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120 lines
6.8 KiB
Markdown
120 lines
6.8 KiB
Markdown
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# BW_MFI: Bill Williams Market Facilitation Index
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> *The market facilitates price movement when it wants to — volume tells you how hard it tried.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Oscillators |
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| **Inputs** | OHLCV bar (TBar) |
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| **Parameters** | None |
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| **Outputs** | Dual series (Mfi, Zone) |
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| **Output range** | MFI: $\geq 0$; Zone: {0,1,2,3,4} |
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| **Warmup** | 2 bars |
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| **PineScript** | [bw_mfi.pine](bw_mfi.pine) |
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- Bill Williams' Market Facilitation Index measures price movement efficiency per unit of volume, then classifies each bar into one of four zones based on MFI and volume direction changes.
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- **Similar:** [MARKETFI](../marketfi/Marketfi.md) (MFI value only, no zones) | **Complementary:** [OBV](../../volume/obv/Obv.md), [FI](../fi/Fi.md) | **Trading note:** Zone 4 (Squat) often precedes breakouts; Zone 1 (Green) confirms trend strength.
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- Self-validated against direct formula computation. MARKETFI provides the same MFI value; zones are the distinguishing feature.
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The Bill Williams Market Facilitation Index extends the basic MFI calculation $\text{MFI} = (H - L) / V$ with a four-zone classification system that compares current MFI and volume to previous bar values. This classification transforms a simple efficiency measure into an actionable market state detector. Zone 4 (Squat) — high volume with compressed range — is Williams' most important signal, indicating a battle between bulls and bears that typically resolves with a breakout. The dual-output design (continuous MFI value plus discrete zone) enables both quantitative analysis and visual bar coloring.
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## Historical Context
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Bill Williams introduced the Market Facilitation Index in *Trading Chaos* (1995), as part of his broader "Profitunity" trading system. Williams argued that traditional volume analysis was incomplete: knowing that volume increased tells you nothing without understanding whether the market *used* that volume to move price. The MFI answers this question directly — it measures how many price points the market moved per unit of volume traded.
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The four-zone classification system was Williams' key innovation over raw MFI. By cross-referencing MFI direction with volume direction, he created a 2×2 matrix that categorizes every bar into one of four market states. This framework appears in both *Trading Chaos* (1995) and *New Trading Dimensions* (1998). The zone names (Green, Fade, Fake, Squat) became part of the standard Williams lexicon and are implemented in most professional trading platforms including MetaTrader, TradingView, and Bloomberg Terminal.
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## Architecture & Physics
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### 1. MFI Calculation
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$$
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\text{MFI}_t = \frac{H_t - L_t}{V_t}
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$$
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where $H_t$, $L_t$, $V_t$ are the high, low, and volume of bar $t$. Zero-volume guard returns 0.0 (no facilitation when no trades occurred). The MFI value is unbounded above and represents price range per unit of volume — higher values indicate more efficient price movement.
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### 2. Zone Classification Matrix
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The zone is determined by comparing current MFI and volume to the previous bar:
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$$
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\text{Zone}_t = \begin{cases}
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1 \text{ (Green)} & \text{if } \text{MFI}_t > \text{MFI}_{t-1} \text{ and } V_t > V_{t-1} \\
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2 \text{ (Fade)} & \text{if } \text{MFI}_t \leq \text{MFI}_{t-1} \text{ and } V_t \leq V_{t-1} \\
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3 \text{ (Fake)} & \text{if } \text{MFI}_t > \text{MFI}_{t-1} \text{ and } V_t \leq V_{t-1} \\
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4 \text{ (Squat)} & \text{if } \text{MFI}_t \leq \text{MFI}_{t-1} \text{ and } V_t > V_{t-1}
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\end{cases}
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$$
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### 3. Zone Interpretation
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| Zone | Name | MFI | Volume | Market State |
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| :--: | :---- | :-: | :----: | :----------- |
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| 1 | Green | ↑ | ↑ | Trend continuation — market moves efficiently with increasing participation |
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| 2 | Fade | ↓ | ↓ | Fading momentum — traders losing interest, trend exhaustion |
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| 3 | Fake | ↑ | ↓ | Fake breakout — price moves on declining volume, unsupported |
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| 4 | Squat | ↓ | ↑ | Accumulation — high volume absorbed by range compression, breakout imminent |
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### 4. Complexity
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O(1) per bar — single division plus two comparisons. No buffers, no period parameter. The zone classification adds only two boolean comparisons to the base MFI calculation.
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## Mathematical Foundation
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### Parameters
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No configurable parameters. MFI is a pure bar-level computation.
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### Output Interpretation
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| Output | Type | Range | Description |
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| :----- | :--- | :---- | :---------- |
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| MFI | double | $\geq 0$ | Price range per unit of volume |
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| Zone | int | {0,1,2,3,4} | Market state classification (0 = first bar, insufficient data) |
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## Performance Profile
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### Operation Count (Streaming Mode)
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| Operation | Count | Cost (cycles) | Subtotal |
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| :-------- | ----: | ------------: | -------: |
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| SUB | 1 | 1 | 1 |
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| DIV | 1 | 15 | 15 |
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| CMP | 2 | 1 | 2 |
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| **Total** | | | **18** |
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### SIMD Analysis
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| Operation | Vectorizable? | Notes |
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| :-------- | :-----------: | :---- |
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| MFI = (H-L)/V | Yes | Element-wise arithmetic |
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| Zone comparison | Limited | Sequential dependency on previous bar |
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| Batch MFI only | Full SIMD | No inter-element dependency |
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### Quality Metrics
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| Metric | Score | Notes |
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| :----- | :---: | :---- |
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| Accuracy | 10/10 | Exact formula, no approximation |
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| Timeliness | 10/10 | Zero lag — current bar only |
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| Smoothness | 3/10 | No smoothing — raw bar-level measure |
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| Signal clarity | 7/10 | Discrete zones are unambiguous |
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| Memory | 10/10 | O(1) — four scalar values |
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## Common Pitfalls
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1. **Zero volume bars:** Holiday/pre-market bars with zero volume produce MFI = 0 and can skew zone classification on the next bar. Filter these bars or use minimum volume thresholds.
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2. **MFI scale varies by instrument:** Raw MFI values are not comparable across instruments with different price levels or volume scales. Use percentage-based normalization for cross-instrument comparison.
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3. **Equal values edge case:** When MFI or volume exactly equals the previous bar, the implementation treats this as "not up" — resulting in Zone 2 (Fade) when both are equal, Zone 4 (Squat) when only volume increases, or Zone 3 (Fake) when only MFI increases.
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4. **First bar has no zone:** Zone 0 indicates insufficient data (first bar). Ensure downstream logic handles this sentinel value.
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## Resources
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- **Williams, B.** *Trading Chaos*. Wiley, 1995. Chapter on Market Facilitation Index.
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- **Williams, B.** *New Trading Dimensions*. Wiley, 1998. Extended MFI zone analysis.
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- **Williams, B.** *Trading Chaos: Second Edition*. Wiley, 2004. Updated zone interpretations.
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