Balance of Power (BOP)

I. What is BOP

The Balance of Power (BOP) is an oscillator that measures the relative strength of buyers versus sellers within each trading period. It quantifies the result of the bull-bear power struggle by calculating the ratio of the difference between the close and open prices to the total bar range.

Historical Background

BOP was invented by Igor Livshin and published in 2001 in Technical Analysis of Stocks and Commodities magazine. Livshin proposed that traditional indicators (such as RSI, MACD) primarily focus on the direction and magnitude of price changes but overlook a key question: within each trading period, are buyers or sellers in control? BOP was designed specifically to answer this question.

Indicator Classification

• Type: Oscillator, displayed in a separate panel
• Category: Momentum / Volume-Price
• Default Parameters: Smoothing period $= 14$ (SMA)
• Value Range: -1 to +1

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II. Mathematical Principles and Calculation

Core Formulas

Raw BOP:

$BOP_t = \frac{C_t - O_t}{H_t - L_t}$

Where:

• $C_t$ = close price
• $O_t$ = open price
• $H_t$ = high price
• $L_t$ = low price

Smoothed BOP:

$BOP_{smooth} = SMA(BOP, n)$

Default $n = 14$.

Step-by-Step Calculation Logic

1. Calculate numerator: Close - Open = direction and size of the candle body
2. Calculate denominator: High - Low = total bar range
3. Divide: Get the ratio of body to range (with sign)
4. Smooth: Apply SMA(14) to raw BOP to reduce daily noise

Value Interpretation

BOP Value	Candle Shape	Meaning
+1	Perfect bullish candle (open = low, close = high)	Buyers in full control
+0.5	Medium bullish candle	Buyers dominant
0	Doji or equal upper/lower shadows	Balanced power
-0.5	Medium bearish candle	Sellers dominant
-1	Perfect bearish candle (open = high, close = low)	Sellers in full control

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III. Python Implementation

import numpy as np
import pandas as pd


def bop(open_price: pd.Series,
        high: pd.Series,
        low: pd.Series,
        close: pd.Series,
        period: int = 14) -> pd.DataFrame:
    """
    Calculate the Balance of Power (BOP)

    Parameters
    ----------
    open_price : pd.Series
        Open price series
    high : pd.Series
        High price series
    low : pd.Series
        Low price series
    close : pd.Series
        Close price series
    period : int
        SMA smoothing period, default 14

    Returns
    -------
    pd.DataFrame
        DataFrame with raw BOP and smoothed BOP
    """
    range_hl = high - low

    # Handle zero denominator
    raw_bop = np.where(
        range_hl == 0,
        0,
        (close - open_price) / range_hl
    )
    raw_bop = pd.Series(raw_bop, index=close.index)

    smooth_bop = raw_bop.rolling(window=period, min_periods=period).mean()

    return pd.DataFrame({
        "BOP_raw": raw_bop,
        "BOP": smooth_bop
    }, index=close.index)


# ========== 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) * 0.5,
        "high":  base + np.abs(np.random.randn(120) * 1.0),
        "low":   base - np.abs(np.random.randn(120) * 1.0),
        "close": base + np.random.randn(120) * 0.4,
        "volume": np.random.randint(1000, 10000, size=120),
    })
    df.set_index("date", inplace=True)

    # Calculate BOP
    bop_df = bop(df["open"], df["high"], df["low"], df["close"])
    df = pd.concat([df, bop_df], axis=1)

    print("=== Balance of Power Results (last 15 rows) ===")
    print(df[["close", "BOP_raw", "BOP"]].tail(15).to_string())

    # Zero line crossover signals
    df["signal"] = np.where(
        (df["BOP"] > 0) & (df["BOP"].shift(1) <= 0), 1,
        np.where(
            (df["BOP"] < 0) & (df["BOP"].shift(1) >= 0), -1, 0
        )
    )

    signals = df[df["signal"] != 0]
    print("\n=== BOP Zero Line Crossover Signals ===")
    print(signals[["close", "BOP", "signal"]].to_string())

    # BOP extreme values
    print(f"\n=== BOP Descriptive Statistics ===")
    print(df["BOP"].describe().to_string())

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IV. Problems the Indicator Solves

1. Bull/Bear Power Assessment

BOP directly quantifies the degree of buyer and seller control within each trading period:

• BOP > 0: Buyers dominant, close above open
• BOP < 0: Sellers dominant, close below open
• BOP absolute value near 1: One side in complete control

2. Zero Line Crossover Signals

Signal	Condition	Meaning
Buy	Smoothed BOP crosses above zero	Buying power begins to dominate
Sell	Smoothed BOP crosses below zero	Selling power begins to dominate

3. Trend Strength Confirmation

During an uptrend:

• BOP consistently positive and increasing — trend is healthy, strong buying power
• BOP begins declining from positive values — buying power weakening, trend may decelerate

4. Divergence Detection

• Bearish Divergence: Price new high + BOP peak declining — although price is rising, buyer dominance is weakening
• Bullish Divergence: Price new low + BOP trough rising — although price is falling, seller control is weakening

5. Quantification of Candlestick Patterns

BOP is essentially a quantitative measure of the relationship between candle body and shadows, serving as a numerical alternative to candlestick pattern analysis.

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V. Advantages, Disadvantages, and Use Cases

Advantages

Advantage	Description
Intuitive logic	Directly quantifies bull/bear power balance, clear meaning
Bounded range	-1 to +1, easy to standardize and compare
Uses all OHLC data	Combines four price dimensions: open, high, low, close
Complementary perspective	Provides open price information that most indicators ignore

Disadvantages

Disadvantage	Description
Dependent on open price	Unreliable when open price quality is poor
Intraday noise	Raw BOP is very volatile, must be smoothed before use
Simple signals	Primarily relies on zero line crossovers, single signal dimension
Less researched	Compared to RSI, MACD, and other indicators, fewer academic and practical studies exist

Use Cases

• Stock markets: Where opening auction mechanisms make the open price meaningful
• Daily timeframe: Daily open and close prices are most representative
• Auxiliary confirmation: As a supplementary validation tool for other technical indicator signals

Comparison with Related Indicators

Comparison	BOP	RSI	OBV
Data Input	O/H/L/C	C	C + Volume
Focus	Bull/Bear power	Rate of change	Volume-price relationship
Value Range	-1 to +1	0-100	Unbounded
Smoothing	SMA	Wilder	Cumulative