Sliding Window Real-Time Volatility via Numerically Stable Updates

In high-frequency quantitative options market-making at Optiver and Jump Trading, estimating the rolling volatility of asset returns is critical for delta hedging and quoting spreads.

Given a stream of real-valued log-returns $x_1, x_2, \dots$, we maintain the unbiased sample standard deviation $s$ over a sliding window of size $K$: $s = \sqrt{\frac{1}{K - 1} \sum_{i=1}^K (x_i - \bar{x})^2}$ The textbook algebraic shortcut $\sum x_i^2 - \frac{(\sum x_i)^2}{K}$ suffers from catastrophic cancellation in floating-point arithmetic when the variance is small relative to the mean squared, frequently producing negative radicands or precision loss.

Your Task

Implement the RollingVolatility class using an $\mathcal{O}(1)$ numerically stable updating formula:

• __init__(window_size: int): Initializes the calculator with capacity $K$.
• update(val: float) -> float | None: Adds return val to the stream. Once exactly $K$ items are in the buffer, returns the sample standard deviation $s$ (rounded to 6 decimal places). If fewer than $K$ items have been received, returns None.

Constraints

• $2 \le K \le 10^5$
• Stream values $-1.0 \le val \le 1.0$
• Per-tick update must execute in strictly $\mathcal{O}(1)$ time with zero catastrophic loss of significance.

Examples

rv = RollingVolatility(window_size=3)
rv.update(0.01) # returns None
rv.update(0.02) # returns None
rv.update(0.03) # returns 0.01
rv.update(0.04) # window becomes [0.02, 0.03, 0.04], returns 0.01