QuanTAlib/lib/reversals/pivotext/Pivotext.md

PIVOTEXT: Extended Traditional Pivot Points

Classic pivots tell you where the crowd expects the market to pause. Extended pivots tell you where the crowd starts to panic.

Property	Value
Category	Reversal
Inputs	OHLCV bar (TBar)
Parameters	None
Outputs	Single series (PIVOTEXT)
Output range	Varies (see docs)
Warmup	2 bars
PineScript	pivotext.pine

• Extended Traditional Pivot Points calculate eleven horizontal support and resistance levels from the previous bar's high, low, and close.
• No configurable parameters; computation is stateless per bar.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

Extended Traditional Pivot Points calculate eleven horizontal support and resistance levels from the previous bar's high, low, and close. The core levels (PP, R1-R3, S1-S3) are identical to classic floor trader pivots. The extension adds R4/R5 and S4/S5 levels that project further beyond the prior bar's range, covering extreme move scenarios such as gap opens, news-driven spikes, and trend continuation through multiple prior-range increments. The formula is pure arithmetic with zero parameters.

Historical Context

Floor trader pivots date to the 1930s when pit traders computed PP = (H + L + C) / 3 and derived three symmetric support/resistance levels from it. The formula was simple enough to compute by hand before the market open, making it one of the earliest systematic approaches to intraday level identification.

Classic pivots (R1-R3, S1-S3) cover the range from roughly 1x to 2x the prior bar's range projected from the high or low. In practice, large gap opens or momentum-driven moves routinely exceed R3/S3. Traders discovered they needed additional levels to bracket these extreme scenarios without switching to entirely different frameworks (Fibonacci extensions, measured moves, etc.).

The extended formula simply continues the same arithmetic progression. R3 uses a 2x multiplier on (PP - L) added to H; R4 uses 3x; R5 uses 4x. The symmetry holds for support levels. This mechanical extension preserves the simplicity of the original system while providing reference levels for moves that exceed the "normal" 1-3 range pivots.

Unlike Fibonacci pivots, Camarilla pivots, or DeMark pivots, the extended traditional formula makes no claim about specific retracement ratios or market microstructure. The levels are pure geometric projections of the prior range. Their value lies in consensus: enough traders watch these levels that they become self-reinforcing reference points.

Architecture and Physics

1. Previous Bar's HLC

The indicator stores the high (H), low (L), and close (C) of the most recently completed bar. On each new bar, these stored values become the basis for computing the current bar's pivot levels, and the new bar's HLC replaces them for the next computation.

2. Central Pivot Point (PP)

PP = \frac{H_{prev} + L_{prev} + C_{prev}}{3}

The arithmetic mean of the previous bar's HLC. This is the gravitational center of the level system. All other levels derive from PP and the prior range.

3. Range and Intermediate Values

range = H_{prev} - L_{prev} ppMinusL = PP - L_{prev} hMinusPP = H_{prev} - PP

The range scales the distance between levels. The asymmetric terms ppMinusL and hMinusPP determine how far resistance extends above H and support extends below L.

4. Classic Levels (R1-R3, S1-S3)

R_1 = 2 \cdot PP - L_{prev} \qquad S_1 = 2 \cdot PP - H_{prev} R_2 = PP + range \qquad S_2 = PP - range R_3 = H_{prev} + 2 \cdot (PP - L_{prev}) \qquad S_3 = L_{prev} - 2 \cdot (H_{prev} - PP)

5. Extended Levels (R4-R5, S4-S5)

R_4 = H_{prev} + 3 \cdot (PP - L_{prev}) \qquad S_4 = L_{prev} - 3 \cdot (H_{prev} - PP) R_5 = H_{prev} + 4 \cdot (PP - L_{prev}) \qquad S_5 = L_{prev} - 4 \cdot (H_{prev} - PP)

The progression is arithmetic: each successive level adds one more ppMinusL (resistance) or hMinusPP (support) increment.

6. Level Ordering Invariant

For any bar where H_{prev} > L_{prev} (non-degenerate range):

S_5 < S_4 < S_3 < S_2 < S_1 \leq PP \leq R_1 < R_2 < R_3 < R_4 < R_5

When the close is exactly at the midpoint of the range, PP = S_1 = R_1 (all collapse to the midpoint), and the S/R levels fan out symmetrically. In general, the close's position within the range determines the asymmetry between resistance and support spacing.

7. Eleven Outputs

All eleven levels are computed simultaneously and remain constant until a new bar arrives. The primary output (Last.Val) returns PP; individual properties expose all eleven levels.

Signal Interpretation

Condition	Interpretation
Price between S1 and R1	Normal range; no directional bias
Price tests R2 from below	First extension test; watch for rejection
Price tests S2 from above	First support extension; potential bounce
Price reaches R3/S3	Classic extreme; high-probability reversal zone
Price breaks R4/S4	Significant momentum; extended trend likely
Price reaches R5/S5	Rare extreme; potential exhaustion or blow-off
Levels cluster tightly	Prior bar had low range; expect volatility expansion
Wide level spacing	Prior bar was volatile; levels may be less precise

Mathematical Foundation

Parameters

Extended Traditional Pivot Points has no configurable parameters. The formula is fixed by definition.

Parameter	Value	Notes
Inputs	H, L, C	Previous bar's high, low, close
Outputs	11	PP, R1, R2, R3, R4, R5, S1, S2, S3, S4, S5
Parameters	0	No tuning required

Warmup Period

W = 2

The indicator requires 2 bars: the first bar provides HLC for storage; the second bar triggers computation from the stored values. Prior to warmup completion, all outputs are NaN.

Derivation Notes

The classic pivot formula is not derived from statistical theory. It is an empirical heuristic that became standardized through widespread adoption. The extension to R4/R5 and S4/S5 follows the same arithmetic progression pattern already established by R3/S3. No new constants or empirical fitting are introduced.

R3 and S3 use a coefficient of 2 on the asymmetric terms. R4/S4 use 3. R5/S5 use 4. The progression could continue indefinitely, but levels beyond R5/S5 are rarely referenced in practice.

All R/S level computations use Math.FusedMultiplyAdd for the multiplier * offset + base pattern, providing single-rounding precision.

Performance Profile

Operation Count (Streaming Mode)

Extended Pivot Points adds R4/S4 levels beyond Classic — O(1) with 4 support/resistance pairs.

Operation	Count	Cost (cycles)	Subtotal
Store prev HLC	3	1 cy	~3 cy
PP = (H + L + C) / 3	1	2 cy	~2 cy
R1..R4 arithmetic + S1..S4	8	2 cy	~16 cy
NaN guard + state update	1	2 cy	~2 cy
Total	O(1)	—	~23 cy

O(1) pure arithmetic. Extended variant generates 4 pairs vs Classic 3 pairs, adding ~4 cy. All levels SIMD-parallel in batch mode.

Implementation Design

Pure arithmetic with no loops, no buffers, no auxiliary data structures. Each Update call performs 1 division (PP), 1 subtraction (range), 2 subtractions (ppMinusL, hMinusPP), 2 additions (R2, S2), and 8 FMA operations (R1, S1, R3-R5, S3-S5), plus 3 comparisons for NaN validation.

Metric	Score	Notes
Complexity	O(1)	Fixed arithmetic; no iteration
Allocations	0	Hot path is allocation-free
Warmup	2 bars	Minimum possible
Accuracy	10/10	Exact arithmetic via FMA; no approximation
Timeliness	10/10	No lag; levels available immediately on new bar
Smoothness	N/A	Discrete levels; smooth/noisy not applicable

State Management

Internal state uses a record struct with local copy pattern for JIT struct promotion. The state tracks previous bar's HLC and last-valid values for NaN/Infinity input substitution. Bar correction via isNew flag enables same-timestamp rewrites without state corruption.

SIMD Applicability

Not applicable for streaming (single bar computation). The BatchAll span API processes multiple bars but the per-bar computation (13 arithmetic operations) is too simple to benefit from vectorization overhead.

FMA Usage

The implementation uses Math.FusedMultiplyAdd for eight of the eleven R/S level computations. R1 and S1 use FMA(2, pp, -pL) and FMA(2, pp, -pH). R3-R5 and S3-S5 use FMA with the precomputed ppMinusL and hMinusPP intermediate values. R2 and S2 are simple additions/subtractions that do not benefit from FMA.

Validation

Self-consistency validation confirms all API modes produce identical results:

Mode	Status	Notes
Streaming (Update)	Passed	Bar-by-bar with isNew support
Batch (Batch(TBarSeries))	Passed	PP values match streaming
Span (Batch(Span))	Passed	PP values match streaming
BatchAll (BatchAll(Span))	Passed	All 11 levels match streaming
Event (Pub subscription)	Passed	Fires on every update

Library	Status	Notes
QuanTAlib	Passed	All modes self-consistent; level ordering invariant holds
Skender	N/A	Does not implement extended traditional variant
TA-Lib	N/A	Not implemented
Tulip	N/A	Not implemented
Ooples	N/A	Not validated

Mathematical correctness is validated by computing expected values from the extended formula for each bar and comparing against the indicator output at precision 10.

Common Pitfalls

1. First bar returns NaN. The indicator needs the previous bar's HLC to compute pivots. The first bar stores HLC but produces no output. This is correct behavior, not a bug. WarmupPeriod = 2.

2. R1-R3/S1-S3 are identical to classic PIVOT. The extended indicator adds R4/R5/S4/S5 but does not modify the classic levels. If you only need 7 levels, use the Pivot class instead to avoid computing unused outputs.

3. R4/R5 and S4/S5 project far from the current range. These levels represent 3x and 4x prior-range extensions. For low-volatility instruments, they may be so distant as to be meaningless. For high-volatility instruments or gap scenarios, they provide the only pre-computed reference levels.

4. Level spacing is asymmetric when close is not at the range midpoint. When the close is near the high, resistance levels are spaced more tightly than support levels, and vice versa. This is by design: the formula reflects where the close sits within the prior range.

5. Zero-range bars collapse all levels to a single price. When H_{prev} = L_{prev} (doji or single-print bar), all eleven levels equal the prior close, and PP equals the prior close. This is mathematically correct but provides no useful levels.

6. TValue input collapses range to zero. When updating with TValue instead of TBar, all OHLC fields equal the single price, producing zero range and all levels equal to that price. Use TBar input for meaningful pivot calculations.

7. R5/S5 are rarely reached. In typical market conditions, price reaching R5 or S5 represents approximately a 4x prior-range move. This occurs during panic selling, short squeezes, or major news events. Do not expect these levels to act as regular support/resistance.

References

• Person, J. L. (2004). A Complete Guide to Technical Trading Tactics: How to Profit Using Pivot Points, Candlesticks & Other Indicators. John Wiley and Sons.
• Wikipedia: Pivot point (technical analysis)
• TradingView: Pivot Points Standard
• Nison, S. (2001). Japanese Candlestick Charting Techniques. Prentice Hall Press. (Discussion of floor trader pivot methodology.)