QuanTAlib/lib/channels/jbands/jbands.md

JBANDS: Jurik Adaptive Envelope Bands

JBANDS expose the internal adaptive envelope mechanism of the Jurik Moving Average (JMA), producing asymmetric bands that snap instantly to new price extremes and decay exponentially during consolidation. Unlike standard volatility bands (Bollinger, Keltner) which maintain symmetric width around a center line, JBANDS feature "snap-and-decay" hysteresis: expansion is instantaneous (plasticity), contraction is gradual (elasticity). The decay rate is dynamically modulated by a two-stage volatility estimator — a 10-bar SMA feeding a 128-bar trimmed mean — making the bands tight during quiet markets and expansive during trends. The center line is the full JMA: a 2-pole IIR filter with phase control and adaptive alpha.

Historical Context

Mark Jurik of Jurik Research developed the Jurik Moving Average and its associated bands in the 1990s as a proprietary commercial tool optimized for real-world trading. Unlike academic indicators, JMA was designed with emphasis on reducing lag while maintaining smoothness, using adaptive volatility modulation to adjust bandwidth dynamically.

The "snap-and-decay" behavior draws from hysteresis in physics — systems that respond differently to increasing versus decreasing inputs. When price moves to a new extreme, the band deforms immediately (plastic response). When price retreats, the band recovers gradually (elastic response). This asymmetry matches empirical market behavior: breakouts are sudden, consolidations are gradual. The two-stage volatility engine (local deviation → SMA → trimmed mean) provides robust reference volatility that resists contamination by outliers.

Architecture & Physics

1. Local Deviation

The maximum absolute distance from price to either band:

d_{\text{local}} = \max(|x_t - \text{Upper}_{t-1}|,\; |x_t - \text{Lower}_{t-1}|) + \epsilon

2. Two-Stage Volatility Estimation

Stage 1: 10-bar SMA of local deviation:

\text{highD}_t = \text{SMA}(d_{\text{local}},\; 10)

Stage 2: 128-bar trimmed mean (discard top/bottom 25% when full, or 25% of available count during warmup):

d_{\text{ref}} = \text{TrimmedMean}(\text{highD values},\; 128)

3. Dynamic Exponent

\text{ratio} = \frac{|x_t - \text{band}|}{d_{\text{ref}}} d = \min\left(\max\left(\text{ratio}^{P_{\text{exp}}},\; 1\right),\; \log_2\sqrt{\frac{P-1}{2}} + 2\right)

Where P_{\text{exp}} = \max(\log_2\sqrt{(P-1)/2},\; 0.5).

4. Adaptive Decay

The adaptation factor uses the precomputed \text{sqrtDiv}:

\alpha_{\text{band}} = \text{sqrtDiv}^{\sqrt{d}}

5. Snap-and-Decay Band Update

\text{Upper}_t = \begin{cases} x_t & \text{if } x_t > \text{Upper}_{t-1} \\ x_t - (x_t - \text{Upper}_{t-1}) \cdot \alpha_{\text{band}} & \text{otherwise} \end{cases} \text{Lower}_t = \begin{cases} x_t & \text{if } x_t < \text{Lower}_{t-1} \\ x_t - (x_t - \text{Lower}_{t-1}) \cdot \alpha_{\text{band}} & \text{otherwise} \end{cases}

6. JMA Center Line (2-Pole IIR)

\alpha_{\text{jma}} = \text{lenDiv}^d c_0 = (1 - \alpha_{\text{jma}}) \cdot x_t + \alpha_{\text{jma}} \cdot c_{0,t-1} c_8 = (x_t - c_0)(1 - \text{lenDiv}) + \text{lenDiv} \cdot c_{8,t-1} a_8 = (\text{phase} \cdot c_8 + c_0 - \text{JMA}_{t-1}) \cdot (1 + \alpha_{\text{jma}}^2 - 2\alpha_{\text{jma}}) + \alpha_{\text{jma}}^2 \cdot a_{8,t-1} \text{JMA}_t = \text{JMA}_{t-1} + a_8

7. Complexity

Dominated by the trimmed mean's partial sort: O(n \log n) for the 128-element buffer. All other operations are O(1). In practice, the 128-element sort is fast due to cache-friendly size.

Mathematical Foundation

Parameters

Parameter	Description	Default	Constraint
period	Nominal lookback length	10	> 0
phase	Controls JMA overshoot/smoothness	0	[-100, 100]
source	Input price series	close

Precomputed Constants (from period and phase)

Constant	Formula
\text{\_PHASE}	\text{phase}/100 + 1.5, clamped to [0.5, 2.5]
\text{\_LEN0}	(P - 1) / 2
\text{\_LOG\_PARAM}	\max(\log_2\sqrt{\text{\_LEN0}} + 2,\; 0)
\text{\_SQRT\_PARAM}	\sqrt{\text{\_LEN0}} \cdot \text{\_LOG\_PARAM}
\text{lenDiv}	\text{\_LEN0} \cdot 0.9 / (\text{\_LEN0} \cdot 0.9 + 2)
\text{sqrtDiv}	\text{\_SQRT\_PARAM} / (\text{\_SQRT\_PARAM} + 1)
P_{\text{exp}}	\max(\text{\_LOG\_PARAM} - 2,\; 0.5)

Pseudo-code

function JBANDS(source, period, phase):
    precompute constants from period and phase

    // 1. Local deviation
    dLocal = max(|source - upper|, |source - lower|) + ε

    // 2. Volatility: 10-bar SMA → 128-bar trimmed mean
    highD = SMA(dLocal, 10)
    dRef = TrimmedMean(highD_history, 128)

    // 3. Dynamic exponent
    ratio = |source - band| / dRef
    d = clamp(ratio^P_exp, 1, LOG_PARAM)

    // 4. Snap-and-decay bands
    adapt = sqrtDiv^√d
    if source > upper: upper = source
    else: upper = source - (source - upper) * adapt
    (symmetric for lower)

    // 5. JMA center line (2-pole IIR)
    alpha = lenDiv^d
    ... (c0, c8, a8 recursion) ...
    jma = prev_jma + a8

    return [jma, upper, lower]

Output Interpretation

Output	Description
middle	JMA center line (adaptive low-lag smoothed price)
upper	Adaptive upper envelope (snaps up, decays down)
lower	Adaptive lower envelope (snaps down, decays up)

Resources

• Jurik, M. Jurik Research. (Proprietary JMA specification and band logic)
• Mandelbrot, B. "The Variation of Certain Speculative Prices." Journal of Business, 36(4), 1963. (Fat-tailed distributions motivating adaptive approaches)