returns vary and updating that estimate as new data arrives. You solve questions by picking the right model (equal weights, EWMA or GARCH), applying its formula to the given returns and parameters, then scaling the result to the horizon asked.

What this chapter covers

This chapter is about turning a return series into a usable estimate of risk. You start with how returns are defined and how daily variance is estimated. Then you meet the main methods: implied volatility from option prices, historical volatility with equal weights, the exponentially weighted moving average (EWMA) model, and GARCH(1,1). The last topics cover forecasting and mean reversion, and how the same ideas apply to covariance and correlation.

The core idea is that volatility is not constant. It clusters: large moves tend to follow large moves. Equal weights ignore this and react slowly. EWMA and GARCH give more weight to recent data, so they respond faster. GARCH adds a long-run variance, so forecasts pull back toward a long-run level instead of staying where they are.

This chapter connects directly to the rest of the paper. Value at Risk and expected shortfall in the valuation and risk models topic need a volatility input. Option pricing in financial markets and products uses volatility as the key unknown. Quantitative analysis supplies the statistics: variance, covariance, maximum likelihood and time-series ideas. If you are comfortable here, many questions elsewhere become easier.

Questions here are formula-driven and predictable, which makes them good marks for a candidate who practises. A typical question gives you a few numbers and asks for an updated EWMA variance, a GARCH forecast or a long-run volatility. There is little ambiguity once you know the recursion and the conditions. The same skills feed into VaR and option questions, so time spent here pays back across the whole 100-question paper. GARP publishes no pass mark, so you want to bank every question that can be solved mechanically.

Measuring and Monitoring Volatility: topics in the order to study them

  1. 1Volatility Basics and Return MeasurementEverything else uses returns and variance, so fix simple vs log returns, the zero-mean assumption and square-root-of-time scaling first.
  2. 2Implied vs Historical VolatilityIt sets the big picture of where volatility numbers come from before you study the models in detail.
  3. 3Equally Weighted (Moving Average) VolatilityIt is the simplest estimator and shows the weakness (slow reaction, ghost effects) that motivates EWMA.
  4. 4EWMA Model and RiskMetricsIt is the first weighted model, with one parameter λ and a simple recursion, and a frequent calculation question.
  5. 5GARCH(1,1) ModelIt builds on EWMA by adding a long-run variance term, so learn it once you know the EWMA recursion.
  6. 6Estimating GARCH Parameters and Model ChecksMaximum likelihood and diagnostic checks make sense only after you know what the model is.
  7. 7Volatility Forecasting and Mean ReversionMulti-day forecasts use the GARCH persistence and long-run variance, so this comes after the model is clear.
  8. 8Correlation and Covariance MonitoringIt reuses the EWMA and GARCH recursions on cross-products of returns, so it is easiest at the end.

How to prepare Measuring and Monitoring Volatility

Treat this chapter as a set of formulas you can apply fast and a set of concepts that tell you which formula fits.

  1. Write the key formulas on one page: variance from returns, EWMA, GARCH(1,1), long-run variance and the mean-reversion forecast. Say each one aloud in words.
  2. Work through one numeric example per model by hand with a financial calculator. Practise the EWMA update: σₙ² = λσₙ₋₁² + (1 − λ)rₙ₋₁².
  3. For GARCH(1,1), σₙ² = ω + αrₙ₋₁² + βσₙ₋₁². Practise finding the long-run variance V = ω ÷ (1 − α − β), and checking that α + β < 1.
  4. Learn the concepts behind the numbers: volatility clustering, why EWMA is a special case of GARCH, what persistence means and how maximum likelihood picks parameters.
  5. Practise forecasting k days ahead: E[σₙ₊ₖ²] = V + (α + β)ᵏ(σₙ² − V). Then convert variance to annualised volatility with the right number of trading days.
  6. Do timed mixed sets. Under four hours for 100 questions, you have about two and a half minutes each, so aim to finish these calculations in about that time.
  7. Finish with a short review of correlation and covariance updating, using the same recursion on the product of two returns.

Common mistakes in Measuring and Monitoring Volatility

  • Mixing up variance and volatility in the recursion.

    Fix: Square the volatility first, apply the formula, then take the square root at the end and check the units.

  • Using the wrong return in the update.

    Fix: Label each input with its day before you plug it in. The newest squared return is rₙ₋₁², from the day before the one being estimated.

  • Forgetting the long-run term in GARCH, or treating EWMA as mean-reverting.

    Fix: Remember that EWMA is GARCH with ω = 0 and α + β = 1. It therefore has no long-run variance and no mean reversion.

  • Computing long-run volatility from ω alone.

    Fix: Use V = ω ÷ (1 − α − β) for variance, then take the square root for volatility.

  • Annualising with the wrong factor.

    Fix: Use the stated number of days, such as 252, and apply √ to the multiplier for volatility but not for variance.

  • Claiming implied volatility is always a better forecast than historical.

    Fix: Say that implied volatility is forward-looking but depends on an option pricing model and can include a risk premium, while historical volatility is backward-looking but directly observed.

Last-day revision: Measuring and Monitoring Volatility

  • Variance is estimated from squared returns; the common simplification assumes a mean of zero for short horizons.
  • Scale volatility to a longer horizon by multiplying by √T, under the assumption of independent returns.
  • Implied volatility is backed out of option prices and is forward-looking; historical volatility uses past returns.
  • Equal weights give every observation the same weight and react slowly to new shocks.
  • EWMA: σₙ² = λσₙ₋₁² + (1 − λ)rₙ₋₁². RiskMetrics used λ = 0.94 for daily data.
  • EWMA has no long-run variance, so forecasts stay flat at the current level.
  • GARCH(1,1): σₙ² = ω + αrₙ₋₁² + βσₙ₋₁², with weights summing: γ + α + β = 1 where ω = γV.
  • Long-run variance V = ω ÷ (1 − α − β); it requires α + β < 1 for stability.
  • Persistence is α + β; higher values mean slower mean reversion.
  • Forecast: E[σₙ₊ₖ²] = V + (α + β)ᵏ(σₙ² − V).
  • Parameters are estimated by maximum likelihood, which maximises the likelihood of the observed returns.
  • Covariance updates the same way as variance, using the product of the two returns in place of the squared return.

Measuring and Monitoring Volatility practice questions

Measuring and Monitoring Volatility in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Measuring and Monitoring Volatility: frequently asked questions

How many questions can I expect from this chapter?

GARP does not publish a fixed count per chapter, so plan on several questions from it. Check the current Study Guide and Learning Objectives, since GARP revises the curriculum every year.

Do I need to memorise the GARCH formulas?

Yes. You should know the GARCH(1,1) update, the long-run variance and the forecast formula without looking them up. They are short, and most calculation questions use them directly.

Is EWMA the same as GARCH?

EWMA is a special case of GARCH(1,1) in which ω = 0, α = 1 − λ and β = λ. Because of this it has no long-run variance and its forecasts do not revert to a mean.

Can I use a financial calculator for these questions?

Yes, for the arithmetic. Store the values and use the square root and power keys for forecasts such as (α + β)ᵏ. The main work is setting up the formula correctly.

Should I use simple or log returns?

Follow the question. Log returns are common in these models and add across time, while simple returns add across assets. For small daily moves the two are very close.