Free lesson · Volatility

Realized variance starts with squared returns

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A volatility estimate summarizes a path’s movement; the sampling rule determines which movement is visible.

For intraday log returns, summing squares gives a realized-variance proxy for the observed window. It differs from demeaned sample variance. At coarse intervals drift matters; at very fine intervals bid/ask bounce and price discreteness distort the estimator.

An annualized daily estimate multiplies daily variance by a stated number of comparable trading days, then takes the square root. Equities and continuously traded digital assets need explicit calendar choices and overnight treatment.

Apply it in a strategy

• Freeze inputs at the stated decision time and record their units.
• Create a reproducible daily risk target and compare sampling choices.
• Recompute the example, then change the material assumption and explain the difference.

Research deliverable

Realized variance starts with squared returns: produce the worked calculation, a timestamped input record and a written decision addressing this limitation: Market microstructure noise, omitted overnight returns and nonrepresentative days can invalidate the interpretation.

These are synthetic mechanics examples, not historical performance or paper replications. Module evidence and research boundaries record the 12 September 2026 review.

Research sources, review dates and limitations

# Python 3.10+; standard library and NumPy only.
# Synthetic teaching inputs; conventions and units are defined in the notation above.
import numpy as np

def realized_volatility(log_returns,days_per_year=252):
    r=np.asarray(log_returns,float)
    if r.ndim!=1 or len(r)==0 or not np.isfinite(r).all() or days_per_year<=0: raise ValueError('Finite returns and positive day count required')
    variance=float(r@r)
    return variance,float(np.sqrt(days_per_year*variance))

assert abs(realized_volatility([.01,-.02])[0]-.0005)<1e-12
print(realized_volatility([.01,-.02]))