Technical Indicator Optimization

Simple Moving Average (SMA)

The SMA of period at time is the unweighted arithmetic mean of the last closing prices:

The SMA acts as a low-pass filter, smoothing out short-term fluctuations to reveal the underlying trend direction. The price position relative to its SMA (above or below) is one of the simplest trend-following signals.

Exponential Moving Average (EMA)

The EMA gives exponentially decaying weight to older observations, making it more responsive to recent price changes:

where the smoothing factor for a span of periods is:

A shorter span produces a higher and faster response. EMA crossovers (e.g., 12-period EMA crossing above 26-period EMA) are widely used trend signals.

SuperTrend

The SuperTrend indicator combines trend direction with volatility (ATR-based) bands. Given a multiplier and ATR period :

The SuperTrend flips between the upper and lower bands based on price action: when the close is above the upper band, the trend is bullish and the lower band becomes the trailing stop; when the close drops below the lower band, the trend reverses to bearish.

Relative Strength Index (RSI)

The RSI, introduced by Wilder (1978), measures the magnitude of recent price changes to evaluate overbought or oversold conditions:

where (Relative Strength) is the ratio of the exponentially smoothed average of up-period gains to the exponentially smoothed average of down-period losses over periods:

RSI oscillates between 0 and 100. Readings above 70 are traditionally considered overbought; below 30 is considered oversold. In the cross-sectional framework, extreme RSI values indicate which assets have the strongest or weakest recent momentum.

Williams %R

Williams %R measures where the current close sits within the recent high-low range over periods:

where and are the highest high and lowest low over the lookback period, and is the current close. The indicator ranges from -100 to 0; values near -100 indicate the asset is near its period low (oversold), while values near 0 indicate it is near its high (overbought).

Commodity Channel Index (CCI)

CCI measures the deviation of the typical price from its moving average, scaled by the mean absolute deviation:

where the typical price is:

The constant 0.015 is chosen so that approximately 75% of CCI values fall between -100 and +100 under a normal distribution. Values beyond this range indicate statistically unusual price movements.

Rate of Change (ROC)

ROC is the simplest momentum indicator, measuring the percentage change in price over periods:

ROC is directly related to the return over the lookback period. In the cross-sectional framework, it captures relative momentum: assets with higher ROC have outperformed their peers over the lookback window.

Average True Range (ATR)

ATR, introduced by Wilder (1978), measures market volatility by decomposing the entire range of an asset's price for a period. The True Range is the greatest of:

ATR is the moving average of the True Range over periods:

ATR does not indicate price direction but measures the degree of price volatility. Higher ATR values indicate greater volatility and can be used for position sizing (inverse ATR weighting) or as an input to volatility-adjusted signals.

Bollinger Bands

Bollinger Bands, introduced by John Bollinger (2001), consist of a middle band (SMA) and upper/lower bands set at standard deviations above and below the SMA:

The Bollinger Band %B indicator normalizes where the price sits within the bands:

Typically and . Values of %B above 1 indicate the price is above the upper band (potentially overbought); below 0 indicates it is below the lower band (potentially oversold). Band width contraction signals low volatility regimes that often precede breakouts.

On-Balance Volume (OBV)

OBV is a cumulative volume-based indicator that relates volume to price changes. It adds volume on up days and subtracts volume on down days:

where is the trading volume at time . The theory behind OBV is that volume precedes price: smart money accumulation or distribution is reflected in volume patterns before price movements materialize. Rising OBV with rising prices confirms the trend; divergences may signal reversals.

Cross-Sectional Z-Score Normalization

At each time , each indicator is computed for every asset in the universe. The raw indicator value is then normalized cross-sectionally to a Z-score:

where and are the cross-sectional mean and standard deviation of indicator across all assets at time :

This normalization makes the indicators comparable across different scales and units, and focuses on the relative ranking of assets within the universe at each point in time.

Signal Aggregation

The composite signal for asset at time is a weighted sum of the individual indicator Z-scores:

where is the weight assigned to indicator (with ). These indicator weights can be set equal (simple average), determined by information coefficient analysis, or optimized using historical performance. Assets with higher composite scores are considered more technically favorable.

LP Formulation

The composite signals are used as expected alpha inputs to a linear programming formulation that maximizes the signal-weighted portfolio return subject to risk and allocation constraints:

The upper bound prevents excessive concentration in any single asset, and the variance constraint controls total portfolio risk. When the variance constraint is included, the problem becomes a quadratically constrained linear program (QCLP), which can be solved efficiently with standard convex optimization solvers.

Rebalancing

The portfolio is rebalanced at regular intervals (e.g., weekly or monthly). At each rebalance date, the indicators are recomputed, Z-scores are recalculated cross-sectionally, and the LP is re-solved to produce updated portfolio weights. Turnover constraints can be added to the LP formulation to control transaction costs:

where is the maximum allowed two-way turnover and are the weights from the previous rebalance.