Optimal Hedge Ratio

Hedge Parameters

Std Dev of Spot Price Changes (σ_S)

Standard deviation of changes in spot price per unit time

Std Dev of Futures Price Changes (σ_F)

Standard deviation of changes in futures price per unit time

Correlation Coefficient (ρ)

Correlation between spot and futures price changes (-1 to 1)

Number of Contracts (Optional)

Spot Position Size (Q_A)

Size of the position being hedged (units or value)

Futures Contract Size (Q_F)

Size of one futures contract (units per contract)

Quick Examples

Optimal Hedge Results

Optimal Hedge Ratio (h*)

Minimum variance hedge ratio

Hedge Effectiveness

Variance reduction achieved (ρ²)

Contracts Needed (N*)

Number of futures contracts

Calculation Breakdown

h* = ρ × (σ_S / σ_F)

Interpretation

Understanding the Optimal Hedge Ratio

Formula

h* = ρ × (σ_S / σ_F)
N* = h* × (Q_A / Q_F)

Key Components

σ_S — Spot Volatility

Standard deviation of price changes in the asset being hedged over the hedge period.

σ_F — Futures Volatility

Standard deviation of price changes in the futures contract used for hedging over the same period.

ρ — Correlation

Pearson correlation between spot and futures price changes. Higher correlation improves hedge effectiveness.

Practical Applications

Hedging commodity price risk using futures contracts
Currency risk management with foreign exchange forwards
Equity portfolio hedging with index futures
Interest rate risk management with bond futures
Cross-hedging when exact futures contract is unavailable

Limitations

Based on historical data; future correlations may differ
Minimises variance but does not eliminate basis risk
Assumes a static hedge — dynamic rebalancing may improve results
Number of contracts must be rounded to a whole number in practice
Does not account for liquidity or transaction costs