Hedge Ratio Calculator — Optimal Futures Hedge (h* = ρ·σS/σF)

Formula

How this is calculated

The minimum-variance hedge ratio h* is derived by minimising the variance of the combined spot-plus-futures portfolio. If S is the spot price and F is the futures price, the optimal ratio h* = ρ × σS / σF, where ρ is the correlation between the price changes ΔS and ΔF, σS is the standard deviation of ΔS, and σF is the standard deviation of ΔF. A ratio of 1.0 means one unit of futures for each unit of spot exposure; a ratio of 0.85 means hedging 85% of the position.

The number of contracts is N* = h* × (total portfolio value) / (futures price × contract size). Because contracts come in whole numbers, N* is rounded to the nearest integer in practice. Hedging effectiveness ρ² (expressed as a percentage here) tells you what fraction of the variance is eliminated by the hedge — if ρ = 0.85, hedging effectiveness = 72.25%, meaning the hedge removes 72% of the price risk but leaves the remaining 28% unhedged (basis risk).

This model assumes that the joint distribution of ΔS and ΔF is bivariate normal and that the hedge ratio is held constant over the hedging horizon. In practice, both σ and ρ change over time; rolling hedges and dynamic re-balancing are used for longer horizons. The inputs should be estimated from historical data over the same frequency as the hedging period (e.g., monthly returns if hedging over one month).