Hedging Index Futures with Spot Using Basis Covariance
Summary
The document examines why index futures may show larger absolute price moves than the underlying index even when they trade in backwardation. It distinguishes futures delta, which describes the price relationship implied by the basis, from a hedge ratio estimated from observed co-movement. The empirical futures basis is expressed as a continuously compounded spread to spot, and futures returns are decomposed into spot returns, basis changes, and time to expiry effects.
For a minimum variance hedge in percentage returns, the suggested ratio is the covariance of futures and spot returns divided by spot return variance. Under the stated approximation, this ratio depends on how basis changes correlate with spot returns; converting back to contract or index point units also requires the futures delta. The answer cites an observed small positive daily correlation for SET50 as support for a modest over-hedge in that sample. This is an empirical illustration, not a universal rule: the relationship can vary with market, horizon, liquidity, and changing basis conditions, and the discussion does not resolve each proposed exit-timing scenario.
Key ideas
- The empirical basis can differ from the theoretical funding-minus-dividend basis because of market frictions and limits to arbitrage.
- Futures returns reflect spot returns, changes in basis, and the passage of time to expiry.
- A return-based minimum variance hedge ratio is estimated from covariance and variance, rather than read directly from futures delta.
- Basis changes correlated with spot returns can shift the hedge ratio above or below one.
- To translate a percentage hedge ratio into absolute exposure, account for the futures delta.
Tags
Full text
# why does index futures swing more than index?
# why does index futures swing more than index?
why does index futures swing (in absolute) more than index, when index futures price is lower than index (Backwardation)?
Say, SET50 Index(Thailand) is at 950, SET50 active Futures will be at around 945 (1 month to expiration). If SET50 moves 10 points, SET50 active Futures will, on average, move more than 10 points.(by comparing (Daily and intra-day standard deviation of $\Delta S$ and $\Delta F$)
- What is an explanation?
Given that $F=Se^{(r-q)(T-t)}$ and $\Delta _F=e^{(r-q)(T-t)}$ and minimum variance hedge ratio($h^*$) = $\rho \frac {\Delta S}{\Delta F}$,
- Is there any relationship between $\Delta _F$ and $h^*$?
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Other related questions
If I want to hedge SET50 with SET50 Futures (that swing more than index) and minimise basis risk, what Delta should I use if
- I want to hedge until futures expiration
- I want to close out hedge position before expiration (say, 5 days) and futures prices are mostly traded fair
- I want to close out hedge position before expiration (say, 5 days) and futures prices are always traded cheap but expected to be converged to fair at 5 day before expiration
## Answer by Chris Taylor (score 5)
https://quant.stackexchange.com/a/31947
The empirical relationship between the futures price $F$ and the spot price $S$ is
$$ F = S e^{b\tau} $$
where $\tau$ is the time to expiry, and $b$ is the empirical basis, i.e. the number that makes the equation hold, given by
$$ b = \frac{1}{\tau}\log(F/S) $$
It can be compared to the theoretical basis,
$$ b_{\rm theor} = r - q $$
where $r$ is the funding rate and $q$ is the expected dividend rate to expiry, but in general you will have $b\neq b_{\rm theor}$ (due to transaction costs, regulation, taxes and other limits to arbitrage).
The percentage futures return can be expressed as
$$ \frac{\delta F}{F} \approx \frac{\delta S}{S} + \tau \cdot \delta b + b \cdot \delta\tau $$
or, with $r_f=\delta F/F$ and $r_s = \delta S/S$,
$$ r_f\approx r_s+ \tau \cdot \delta b + b \cdot \delta\tau $$
The futures delta is, as you said,
$$ \Delta_F = e^{b\tau} $$
which is, in general, less than one if the futures are in backwardation (i.e. $b < 0$) which is true about 75% of the time for SET50 futures. However, as you correctly point out, when the spot moves by 10 points, the futures tend to move by more than 10 points, not less (a rough calculation suggests that the futures move by around 10.5 points for every 10 point move in the underlier).
If you want to hedge the futures by holding an offsetting amount of the spot, one option is to hold $\Delta_F$ of the spot. Alternatively you can take into account the comovement of the futures and the spot, and compute the hedge ratio $\beta$ to minimize the square of $r_f - \beta \cdot r_s$ (note that here we are talking in percentage terms, rather than absolute terms - to convert the hedge ratio back to absolute terms, you should multiply by $\Delta_F$, i.e. you would hold $\beta\cdot\Delta_F$ of the spot for each unit of the futures).
$$ \begin{align} \beta & = \frac{{\rm Cov}(r_f,r_s)}{{\rm Var}(r_s)} \\ & \approx \frac{{\rm Var}(r_s) + \tau \cdot {\rm Cov}(r_s, \delta b)}{{\rm Var}(r_s)} \\ & = 1 + \tau \frac{{\rm Cov}(r_s, \delta b)}{{\rm Var}(r_s)} \\ & = 1 + \tau \cdot \rho_{b,s} \frac{\sigma_b}{\sigma_s} \end{align} $$
where $\rho_{b,s}$ is the correlation between changes in the empirical basis and changes in the spot, and $\sigma_b$, $\sigma_s$ are the volatilities of the basis and the spot.
This tells us that the hedge ratio will be greater than one if changes in the spot are positively correlated with changes in the empirical basis, and less than one if changes in the spot are negatively correlated with changes in the empirical basis.
I measure a daily correlation of around 0.05 between changes in the spot and changes in the basis, indicating that you should over-hedge the futures with the spot, as opposed to a hedge ratio of 1 that you would use if you didn't take spot/basis correlation into account.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.