QuanTAlib/lib/oscillators/smi/Smi.md

SMI: Stochastic Momentum Index

"The stochastic tells you where price is in the range. The SMI tells you how enthusiastically it got there." — William Blau (paraphrased)

Introduction

The Stochastic Momentum Index (SMI) measures where the close sits relative to the midpoint of the recent high-low range, then double-smooths the result with cascaded EMAs. Unlike the classic Stochastic Oscillator which measures distance from the low, SMI measures distance from the midpoint. This centering around zero produces cleaner crossover signals and reduces false readings during trending markets. Range: -100 to +100, with values beyond ±40 indicating extreme momentum.

Historical Context

William Blau introduced the SMI in his 1995 book "Momentum, Direction, and Divergence" as an improvement over George Lane's classic Stochastic Oscillator. Blau's key insight: measuring distance from the range midpoint rather than from the low eliminates the asymmetric bias that plagues traditional stochastics. When price closes at the exact middle of its range, classic Stochastic reads 50 — an arbitrary number that says nothing. SMI reads 0 — neutral, centered, semantically honest.

Tushar Chande and Stanley Kroll proposed a variant in "The New Technical Trader" (1994) that smooths numerator and denominator separately before computing the ratio. This subtle difference in order of operations produces different behavior during volatile periods: Blau's method smooths the ratio directly, which can compress extreme values; Chande/Kroll's method preserves the ratio's sensitivity by smoothing its components independently.

QuanTAlib implements both methods via the blau parameter (default: true for Blau's method).

Architecture and Physics

1. Rolling Highest High and Lowest Low

Using O(1) amortized MonotonicDeque for the kPeriod window:

HH_t = \max_{i=0}^{N-1} \text{High}_{t-i} LL_t = \min_{i=0}^{N-1} \text{Low}_{t-i}

2. Midpoint and Half-Range

\text{midpoint}_t = \frac{HH_t + LL_t}{2} \text{rangeHalf}_t = \frac{HH_t - LL_t}{2}

3. Blau Method (Default)

Compute the raw ratio first, then double-smooth:

\text{raw}_t = \begin{cases} 100 \times \frac{\text{Close}_t - \text{midpoint}_t}{\text{rangeHalf}_t} & \text{if } \text{rangeHalf}_t > 0 \\ 0 & \text{otherwise} \end{cases} K_t = \text{EMA}_2(\text{EMA}_1(\text{raw}_t, \text{kSmooth}), \text{kSmooth}) D_t = \text{EMA}(K_t, \text{dSmooth})

4. Chande/Kroll Method

Smooth numerator and denominator separately, then compute the ratio:

\text{num}_t = \text{Close}_t - \text{midpoint}_t \text{den}_t = \text{rangeHalf}_t K_t = 100 \times \frac{\text{EMA}_2(\text{EMA}_1(\text{num}))}{\text{EMA}_2(\text{EMA}_1(\text{den}))} D_t = \text{EMA}(K_t, \text{dSmooth})

5. EMA with Warmup Compensation

Each EMA stage uses exponential warmup compensation:

\alpha = \frac{2}{N + 1}, \quad d = 1 - \alpha \text{EMA}_t = d \cdot \text{EMA}_{t-1} + \alpha \cdot x_t e_t = d \cdot e_{t-1}, \quad c_t = \frac{1}{1 - e_t} \text{compensated}_t = \text{EMA}_t \cdot c_t

The compensator corrects the initialization bias during warmup, converging to 1.0 as e_t \to 0.

Mathematical Foundation

Blau's Z-Domain Transfer Function

The double-EMA smoothing of the raw ratio has transfer function:

H(z) = \left(\frac{\alpha}{1 - dz^{-1}}\right)^2

This is a cascade of two identical first-order IIR sections, providing -12 dB/octave rolloff in the stopband. The cascade attenuates noise more aggressively than a single EMA of equivalent period, at the cost of additional group delay.

Chande/Kroll Ratio Properties

The separate smoothing approach preserves a fundamental property: when numerator and denominator oscillate at the same frequency, their ratio remains unattenuated. Blau's method, by smoothing the ratio directly, can compress oscillations that the Chande/Kroll approach preserves.

Parameter Mapping

Parameter	Default	Range	Effect
kPeriod (N)	10	1-500	Lookback window for highest/lowest. Larger values produce a wider reference range, reducing sensitivity.
kSmooth	3	1-100	EMA period for the double-smoothing of K. Larger values smooth more aggressively, increasing lag.
dSmooth	3	1-100	EMA period for the signal line D. Controls signal line responsiveness.
blau	true	bool	true for Blau method (smooth ratio); false for Chande/Kroll (smooth components).

Warmup Period

\text{WarmupPeriod} = \text{kPeriod} + \text{kSmooth} + \text{dSmooth}

The indicator becomes IsHot after kPeriod bars (sufficient for the deque window). Full convergence of all three EMA stages requires the full warmup period.

Performance Profile

Operation	Complexity	Notes
MonotonicDeque push	O(1) amortized	Deque maintenance for highest/lowest
Midpoint/rangeHalf	O(1)	Two arithmetic operations
EMA stage 1	O(1)	FMA-optimized
EMA stage 2	O(1)	FMA-optimized
Signal EMA	O(1)	FMA-optimized
Total per bar	O(1)	Zero allocations in hot path

SIMD Analysis

SIMD is not applied in streaming Update due to the recursive EMA dependencies. The Batch span API delegates highest/lowest computation to their respective SIMD-enabled Batch methods, then processes the EMA cascade sequentially.

Quality Metrics

Metric	Score (1-10)	Notes
Noise rejection	7	Double EMA smoothing provides good noise attenuation
Lag	5	Three cascaded EMA stages accumulate group delay
Sensitivity	8	Midpoint centering produces sharper zero crossings than classic Stochastic
Range bound	9	Naturally bounded -100 to +100 by construction
Cross-instrument	8	Percentage-based output is comparable across instruments

Interpretation

Overbought/Oversold Levels

• Above +40: Overbought zone. Close is significantly above the range midpoint. Reversal probability increases.
• Below -40: Oversold zone. Close is significantly below the range midpoint. Bounce probability increases.
• Between -20 and +20: Neutral zone. No strong momentum bias.

K and D Crossovers

• K crosses above D: Bullish momentum shift. Momentum is accelerating upward.
• K crosses below D: Bearish momentum shift. Momentum is decelerating or reversing.

Divergence Analysis

• Bullish divergence: Price makes lower lows while SMI K makes higher lows. Range-normalized momentum is contracting despite new price lows.
• Bearish divergence: Price makes higher highs while SMI K makes lower highs. Despite new highs, momentum relative to range is weakening.

Blau vs Chande/Kroll Selection

• Blau (default): Better for trend-following. Smoother output, fewer whipsaws. The ratio compression during high volatility acts as a natural dampener.
• Chande/Kroll: Better for mean-reversion. Preserves component oscillation sensitivity. More responsive during volatile reversals but noisier in trends.

Validation

Library	Validated	Notes
Skender	✔️	SMI available via GetSmi()
TA-Lib	-	No SMI implementation
Tulip	-	No SMI implementation
Ooples	-	Not verified
Self-consistency	✔️	Batch/streaming/span agree within 10^{-6} tolerance

Cross-validation: Streaming, batch (TBarSeries), and span paths produce identical results. Both Blau and Chande/Kroll methods are verified independently.

Common Pitfalls

1. Confusing SMI with classic Stochastic: SMI measures distance from midpoint (range: -100 to +100). Classic Stochastic measures distance from the low (range: 0 to 100). Using Stochastic thresholds (20/80) for SMI produces incorrect signals.
2. Ignoring the method parameter: Blau and Chande/Kroll produce meaningfully different results. Switching methods mid-analysis invalidates comparisons.
3. Zero range handling: When highest equals lowest (constant price over kPeriod), rangeHalf is zero. Division by zero is guarded (returns 0.0), but a sustained zero reading may mask meaningful price action outside the deque window.
4. Cascaded EMA warmup: Three EMA stages each need convergence time. The first few values after IsHot are less reliable than values after the full WarmupPeriod. Trading signals should wait for full convergence.
5. Period selection interaction: kPeriod controls the reference range width; kSmooth controls noise filtering of K; dSmooth controls signal line lag. These three parameters interact. Increasing kPeriod without adjusting smoothing produces a wider range reference with insufficient filtering, yielding noisy K values.
6. Not a standalone signal: SMI measures momentum position within a range. Combine with trend filters for directional context. SMI works best in ranging markets; in strong trends, it can remain in overbought/oversold territory for extended periods.
7. Bar correction with MonotonicDeque: The isNew=false path rebuilds the deque from the circular buffer. Frequent corrections (high-frequency bar updates) are supported but carry O(N) rebuild cost per correction, where N is kPeriod.

References

• Blau, William. "Momentum, Direction, and Divergence." Wiley, 1995.
• Chande, Tushar S. and Kroll, Stanley. "The New Technical Trader." Wiley, 1994.
• Lane, George. "Stochastics." Technical Analysis of Stocks & Commodities, 1984. (Original Stochastic Oscillator)
• PineScript reference implementation: smi.pine