QuanTAlib/lib/momentum/rsi/Rsi.md

RSI: Relative Strength Index

Momentum is the premier anomaly.

Property	Value
Category	Momentum
Inputs	Source (close)
Parameters	period (default 14)
Outputs	Single series (Rsi)
Output range	Varies (see docs)
Warmup	period + 1 bars
PineScript	rsi.pine

• The Relative Strength Index measures the speed and magnitude of price changes.
• Similar: CMO, Stoch | Complementary: MACD for trend confirmation | Trading note: Wilder's RSI; 70/30 overbought/oversold. Divergences signal reversals. 14-period standard.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

The Relative Strength Index measures the speed and magnitude of price changes. Introduced by J. Welles Wilder Jr. in 1978, it oscillates between 0 and 100, identifying overbought and oversold conditions. The "Relative Strength" name is misleading: RSI measures internal strength (price versus itself) not relative strength (asset versus benchmark). Wilder knew this. He kept the name anyway. Marketing, perhaps.

Historical Context

Wilder introduced RSI in New Concepts in Technical Trading Systems, the same book that gave us ATR, ADX, and the Parabolic SAR. He was a mechanical engineer turned real estate developer turned trader. This background shows: RSI has the elegance of an engineered solution rather than the messiness of a discovered pattern.

The key insight: by normalizing gains against losses, RSI produces a bounded oscillator immune to the scale problems plaguing other momentum measures. A stock moving from $10 to $11 and a stock moving from $100 to $110 produce identical RSI readings. This scale independence made RSI the default momentum oscillator across every asset class from corn futures to cryptocurrency.

What most implementations get wrong: the smoothing method. Wilder specified RMA (Wilder's Moving Average, also called SMMA), not SMA or EMA. RMA has infinite memory. This gives RSI its characteristic "stickiness" near extremes.

Architecture & Physics

RSI is built on three concepts: change classification, exponential smoothing, and normalization.

1. Change Classification

Each bar's price change is classified as either a gain or a loss:

\text{Gain}_t = \max(P_t - P_{t-1}, 0)

\text{Loss}_t = \max(P_{t-1} - P_t, 0)

A flat bar (no change) produces zero for both. This is correct: no movement means no momentum in either direction.

2. RMA Smoothing (Wilder's Method)

Gains and losses are smoothed separately using RMA:

\text{AvgGain}_t = \frac{\text{AvgGain}_{t-1} \times (N-1) + \text{Gain}_t}{N}

\text{AvgLoss}_t = \frac{\text{AvgLoss}_{t-1} \times (N-1) + \text{Loss}_t}{N}

RMA is an EMA variant with \alpha = 1/N instead of \alpha = 2/(N+1). The decay is slower, the memory longer. A 14-period RSI "remembers" price action from hundreds of bars ago, with exponentially diminishing influence.

3. Normalization to [0, 100]

The Relative Strength ratio and final normalization:

RS = \frac{\text{AvgGain}}{\text{AvgLoss}}

RSI = 100 - \frac{100}{1 + RS}

4. Edge Case Handling

When AvgLoss approaches zero (sustained uptrend):

• If AvgGain also near zero → RSI = 50 (no momentum either direction)
• If AvgGain positive → RSI = 100 (maximum bullish)

When AvgGain approaches zero (sustained downtrend):

• RSI approaches 0 (maximum bearish)

QuanTAlib uses \epsilon = 10^{-10} as the zero threshold.

Mathematical Foundation

Transfer Function

RSI is a nonlinear function of two parallel RMA filters. The transfer function for each RMA:

H_{RMA}(z) = \frac{\alpha}{1 - (1-\alpha)z^{-1}}

where \alpha = 1/N.

The ratio and normalization introduce nonlinearity, preventing closed-form frequency analysis. RSI responds to both frequency and amplitude of price changes.

Half-Life Analysis

For RMA with \alpha = 1/N:

t_{1/2} = \frac{\ln(2)}{\ln(1/(1-\alpha))} \approx (N-1) \times \ln(2) \approx 0.693N

A 14-period RSI has half-life of approximately 10 bars. Price action from 70 bars ago still contributes ~1% to the current reading.

Warmup Period

RSI requires N + 1 bars minimum: one bar to establish the first change, then N bars for the RMA to initialize. Full convergence (within 1% of steady-state) requires approximately 4.6N bars.

Performance Profile

Operation Count (Streaming Mode)

Operation	Count	Cost (cycles)	Subtotal
SUB (change calculation)	1	1	1
CMP (gain/loss branch)	2	1	2
MUL (RMA decay × N-1)	2	3	6
ADD (RMA numerator)	2	1	2
DIV (RMA by N)	2	15	30
DIV (RS = gain/loss)	1	15	15
ADD (1 + RS)	1	1	1
DIV (100 / (1+RS))	1	15	15
SUB (100 - result)	1	1	1
Total	13	—	~73 cycles

The three divisions dominate (~82% of cycles). FMA optimization provides minimal benefit here.

SIMD Analysis (Batch Mode)

The Calculate method vectorizes gain/loss classification:

Operation	Scalar	SIMD (AVX2)	Speedup
Change calculation	N SUB	N/4 VSUBPD	4×
Gain classification	N CMP + conditional	N/4 VMAXPD	4×
RSI final calculation	N	N/4 VDIVPD	4×

The RMA smoothing remains sequential (recursive dependency).

Benchmark Results

Test environment: Apple M4, .NET 10.0, AdvSIMD, 500,000 bars.

Metric	Value	Notes
Span throughput	18 ns/bar	Including SIMD gain/loss
Streaming throughput	~25 ns/bar	Single Update() call
Allocations (hot path)	0 bytes	ArrayPool for batch temps
Complexity	O(1)	Per bar
State size	~80 bytes	Two RMA instances + prev value

Comparative Performance

Library	Time (500K bars)	Allocated	Relative
QuanTAlib (Span)	~9 ms	0 B	baseline
TA-Lib	~8 ms	40 B	0.89×
Tulip	~8 ms	0 B	0.89×
Skender	~95 ms	28 MB	10.6× slower

Quality Metrics

Metric	Score	Notes
Accuracy	10/10	Matches Wilder's definition exactly
Timeliness	8/10	Responsive, but RMA adds some lag
Overshoot	10/10	Bounded [0, 100], cannot overshoot by design
Smoothness	8/10	RMA provides good filtering

Validation

Validated against external libraries in Rsi.Validation.Tests.cs. Tests run against 5,000 bars with tolerance of 1e-9.

Library	Batch	Streaming	Span	Notes
TA-Lib	✅	✅	✅	Matches TA_RSI after warmup
Skender	✅	✅	✅	Matches GetRsi
Tulip	✅	✅	✅	Matches rsi
Ooples	✅	—	—	Matches CalculateRelativeStrengthIndex

Common Pitfalls

1. Warmup Initialization: The first RSI value requires N+1 bars of data. During the first bar, there is no previous price to calculate change. QuanTAlib returns the RSI value from bar 1 onward, but values stabilize fully after approximately 4.6N bars.

2. RMA vs SMA Confusion: Many online calculators use SMA for the first average, then switch to RMA. This produces different values for the first ~3N bars. QuanTAlib uses RMA throughout for consistency.

3. Overbought/Oversold Interpretation: RSI above 70 means "overbought" but does not mean "sell." In strong uptrends, RSI can stay above 70 for extended periods. The market can remain irrational longer than the RSI can remain extreme.

4. Period Selection: The 14-period default works for daily charts. For intraday trading, consider 7 or 9 periods. For weekly charts, consider 21 or 28. Match the period to your signal horizon.

5. Divergence False Signals: RSI divergence (price makes new high, RSI does not) is a popular strategy but produces many false signals in trending markets. Use confirmation.

6. Bar Correction Handling: When isNew=false, RSI correctly restores the previous price state before recalculating. Both internal RMA instances are also rolled back. Incorrect isNew usage causes RSI to calculate changes from wrong reference points.

7. Scale Independence Assumption: RSI is scale-independent for price levels but not for volatility regimes. A stock with 5% daily swings produces different RSI dynamics than one with 0.5% swings, even at the same period setting.

Usage Examples

// Streaming: one bar at a time
var rsi = new Rsi(14);
foreach (var bar in liveStream)
{
    var result = rsi.Update(new TValue(bar.Time, bar.Close));
    Console.WriteLine($"RSI: {result.Value:F2}");
}

// Batch processing with Span (zero allocation)
double[] prices = LoadHistoricalData();
double[] rsiValues = new double[prices.Length];
Rsi.Calculate(prices.AsSpan(), rsiValues.AsSpan(), period: 14);

// Batch processing with TSeries
var series = new TSeries();
// ... populate series ...
var results = Rsi.Batch(series, period: 14);

// Event-driven chaining
var source = new TSeries();
var rsi14 = new Rsi(source, 14);
source.Add(new TValue(DateTime.UtcNow, 100.0));  // RSI updates automatically

// Prime with historical data
var rsi = new Rsi(14);
rsi.Prime(historicalPrices);  // Ready for live data

References

• Wilder, J. W. (1978). New Concepts in Technical Trading Systems. Trend Research. Chapter: Relative Strength Index.
• Constance Brown. (1999). Technical Analysis for the Trading Professional. McGraw-Hill. (RSI divergence patterns)
• Cutler, David. (1991). "RSI Revisited." Technical Analysis of Stocks & Commodities. (Smoothed RSI variants)