RiskMetrics EWMA Volatility in Numba

Approach

σ²_0 = r_0²
σ²_t = λ σ²_{t−1} + (1 − λ) r_{t−1}²     t = 1..n−1
σ_t  = sqrt(σ²_t)

σ_t depends on r_0 .. r_{t−1} only.
0 ≤ λ ≤ 1.  λ = 1 repeats σ²_0.  λ = 0 gives σ²_t = r_{t−1}².

When to use it

• A causal volatility estimate on a long return series, for scaling positions or for comparing with realized volatility.
• Inside another @njit loop. A Python EWMA over millions of bars spends its time in the interpreter; this one does not, after the first compile.
• For a variance that also reacts to a long-run level and is estimated rather than set, see GARCH(1,1).

Pitfalls

• Initial value. σ²_0 = r_0² is one observation. A large first return leaves a long tail when λ is close to 1. Some desks seed the recursion with a sample variance of an earlier window instead.
• Look-ahead. Using r_t² inside σ_t would mix the return you are about to score with the volatility you score it by. This recursion does not do that. Do not "fix" the index.
• λ is not estimated here. 0.94 is the RiskMetrics daily convention, not a maximum-likelihood estimate for your sample. The half-life of a shock is about ln(0.5)/ln(λ) periods (about 11 days at 0.94).
• NaNs and the first call. A NaN poisons every later value. The first call compiles; cache=True reuses the compiled code afterwards.

Python reference implementation

Requires numpy, numba. Tested before publication; the generator re-executes it on every rebuild. MIT licensed. Download numba-ewma-volatility.py · All snippets

import numpy as np
from numba import njit

@njit(cache=True)
def ewma_vol(returns, lam=0.94):
    """EWMA volatility. var[t] uses returns[:t] only (r[t] is excluded).

    lam is the weight on the previous variance (RiskMetrics daily default 0.94).
    """
    if lam < 0.0 or lam > 1.0:
        raise ValueError("lam must be in [0, 1]")
    r = returns
    n = r.shape[0]
    var = np.empty(n, dtype=np.float64)
    if n == 0:
        return var
    var[0] = r[0] * r[0]
    one = 1.0 - lam
    for t in range(1, n):
        prev = r[t - 1]
        var[t] = lam * var[t - 1] + one * prev * prev
    return np.sqrt(var)

FAQ

Why does the variance at t use the return at t−1?

So that the volatility sitting on row t was knowable before r_t was. That is the RiskMetrics recursion. Putting r_t into var[t] would make any later Sharpe-style scaling use information from the return it is scaling.

Is this GARCH?

No. GARCH(1,1) has a constant term and the two slopes are estimated. EWMA is the special case ω = 0 and α + β = 1, with λ chosen rather than fitted. It has no finite unconditional variance.