Williams %R

I. What is Williams %R

Williams %R (also known as %R or W%R) is a momentum oscillator that measures the position of the current closing price relative to the price range over a given period. It is very similar to Stochastic %K, but %R expresses the “distance from the top of the range,” so its values are negative (-100 to 0), using inverse logic.

Historical Background

Williams %R was invented by the famous American trader Larry Williams. Larry Williams won the 1987 World Cup Trading Championship with a record-breaking return of 11,376%. The %R indicator became popular among short-term traders due to its simple calculation and direct signals.

Indicator Classification

• Type: Oscillator, displayed in a separate panel
• Category: Momentum
• Default Parameters: Period $n = 14$
• Value Range: -100 to 0

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II. Mathematical Principles and Calculation

Core Formula

$\%R = \frac{HH_n - C_t}{HH_n - LL_n} \times (-100)$

Where:

• $C_t$ = current closing price
• $HH_n$ = highest high over the past $n$ periods
• $LL_n$ = lowest low over the past $n$ periods
• $n$ = lookback period (default 14)

Equivalent Expression

$\%R = -100 + \frac{C_t - LL_n}{HH_n - LL_n} \times 100$

It can be seen that $\%R = \%K_{fast} - 100$.

Step-by-Step Calculation Logic

1. Determine lookback period $n$ (default 14)
2. Calculate range extremes: Find the highest high $HH_n$ and lowest low $LL_n$ over the past $n$ periods
3. Calculate distance from top: $HH_n - C_t$
4. Divide by range width: $(HH_n - C_t) / (HH_n - LL_n)$
5. Multiply by -100: Map the result to the -100 to 0 range

Value Interpretation

%R Value	Meaning
0	Closing price equals the range high
-50	Closing price is at the midpoint of the range
-100	Closing price equals the range low

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III. Python Implementation

import numpy as np
import pandas as pd


def williams_r(high: pd.Series,
               low: pd.Series,
               close: pd.Series,
               period: int = 14) -> pd.Series:
    """
    Calculate Williams %R

    Parameters
    ----------
    high : pd.Series
        High price series
    low : pd.Series
        Low price series
    close : pd.Series
        Close price series
    period : int
        Lookback period, default 14

    Returns
    -------
    pd.Series
        Williams %R values (-100 to 0)
    """
    highest_high = high.rolling(window=period, min_periods=period).max()
    lowest_low = low.rolling(window=period, min_periods=period).min()

    wr = (highest_high - close) / (highest_high - lowest_low) * (-100)

    return wr


# ========== Usage Example ==========
if __name__ == "__main__":
    np.random.seed(42)
    dates = pd.date_range("2024-01-01", periods=120, freq="D")
    base = 100 + np.cumsum(np.random.randn(120) * 1.0)

    df = pd.DataFrame({
        "date": dates,
        "open":  base + np.random.randn(120) * 0.3,
        "high":  base + np.abs(np.random.randn(120) * 0.8),
        "low":   base - np.abs(np.random.randn(120) * 0.8),
        "close": base,
        "volume": np.random.randint(1000, 10000, size=120),
    })
    df.set_index("date", inplace=True)

    # Calculate Williams %R
    df["WR_14"] = williams_r(df["high"], df["low"], df["close"], period=14)

    print("=== Williams %R Results (last 15 rows) ===")
    print(df[["close", "WR_14"]].tail(15).to_string())

    # Overbought/Oversold signals
    df["signal"] = np.where(
        df["WR_14"] > -20, -1,    # Overbought (near 0)
        np.where(df["WR_14"] < -80, 1, 0)  # Oversold (near -100)
    )

    print("\n=== Overbought Signals (%R > -20) ===")
    overbought = df[df["signal"] == -1]
    print(overbought[["close", "WR_14"]].head(10).to_string())

    print("\n=== Oversold Signals (%R < -80) ===")
    oversold = df[df["signal"] == 1]
    print(oversold[["close", "WR_14"]].head(10).to_string())

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IV. Problems the Indicator Solves

1. Overbought / Oversold Detection

Zone	Condition	Meaning
Overbought	%R > -20	Price near range top, pullback likely
Oversold	%R < -80	Price near range bottom, bounce likely
Neutral	-80 <= %R <= -20	No extreme signal

2. Trend Confirmation

• %R operating between -50 and 0 — bullish trend
• %R operating between -100 and -50 — bearish trend
• %R crossing back and forth around -50 — oscillating, no trend

3. Divergence Signals

• Bearish Divergence: Price makes a new high, %R fails to reach a higher level — upward momentum exhausted
• Bullish Divergence: Price makes a new low, %R fails to reach a lower level — downward momentum exhausted

4. Failure Swing

Similar to RSI:

• %R pulls back from overbought zone, rallies but fails to re-enter overbought — short signal
• %R bounces from oversold zone, declines but fails to re-enter oversold — long signal

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V. Advantages, Disadvantages, and Use Cases

Advantages

Advantage	Description
Extremely simple	Simple formula, fast computation
Highly responsive	No built-in smoothing, directly reflects price position within range
Intuitive thresholds	-20 and -80 boundaries are clear
Complements Stochastic	Provides a reverse perspective, convenient for cross-validation

Disadvantages

Disadvantage	Description
High noise	No smoothing leads to frequent false signals
Saturates in trends	Stays in overbought/oversold zones during strong trends
Negative value intuition	-100 to 0 inverse logic is less intuitive than 0-100
Needs confirmation	Not recommended as a standalone trading decision basis

Use Cases

• Short-term trading: Suited for quick in-and-out trading styles
• Range-bound markets: Overbought/oversold signals work well within clear support/resistance zones
• Alternative to Stochastic: When a more raw, sensitive signal is needed

Comparison with Related Indicators

Comparison	Williams %R	Stochastic	RSI
Value Range	-100 to 0	0-100	0-100
Smoothing	None	Dual SMA smoothing	Wilder Smoothing
Sensitivity	Very high	High	Medium
Noise	High	Medium	Low
Math Relationship	%K - 100	Independent	Independent