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Free lesson · FX and commodities

Minimum-variance cross hedging and basis risk

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Start with the idea

A related futures contract can offset part of a physical exposure’s price changes without tracking it perfectly.

Symbols, units & horizon
  • ΔS,ΔF: physical spot and hedge futures price changes over the same hedge interval, compatible currency per physical unit
  • h*: dimensionless minimum-variance ratio for hedged change ΔS−hΔF
  • Q: physical units owned
  • m: units per futures contract
  • n*: contracts to short before rounding

When and why to use this

Size a cross hedge and measure residual basis exposure for commodity producers or consumers.

A related futures contract can offset part of a physical exposure’s price changes without tracking it perfectly.

When spot and futures prices share compatible per-unit scales, the variance-minimizing hedge ratio is their price-change covariance divided by futures price-change variance. A producer long physical inventory uses a short futures quantity proportional to that ratio.

Estimate using only prior data over the hedge horizon. Contract grade, location, seasonality and structural supply disruptions can change the relationship. Rounding to whole contracts changes the achieved hedge and should be reported.

h∗=Cov⁡(ΔS,ΔF)Var⁡(ΔF),n∗=h∗Qm
Variance-minimizing calculus under an estimated covariance model

Minimum-variance cross hedging and basis risk

  1. Expand Var(ΔS−hΔF)=Var(ΔS)+h²Var(ΔF)−2hCov(ΔS,ΔF).
  2. Differentiate with respect to h and set to zero, assuming positive futures variance.
  3. Convert desired hedged physical units hQ into contracts by dividing by m.
Work it by hand

Covariance 6 and futures variance 9 give h=2/3. For Q=30,000 units and multiplier 1,000, short 20 contracts.

Apply it in a strategy

  • Freeze inputs at the stated decision time and record their units.
  • Size a cross hedge and measure residual basis exposure for commodity producers or consumers.
  • Recompute the example, then change the material assumption and explain the difference.

Research deliverable

Minimum-variance cross hedging and basis risk: produce the worked calculation, a timestamped input record and a written decision addressing this limitation: Historical covariance can fail during location, quality or liquidity shocks; minimum variance does not guarantee a fixed sale price.

These are synthetic mechanics examples, not historical performance or paper replications. Module evidence and research boundaries record the 12 September 2026 review.

Evidence and boundaries · reviewed 12 September 2026

The records below distinguish research status, access depth and data dates. Abstract-only review identifies research questions; it does not establish a replicated empirical claim.

Further reading: Banerjee, Boneva, Pinter & Sushko: Monetary policy transmission to exchange rates: the role of currency carry trades ↗

BIS Bulletin 124; institutional primary research. Version: 6 May 2026. Review: 2026-09-12; Publication summary only; PDF methods not inspected. Markets: FX and leveraged currency positions. Data dates: Exact estimation endpoints not available in inspected summary. Limitation: Motivates stress testing exchange rates jointly with leveraged positioning. It is not a recommendation to hold carry or evidence of executable arbitrage.

Further reading: Du, Nesmith & Heppe: Does Financial Stress Affect Commodity Futures Traders’ Positions? ↗

Federal Reserve working paper 2025-082. Version: September 2025, revised November 2025. Review: 2026-09-12; Abstract on Federal Reserve research index only. Markets: US commodity futures. Data dates: 2008 global financial crisis and COVID-19 crisis; exact window endpoints not inspected. Limitation: Different crisis mechanisms motivate multiple stress paths; aggregate position findings are not causal forecasts of individual futures returns.

Research sources, review dates and limitations

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

# Python 3.10+; standard library and NumPy only.
# Synthetic teaching inputs; conventions and units are defined in the notation above.
def cross_hedge(covariance,futures_variance,physical_quantity,multiplier):
    if futures_variance<=0 or physical_quantity<0 or multiplier<=0: raise ValueError('Invalid variance, quantity or multiplier')
    ratio=covariance/futures_variance
    return ratio,ratio*physical_quantity/multiplier

assert cross_hedge(6,9,30000,1000)==(2/3,20)
print(cross_hedge(6,9,30000,1000))

Continue learning

FX & Commodities: Quotes, Carry, Curves & Hedges — all lessons
  1. Invert both units and bid/ask sides
  2. Triangular conversion with executable sides
  3. Covered interest parity by matching currency cash flows
  4. Unhedged carry leaves exchange-rate risk
  5. Storage and convenience yield in a commodity forward
  6. Futures profit and margin cash demands
  7. Rolling futures without inventing a cash profit
  8. Minimum-variance cross hedging and basis risk

Quantitative finance and development glossary · Python resources and libraries · Research sources and limitations