Historical Volatility Calculator

Methodology, equations and examples

Volatility is the standard deviation of an asset's returns, usually quoted per year. Option prices give the market's forward-looking estimate, its implied volatility. A price history gives a backward-looking one, and this calculator measures it three ways. The close-to-close figure weights every return in the series equally. EWMA weights recent returns more. GARCH(1,1) also pulls the estimate back toward a long-run level, so it can forecast how volatility will change.

Returns

Each return is a log return, r_t = ln(P_t / P_t−1), where P_t is the close at period t. Log returns add up over time, so a year's return is the sum of its daily returns. The calculator works with the returns less their mean, ε_t = r_t − r̄. For daily data the mean is small and this changes little; for monthly data it matters more.

Close-to-close historical volatility

The historical volatility is the sample standard deviation of the returns:

s = √( Σ (r_t − r̄)² / (n − 1) )

It is annualized by multiplying by the square root of the periods per year, √252 for trading days, √52 for weeks and √12 for months. This assumes returns are independent from one period to the next, so their variances add.

Example: closes of 100, 102, 101 and 103.5 give returns of 1.980 %, −0.985 % and 2.445 %. Their mean is 1.147 % and their sample standard deviation 1.861 %, or 1.861 % × √252 = 29.54 % a year.

EWMA

The exponentially weighted moving average (EWMA) makes each period's variance a blend of the last variance and the last squared return:

h_t+1 = λ h_t + (1 − λ) ε_t²

h_t is the variance of period t's return, estimated from the returns before it. The weight on a return falls by a factor of λ each period, so with λ = 0.94 a return 11 periods back counts about half as much as the latest one. RiskMetrics uses λ = 0.94 for daily data. The calculator starts the recursion from the sample variance and reports h for the period after the last price.

Example (Hull): with λ = 0.90, yesterday's volatility estimate 1 % and yesterday's return 2 %, today's variance is 0.90 × 0.0001 + 0.10 × 0.0004 = 0.00013, a volatility of 1.14 % a day.

GARCH(1,1)

GARCH(1,1), the generalized autoregressive conditional heteroskedasticity model, adds a constant to the EWMA recursion:

h_t+1 = ω + α ε_t² + β h_t

with ω > 0, α ≥ 0, β ≥ 0 and α + β < 1. The constant draws the variance toward the long-run variance V_L = ω / (1 − α − β). EWMA is the special case ω = 0, α = 1 − λ, β = λ, which has no long-run level.

• α is the weight on the latest shock: a high α makes volatility jump after a large move.
• β is the weight on the last variance: a high β makes volatility slow to change.
• α + β is the persistence. A shock to the variance decays by that factor each period, so its half-life is ln 0.5 / ln(α + β) periods.

Example (Hull): with ω = 0.000002, α = 0.13 and β = 0.86, the long-run variance is 0.000002 / 0.01 = 0.0002, a volatility of 1.41 % a day. If yesterday's volatility estimate was 1.6 % and yesterday's return 1 %, today's variance is 0.000002 + 0.13 × 0.0001 + 0.86 × 0.000256 = 0.00023516, a volatility of 1.53 % a day.

Fitting by maximum likelihood

The calculator chooses ω, α and β to make the observed returns as likely as possible, assuming each ε_t is normal with variance h_t. That maximizes the log-likelihood

ln L = −½ Σ [ ln 2π + ln h_t + ε_t² / h_t ]

It starts at α = 0.08 and β = 0.90 with ω set so that the long-run variance equals the sample variance, and searches with the Nelder-Mead simplex method. The search runs over ω = e^a, α + β = 1 / (1 + e^(−b)) and α / (α + β) = 1 / (1 + e^(−c)), so every trial satisfies the constraints. It then restarts once from its answer, to make sure it has not stopped early. If the search does not converge, the results say so.

A fit needs enough data. With a few hundred returns, α and β on their own are uncertain, though their sum is usually well determined. Several years of daily prices give steadier estimates. The calculator needs at least 100 returns before it fits GARCH.

Forecasting volatility

The expected variance k periods ahead decays geometrically toward the long-run level:

E[h_n+k] = V_L + (α + β)^(k−1) (h_n+1 − V_L)

Averaging this over the next T periods gives the variance to use for, say, pricing an option that expires in T periods:

average variance = V_L + (1 − (α + β)^T) / (T (1 − α − β)) × (h_n+1 − V_L)

The forecast table annualizes each average. Near-term horizons stay close to today's volatility; long horizons approach the long-run figure. This is the term structure of volatility the model implies.

The example

"Show an example" fills in 1,000 daily closes simulated on this page from a GARCH(1,1) with ω = 0.000002, α = 0.08, β = 0.90 and a mean return of 0.03 % a day. The long-run volatility of that model is √(0.000002 / 0.02 × 252) = 15.87 % a year, and a shock has a half-life of ln 0.5 / ln 0.98 = 34.3 days. The fit recovers a persistence of 0.979 against the true 0.98, with α = 0.066 and β = 0.913. Its long-run volatility comes out at 13.41 %, lower than the truth, because 999 returns pin down the slow decay of volatility less well than its persistence. A different seed would give a different, equally valid sample.

Limits

All three methods treat each period's return as normal given its variance. Real returns have fatter tails, so the GARCH fit tends to understate how often large moves happen. Volatility from past prices describes the past. It is a starting point for judging an implied volatility or a risk figure, not a prediction of what the market will do. For study and comparison; not financial advice.