7.6 Hedging with futures

7.6.1Setup#

A hedger is exposed to $Q_A$ units of an asset with price $S$ over a horizon $[t_1, t_2]$. She sells $N$ futures contracts, each on $Q_F$ units, with price $F$. The hedge ratio is $h = NQ_F/Q_A$: futures units per unit of exposure. Write $\Delta S = S_{t_2} - S_{t_1}$ and $\Delta F = F_{t_2} - F_{t_1}$. Per unit of exposure, the change in value of the hedged position is

If $S$ is exactly the futures' underlying and $t_2 = T$, then $\Delta S = \Delta F + b_{t_1}$, a constant, and $h = 1$ removes all risk. Otherwise $\Delta S$ and $\Delta F$ are imperfectly correlated and the best $h$ is a statistical question.

7.6.2The minimum-variance hedge ratio#

Let $\sigma_S^2 = \operatorname{Var}(\Delta S)$, $\sigma_F^2 = \operatorname{Var}(\Delta F)$ and $\rho = \operatorname{Corr}(\Delta S, \Delta F)$. Then

Proof

(7.14) is a convex quadratic in $h$ with second derivative $2\sigma_F^2 > 0$. Setting $d\operatorname{Var}/dh = -2\rho\sigma_S\sigma_F + 2h\sigma_F^2 = 0$ gives $h^* = \rho\sigma_S/\sigma_F$. Substituting, $\operatorname{Var}(h^*) = \sigma_S^2 - 2\rho^2\sigma_S^2 + \rho^2\sigma_S^2 = \sigma_S^2(1 - \rho^2)$, which gives (7.16). Completing the square, $\operatorname{Var}(h) = \sigma_F^2(h - h^*)^2 + \sigma_S^2(1-\rho^2)$.∎

Three remarks.

1. $h^*$ is the population slope of a regression of $\Delta S$ on $\Delta F$, and $\rho^2$ is that regression's $R^2$. Estimating the hedge ratio is ordinary least squares (Johnson 1960; Ederington 1979).
2. The penalty for a wrong ratio is quadratic and scaled by $\sigma_F^2$. Errors of 5% in $h$ cost little; using $h = 1$ when $h^* = 0.6$ can leave more variance than not hedging at all, since $\operatorname{Var}(1) > \sigma_S^2$ whenever $h^* < 1/2$.
3. The optimal number of contracts is

7.6.3Hedging an equity portfolio with index futures#

For an equity portfolio it is more natural to work in returns. Let $R_P$ be the portfolio return and $R_F$ the futures return over the horizon, $V_P$ the portfolio value and $V_F = F \times \text{lot}$ the value of one contract. Selling $N$ contracts changes the hedged value by $V_PR_P - NV_FR_F$. Applying Prop. 7.8 to $\Delta S = V_PR_P$ and $\Delta F = V_FR_F$ (per contract),

where $\beta$ is the regression slope of portfolio returns on futures returns. This is the CAPM beta (M06) if the futures return tracks the market index return closely, which it does for Nifty futures because the basis is small relative to index moves.

Proof

After trading $N$ contracts, the position's return sensitivity to $R_F$ is $(\beta V_P + NV_F)/V_P$. Setting it equal to $\beta^\star$ and solving gives (7.18).∎

7.6.4Tailing the hedge#

"Tailing" is used for two related adjustments.

1. Value-based ratio. If $\rho$, $\sigma_S$, $\sigma_F$ are estimated from daily returns (the usual case), the contract count is $N^* = \hat h\,V_A/V_F$ with current values, and it drifts as prices move. Hull calls this the tailing adjustment and recommends recomputing it as values change.
2. Interest tail. A futures hedge is paid daily while the hedged exposure settles at $T$. From the proof of Prop. 7.5, a hedge that replicates a forward holds $e^{-r(T - t)}$ futures per unit of forward exposure. The tailed count is

For worked example 7.10 with $r = 5.5\%$ and 54 days, $e^{-0.055 \times 54/365} = 0.99190$, so $37.34 \to 37.04$. The interest tail changes the hedge by less than one lot on an index hedge of a month or two. It matters for large, long-dated hedges, such as a refiner hedging a year of production, and it is the mechanism behind the futures–forward gap of 7.2.4.

7.6.5Practitioner notes on estimation#

• Use futures returns, not index returns, on the right-hand side. That is the instrument you trade, and its return includes basis changes.
• Match the horizon. A hedge held for a month should ideally use the correlation of monthly changes. Daily estimates are more precise but assume the correlation structure scales with horizon, which fails when there is lagged price adjustment (thinly traded small caps react to the index with a delay, so daily betas understate monthly betas).
• Sampling error. With $n$ observations, the standard error of $\hat h$ is approximately $(\sigma_S/\sigma_F)\sqrt{(1-\rho^2)/n}$. With $\sigma_S/\sigma_F = 1$, for $\rho = 0.95$ and $n = 20{,}000$ this is 0.002; for 250 daily observations and $\rho = 0.7$, it is about 0.045. The lab m07_hedge.py checks this on simulated data.
• Betas move. Estimates drift with regime; fat tails (M02) make OLS sensitive to a few days. Re-estimate on rolling windows and compare with a robust estimator.
• The stack-and-roll problem. A long-dated exposure hedged with short-dated futures has two extra risks: the roll cost (7.13) on every roll, and the mismatch between the sensitivity of a long-dated price and the front-month price. 7.8.1 shows both.