QuanTAlib/lib/oscillators/ac/Ac.md

AC: Acceleration Oscillator

"Knowing speed is useful. Knowing whether you're speeding up or slowing down is what keeps you alive."

Introduction

The Acceleration Oscillator (AC) is Bill Williams' second-derivative momentum indicator. Where the Awesome Oscillator (AO) measures the speed of market momentum, AC measures whether that momentum is accelerating or decelerating. AC is computed as AO minus a 5-period SMA of AO. Zero crossings and color changes signal shifts in market driving force before price reverses.

Historical Context

Bill Williams introduced AC alongside AO in his "Trading Chaos" methodology. While AO already strips trend by subtracting a slow SMA from a fast SMA (both applied to the bar midpoint), traders found they needed earlier warning of momentum shifts. AC provides exactly that: the rate of change of AO itself. When AC crosses zero from below, the market's driving force is accelerating upward, often preceding AO's own zero crossing by several bars.

Calculation

The AC indicator is calculated in two stages:

Stage 1: Awesome Oscillator

\text{Median Price} = \frac{\text{High} + \text{Low}}{2} \text{AO} = \text{SMA}(\text{Median Price}, \text{fast}) - \text{SMA}(\text{Median Price}, \text{slow})

Stage 2: Acceleration

\text{AC} = \text{AO} - \text{SMA}(\text{AO}, \text{acPeriod})

Default parameters: fast = 5, slow = 34, acPeriod = 5.

Interpretation

• AC > 0 and rising (green): Bullish acceleration. Momentum is strengthening.
• AC > 0 and falling (red): Bullish deceleration. Momentum still positive but weakening.
• AC < 0 and falling (green to red): Bearish acceleration. Momentum is weakening further.
• AC < 0 and rising (red to green): Bearish deceleration. Downward momentum is weakening.
• Zero crossings: Often precede AO zero crossings, providing earlier entry/exit signals.

Bill Williams' Trading Rules

1. Buy signal: AC is green (rising) for two consecutive bars above zero, or three consecutive green bars below zero.
2. Sell signal: AC is red (falling) for two consecutive bars below zero, or three consecutive red bars above zero.

Parameters

Parameter	Default	Range	Description
fastPeriod	5	> 0	Fast SMA period for AO calculation
slowPeriod	34	> fast	Slow SMA period for AO calculation
acPeriod	5	> 0	SMA period applied to AO values

API

Streaming

var ac = new Ac(fastPeriod: 5, slowPeriod: 34, acPeriod: 5);
TValue result = ac.Update(bar, isNew: true);

Batch (TBarSeries)

TSeries results = Ac.Batch(barSeries);

Batch (Span)

Ac.Batch(highSpan, lowSpan, outputSpan, fastPeriod: 5, slowPeriod: 34, acPeriod: 5);

Calculate

var (results, indicator) = Ac.Calculate(barSeries, fastPeriod: 5, slowPeriod: 34, acPeriod: 5);

Usage

// Streaming
var ac = new Ac();
foreach (var bar in bars)
{
    var result = ac.Update(bar);
    if (ac.IsHot && result.Value > 0)
    {
        // Bullish momentum accelerating
    }
}

// Event-driven chaining
ac.Pub += (sender, e) => Console.WriteLine($"AC: {e.Value.Value:F4}");

Performance

Operation	Complexity	Allocations
Update (streaming)	O(1)	Zero
Batch (Span)	O(n)	ArrayPool
Warmup period	slow + ac - 1	—

AC uses three internal SMA instances. Each SMA uses a RingBuffer for O(1) sliding window computation. The Batch path uses SIMD-accelerated subtraction via SimdExtensions.Subtract.

Validation

AC is validated via self-consistency (AC = AO - SMA(AO, acPeriod)) and batch/streaming equivalence. No external library implements AC with identical SMA methodology for cross-library validation.

Test	Status
AC = AO - SMA(AO) identity	Pass
Batch/streaming match	Pass
Span/TBarSeries match	Pass
Determinism	Pass
Constant input convergence	Pass (→ 0)
Large dataset stability	Pass (5000 bars)

Sources

• Williams, Bill. "Trading Chaos." Wiley, 1995.
• Williams, Bill. "New Trading Dimensions." Wiley, 1998.
• Investopedia: Accelerator Oscillator
• TradingView: AC