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Compute the historical volatility and the realized return of a stock.

FRM® Part I · Valuation and Risk Models · The Black-Scholes-Merton Model · FRM1.VR.15.b

The idea

Historical volatility is estimated from continuously compounded returns between observations. With n returns, the volatility per interval is their sample standard deviation s. Each return is u_i = ln(S_i ÷ S_(i−1)). The standard deviation s is the square root of the sample variance. Sample variance = Σ(u_i − mean)² ÷ (n − 1). It is annualised by dividing by the square root of the interval measured in years. For daily data with 252 trading days, that means multiplying s by the square root of 252. The same data give the realised return. The mean log return times the number of periods per year estimates μ − σ²÷2. The expected return μ is therefore estimated by adding half the annualised variance. The standard error of a volatility estimated from n returns is roughly σ divided by the square root of 2n. Do not divide by n, and do not multiply by 252 when annualising.

Formulas

Annualised volatility from a sample of returns

σ = s ÷ √τ

s
the sample standard deviation of the log returns, divisor n − 1
τ
the interval between observations in years, 1 ÷ 252 for trading days

For daily data this is s × √252. Multiplying by 252, or using calendar days, overstates it.

Standard error of a volatility estimate

SE = σ ÷ √(2n)

σ
the estimated volatility
n
the number of returns used

An approximation. Using σ ÷ √n, the standard error of a mean, overstates it.

A practice question

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

In estimating historical volatility from a series of prices, the return between successive observations S(i-1) and S(i) is measured as:

  1. [S(i) - S(i-1)] / S(i)Why not: Divides the price change by the ending price instead of using the log of the price ratio.
  2. ln(S(i)) / ln(S(i-1))Why not: Takes the ratio of two logs rather than the log of a ratio.
  3. S(i) / S(i-1)Why not: Uses the price relative, which is close to 1, not a return; the log of it is the continuously compounded return.
  4. ln(S(i) / S(i-1)) - correct

BSM models ln(S_T/S_0) as normal, so volatility is estimated from the log price ratios u_i = ln(S_i/S_(i-1)).

A harder one

Six consecutive daily closing prices of a stock are 50.00, 50.80, 50.20, 51.30, 50.90 and 51.70. Using continuously compounded returns and the sample standard deviation, the annualised historical volatility with 252 trading days is closest to:

  1. 21.8%Why not: Divides the sum of squared deviations by n = 5 instead of n - 1 = 4.
  2. 24.4% - correct
  3. 1.5%Why not: Reports the daily standard deviation without annualising it.
  4. 29.3%Why not: Annualises with 365 calendar days instead of 252 trading days.

The five log returns have a sample standard deviation of 1.536% (divisor n - 1 = 4). Annualised: 1.536% x sqrt(252) = 24.4%.

The hardest tier

Daily continuously compounded returns on a stock have a mean of 0.04% and a standard deviation of 1.5%. Using 252 trading days, the estimate of the stock's expected annual return mu in the BSM model, where the annualised mean log return equals mu - sigma²/2, is closest to:

  1. 12.9% - correct
  2. 10.1%Why not: Takes the annualised mean log return as mu; it estimates mu - sigma²/2 and must be increased by the half variance.
  3. 7.2%Why not: Subtracts the half variance instead of adding it.
  4. 22.0%Why not: Adds half the annualised volatility, 11.9%, instead of half the annualised variance.

Annual mean log return = 0.0004 x 252 = 10.08%. Annual volatility = 1.5% x sqrt(252) = 23.81%, so sigma²/2 = 2.83%. mu = 10.08% + 2.83% = 12.9%.

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