An Exponential Moving Average (EMA) is a type of moving average that applies exponentially decreasing weights to older data points, giving greater importance to recent observations. Unlike a Simple Moving Average that requires storing an entire window of historical values, an EMA can be updated incrementally with a single stored value, making it significantly more gas-efficient for on-chain implementations.

Formula

1EMA_new = α × Price_current + (1 - α) × EMA_previous

Where α (alpha) is the smoothing factor, typically calculated as:

1α = 2 / (n + 1)

Where n is the number of periods. A higher α makes the EMA more responsive to recent data.

On-chain implementation

EMAs are more gas-efficient than SMAs because they only require storing one value:

1uint256 public ema;
2uint256 public constant ALPHA_NUMERATOR = 2;
3uint256 public constant ALPHA_DENOMINATOR = 11; // α = 2/11 for 10-period EMA
4uint256 public constant PRECISION = 1e18;
5
6function updateEMA(uint256 newPrice) external returns (uint256) {
7    if (ema == 0) {
8        ema = newPrice;
9        return ema;
10    }
11
12    // EMA = α * newPrice + (1 - α) * oldEMA
13    ema = (ALPHA_NUMERATOR * PRECISION * newPrice / ALPHA_DENOMINATOR
14        + (ALPHA_DENOMINATOR - ALPHA_NUMERATOR) * PRECISION * ema / ALPHA_DENOMINATOR)
15        / PRECISION;
16
17    return ema;
18}

DeFi use cases

• Synthetix: Uses EMA for exchange rate calculations, balancing manipulation resistance with responsiveness.
• Interest rate models: Lending protocols apply EMAs to utilization rates for smoother interest rate adjustments.
• Volatility estimation: EMAs of price changes help estimate on-chain volatility for risk parameters.

Trade-offs

Security considerations

• Initialization: The first EMA value has outsized influence on subsequent calculations. Protocols should seed the EMA with a trusted price.
• Update frequency: Infrequent updates can cause the EMA to become stale, creating staleness risks.
• Precision loss: The multiplication and division steps in EMA calculations can accumulate rounding errors over time.
• Alpha selection: Too high an alpha makes the EMA susceptible to short-term manipulation; too low makes it unresponsive to legitimate price movements.

EMAs offer an efficient on-chain alternative to SMAs, trading some simplicity for better gas efficiency and faster trend detection.