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Miha Kralj
143 lines
6.4 KiB
Markdown
143 lines
6.4 KiB
Markdown
# VWAPSD: VWAP with Standard Deviation Bands
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> *Standard deviation bands around VWAP measure institutional consensus — proximity signals fair value, distance signals opportunity.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Channel |
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| **Inputs** | OHLCV bar (TBar) |
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| **Parameters** | `numDevs` (default DefaultNumDevs) |
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| **Outputs** | Multiple series (Upper, Lower, Vwap, StdDev, Width) |
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| **Output range** | Tracks input |
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| **Warmup** | `2` bars |
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| **PineScript** | [vwapsd.pine](vwapsd.pine) |
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- VWAP with Standard Deviation Bands combines the Volume Weighted Average Price with a single configurable standard deviation band pair, providing a ...
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- **Similar:** [VwapBands](../vwapbands/vwapbands.md), [BBands](../bbands/bbands.md) | **Complementary:** Cumulative volume delta | **Trading note:** VWAP standard deviation bands; 1σ/2σ/3σ levels used for institutional mean-reversion.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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VWAP with Standard Deviation Bands combines the Volume Weighted Average Price with a single configurable standard deviation band pair, providing a simpler alternative to VWAPBANDS (which uses dual $\pm 1\sigma$ and $\pm 2\sigma$ levels). Three running sums enable O(1) streaming updates. A session reset mechanism clears accumulations at configurable intervals, keeping the indicator anchored to current market structure. The configurable deviation parameter allows traders to select their desired confidence level ($1\sigma$ ≈ 68%, $2\sigma$ ≈ 95%, $3\sigma$ ≈ 99.7%).
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## Historical Context
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VWAP emerged in the 1980s as institutional traders needed a benchmark reflecting actual market participation. Berkowitz, Logue, and Noser (1988) established VWAP as the standard for measuring execution quality. The concept is straightforward: weight each price by the volume traded at that price, producing an average that reflects where the most conviction-backed trading occurred.
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The standard deviation extension follows the same reasoning as Bollinger Bands but applied to volume-weighted statistics. By adding bands at $n$ standard deviations from VWAP, the indicator creates a statistically grounded channel that adapts to actual volume-weighted volatility.
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VWAPSD differs from VWAPBANDS only in output structure: VWAPSD emits one band pair at a configurable distance, while VWAPBANDS always emits two band pairs ($\pm 1\sigma$ and $\pm 2\sigma$). The underlying VWAP and variance calculations are identical.
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## Architecture & Physics
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### 1. Running Sum Accumulation
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Three cumulative sums, reset at session boundaries:
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$$
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\Sigma_{pv} = \sum_{i=1}^{n} P_i \cdot V_i, \quad \Sigma_v = \sum_{i=1}^{n} V_i, \quad \Sigma_{p^2v} = \sum_{i=1}^{n} P_i^2 \cdot V_i
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$$
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where $P_i$ is the source price (typically HLC3) and $V_i$ is volume. Zero-volume bars are skipped to prevent distortion.
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### 2. VWAP (Center Line)
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$$
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\text{VWAP}_t = \frac{\Sigma_{pv}}{\Sigma_v}
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$$
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### 3. Volume-Weighted Standard Deviation
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Using the computational identity $\text{Var}(X) = E[X^2] - (E[X])^2$:
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$$
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\sigma^2 = \frac{\Sigma_{p^2v}}{\Sigma_v} - \text{VWAP}^2
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$$
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$$
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\sigma = \sqrt{\max(0,\;\sigma^2)}
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$$
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### 4. Band Construction
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$$
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U_t = \text{VWAP}_t + k \cdot \sigma_t
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$$
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$$
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L_t = \text{VWAP}_t - k \cdot \sigma_t
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$$
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where $k$ is the number of standard deviations (default 2.0).
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### 5. Session Reset
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On a reset condition, all running sums restart from zero. Configurable reset intervals include intraday (1m through 4H), daily, weekly, monthly, quarterly, semi-annual, annual, or never.
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### 6. Complexity
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Streaming: $O(1)$ per bar. Three additions to running sums, one division, one square root. Memory: three doubles for running sums plus scalar state (~64 bytes per instance).
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## Mathematical Foundation
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### Parameters
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| Symbol | Name | Default | Constraint | Description |
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|--------|------|---------|------------|-------------|
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| $k$ | numDevs | 2.0 | $0.1$ – $5.0$ | Number of standard deviations for bands |
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### VWAPSD vs VWAPBANDS
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| Aspect | VWAPSD | VWAPBANDS |
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|--------|--------|-----------|
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| Band pairs | 1 (configurable $k\sigma$) | 2 ($\pm 1\sigma$ and $\pm 2\sigma$) |
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| Default deviation | 2.0 | 1.0 (inner); 2.0 (outer) |
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| VWAP calculation | Identical | Identical |
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| Variance calculation | Identical | Identical |
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### Output Interpretation
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| Output | Interpretation |
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|--------|---------------|
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| Price above VWAP | Buyers paying above fair value; bullish intraday bias |
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| Price below VWAP | Sellers accepting below fair value; bearish intraday bias |
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| Price at upper band | Overextended above volume-weighted mean by $k\sigma$ |
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| Price at lower band | Overextended below volume-weighted mean by $k\sigma$ |
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| Band width expanding | Intraday volume-weighted dispersion increasing |
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| Band width near zero | Very tight price clustering around VWAP |
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## Performance Profile
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### Operation Count (Streaming Mode)
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VWAPSD is slightly simpler than VWAPBANDS (one band pair instead of two), with identical VWAP and variance computation:
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| MUL (price × vol for sum_pv) | 1 | 3 | 3 |
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| MUL (price² × vol for sum_pv2) | 2 | 3 | 6 |
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| ADD (3 running sums) | 3 | 1 | 3 |
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| DIV (sum_pv / sum_vol for VWAP) | 1 | 15 | 15 |
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| DIV (sum_pv2 / sum_vol for E[X²]) | 1 | 15 | 15 |
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| MUL (VWAP² for variance) | 1 | 3 | 3 |
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| SUB (E[X²] - VWAP²) | 1 | 1 | 1 |
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| SQRT (σ) | 1 | 20 | 20 |
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| MUL (k × σ) | 1 | 3 | 3 |
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| ADD/SUB (VWAP ± k·σ) | 2 | 1 | 2 |
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| **Total (hot)** | **14** | — | **~71 cycles** |
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Saves ~5 cycles vs VWAPBANDS by emitting 2 bands instead of 4. Session reset adds one CMP per bar.
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### Batch Mode (SIMD Analysis)
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Cumulative sums are inherently sequential. Band arithmetic is vectorizable:
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| Optimization | Benefit |
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| :--- | :--- |
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| Running sum accumulation | Sequential (prefix sum dependency) |
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| Variance → SQRT → bands | Vectorizable in a batch post-pass |
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| Session reset detection | Sequential (comparison per bar) |
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## Resources
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- Berkowitz, S., Logue, D. & Noser, E. (1988). "The Total Cost of Transactions on the NYSE." *The Journal of Finance*, 43(1), 97–112.
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- Kissell, R. (2013). *The Science of Algorithmic Trading and Portfolio Management*. Academic Press.
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