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Define volatility, variance rate and implied volatility and set them apart.

FRM® Part I · Quantitative Analysis · Measuring Returns, Volatility, and Correlation · FRM1.QA.12.b

The idea

Volatility is the standard deviation of an asset's returns, quoted per unit of time, usually per year. The variance rate is the variance per unit of time, so it equals the volatility multiplied by itself. Because the variances of independent returns add, the variance rate scales linearly with the horizon and volatility scales with the square root of time. The same scaling gives the volatility over a horizon of T days: Horizon volatility = daily volatility × √T. With 252 trading days in a year, annualise by scaling the daily figure: Annual volatility = daily volatility × √252. Implied volatility is different in kind. It is the volatility that, put into an option pricing model such as Black-Scholes-Merton, reproduces the option's observed market price. It looks forward and reflects market supply and demand, while historical volatility looks back at realised data. A common mistake is to multiply daily volatility by the number of days instead of by its square root.

Formulas

Volatility over T days from daily volatility

σT = σdaily × √T

σdaily
the standard deviation of daily returns
T
the number of trading days in the horizon, 252 for a year

Assumes independent returns, so variances add and volatility grows with the square root of time. Multiplying by T itself is the common error.

A practice question

From the question bank, with the reason each wrong option is wrong.

Implied volatility is best defined as:

  1. the standard deviation of the underlying asset's returns over a recent historical window.Why not: This is historical (realised) volatility, which looks back at data rather than at option prices.
  2. the volatility of the option's own price, measured from its daily changes.Why not: Confuses the volatility of the option price with the volatility parameter of the underlying in the model.
  3. the long-run average volatility to which a time-series model's forecasts revert.Why not: This describes mean-reverting volatility in a time-series model, not a value inferred from option prices.
  4. the volatility input that makes an option pricing model reproduce the observed market price of the option. - correct

Implied volatility is found by inverting a model such as Black-Scholes-Merton: choose sigma so that the model price equals the market price. It is forward-looking and market-based.

A harder one

An analyst estimates a daily return standard deviation of 1.2% and uses 252 trading days per year. The annualised variance rate is closest to:

  1. 0.0363 - correct
  2. 0.00014Why not: This is the daily variance, 0.012², which has not been scaled to a year.
  3. 0.3024Why not: Multiplies the daily volatility by 252 instead of multiplying the daily variance by 252.
  4. 0.1905Why not: This is the annualised volatility, 0.012 x sqrt(252), not the variance rate.

Daily variance = 0.012² = 0.000144. Variance rate scales with time: 0.000144 x 252 = 0.0363. Its square root, 19.05%, is the annual volatility.

The hardest tier

A European call and a European put on the same non-dividend-paying stock have the same strike and expiry. The call's Black-Scholes-Merton implied volatility is 21%. What is the implied volatility of the put, assuming no arbitrage?

  1. Higher than 21%, because puts carry a premium for crash protection.Why not: Any skew premium shows up across strikes, not between a call and a put at the same strike; parity fixes them together.
  2. 21%, because put-call parity makes the two implied volatilities identical. - correct
  3. Lower than 21%, because the put is cheaper than the call when the stock drifts upward.Why not: Confuses the lower price of the put with a lower volatility; both prices embed the same volatility.
  4. It cannot be determined, because implied volatility is specific to the option type.Why not: The model volatility is a property of the underlying, and parity ties put and call prices together.

Put-call parity links the call and put prices. Since the model price of each is increasing in a single sigma, the sigma that fits the call also fits the put. Implied volatility is a property of strike and expiry, not of option type.

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