3. Hedging Strategies Using Futures

Created on January 11, 2026

2026   ·   options   derivatives   hedging   ·   hull-options

This chapter focuses on the practical application of futures: protecting against price movements. While the theory suggests a perfect “lock” on prices, real-world constraints like basis risk and asset mismatches make hedging an exercise in variance minimization.

Short vs. Long Hedges

  • Short Hedge: Occurs when you already own an asset (or will produce it, like a farmer with crops) and intend to sell it later. You sell futures at price $F$ to protect against a price decrease.
    • Sale of physical asset at time $T$: $+S_T$
    • Payoff from Short Future: $+(F - S_T)$
    • Net Price Received: $S_T + F - S_T = F$
    • Result: If prices fall, the gain on the futures offsets the loss on the physical sale.
  • Long Hedge: Occurs when you know you must buy an asset in the future (like an airline needing fuel). You buy futures at price $F$ to protect against a price increase.
    • Purchase of physical asset at time $T$: $-S_T$
    • Payoff from Long Future: $+(S_T - F)$
    • Net Price Paid: $-S_T + S_T - F = -F$
    • Result: If prices rise, the gain on the futures offsets the higher cost of the physical purchase.

Basis Risk

In a perfect world, the spot price ($S$) and futures price ($F$) converge at expiry. In reality, we often close a hedge before expiry or at a different location. The Basis is defined as:

\[\text{Basis} = \text{Spot Price} - \text{Futures Price}\]
  • Strengthening: If the Basis increases ($S - F \uparrow$), it benefits a short hedger but hurts a long hedger.
  • Weakening: If the Basis decreases ($S - F \downarrow$), it benefits a long hedger but hurts a short hedger.

Cross Hedging & Minimum Variance Hedge Ratio

When the asset being hedged is not the same as the underlying of the futures contract, we face cross hedging risk. We must find the optimal hedge ratio ($h^*$) that minimizes the variance of the hedged portfolio.

Derivation

The change in the value of a hedged position is $\Delta P = \Delta S - h \Delta F$. To minimize risk, we minimize the variance:

\[\begin{aligned} \operatorname{Var}(\Delta P) &= \operatorname{Var}(\Delta S) + \operatorname{Var}(h\Delta F) - 2 \operatorname{Cov}(\Delta S, h\Delta F) \\ &= \sigma_S^2 + h^2 \sigma_F^2 - 2h \rho \sigma_S \sigma_F \end{aligned}\]

Taking the derivative with respect to $h$ and setting it to zero:

\[\frac{d}{dh} \operatorname{Var}(\Delta P) = 2h \sigma_F^2 - 2 \rho \sigma_S \sigma_F = 0\]

Solving for the Minimum Variance Hedge Ratio ($h^*$):

\[h^* = \rho \frac{\sigma_S}{\sigma_F} = \frac{\operatorname{Cov}(\Delta S, \Delta F)}{\sigma_F^2}\]

Optimal Number of Contracts ($N^*$)

Once $h^*$ is known, we calculate the number of contracts needed by adjusting for the size of the position ($Q_A$) and the size of one contract ($Q_F$):

\[N^* = \frac{h^* Q_A}{Q_F}\]

Tailing the Hedge: Because futures are settled daily, we usually “tail” the hedge to account for the interest earned on margin cash flows. We use the total dollar value ($V$):

\[N^* = \hat{h} \frac{V_A}{V_F}\]

Hedging an Equity Portfolio

To hedge a diversified stock portfolio, we use stock index futures. Instead of calculating a custom $h^*$, we use the portfolio’s Beta ($\beta$), which measures its sensitivity to the market index:

\[N^* = \beta \frac{V_A}{V_F}\]
  • If $\beta > 1$, the portfolio is more volatile than the market, requiring more futures contracts for protection.
  • If $\beta < 1$, fewer contracts are required.