Many of the participants in the derivatives markets are hedgers. Their approach to the market seeks to reduce a specific risk that they encounter. This risks can arise from the fluctuations of the commodity prices, change of the interest rates interest or any other factor. All hedgers want to achieve a *perfect hedge *that entirely eliminates the risk. In practice, such hedges are very rare. Therefore, market practitioners are trying to construct a hedge which resembles a perfect one as close as possible. In this blog post, we will explore the basics of the static hedging using futures market.

Hedges in the futures market are classified as either *long *or *short. *The purpose of both is to offset the loss resulting from the price fluctuation against main investment or business activity. The basic hedge is implemented via the purchase or sale of the futures contract towards the unfavorable price event.

Ideally, hedger wants to know the date in the future when the asset will be bought or sold, so that it is possible to remove almost all price risk. Moreover, the following need to be considered:

- The asset underlying the futures contract may not be exactly the same as the asset whose price is hedged.
- The hedge may require the futures contract to be closed out before its delivery date.
- The futures price might be different from the spot price of the underlying asset.

These issues lead to the concept of *basis risk*.

## Basis Risk

In hedging, the basis can be defined as follows:

\[Basis = Futures\space price - Spot\space price\]

If the futures underlying asset is the same as the hedged one, the basis should be zero at the expiry of the futures contract. Prior to the expiration, the basis can be positive (contango) or negative (backwardation). As time passes, the spot price does not necessarily change by the same value, thus changing the basis.

Note, that the basis change can result in the improvement or worsening of the hedger's position. For example, the increase in the basis during the short hedge (plan to sell the asset) will result in the improvement of the position because of the higher price for the asset after the futures gains or losses are considered. For the long hedge (plan to buy the asset), the reverse is true. Strengthening basis will result in the higher price for the asset.

The key factor affecting basis risk is the choice of the futures contract to be used for hedging. It is essential to analyze futures contracts to find the one that has a price most closely correlated with the price of the hedged asset.

## Hedging an equity portfolio

In this practical part of the post, we will explore an example of how stock index futures can be used to hedge a diversified equity portfolio. Let's define $V_a$ to be the current value of the portfolio, and $V_f$ futures contract value (the futures price times the contract size).

The hedging ratio is the ratio of the value of the taken futures position to the size of the exposure. If the portfolio resembles the index, the hedge ratio can be assumed to be 1.0 and the number of contracts to be shorted is calculated as \[N=\frac{V_a}{V_f}\]

Consider a portfolio valued \$220,000 that resembles S&P 500. The E-mini S&P 500 index futures price is 2,200, and each futures contract is \$50 times the index. Therefore, the $V_a = 220,000$ and $V_f = 2,200 * 50 = 110,000$ and the number of shorted contracts to hedge the portfolio is 2.

If the portfolio is not similar to the index, we can calculate the beta ($\beta$) which is then used as a hedge ratio in the already shown formula. \[N=\beta\frac{V_a}{V_f}\]

One of the ways to get beta is to run a regression between the excess return of the portfolio and index over the risk-free rate, taking the slope of the resulting line equation.

It is also possible to change the beta of the portfolio via taking positions in futures market. To change the beta from $\beta_1$ to $\beta_2$, where $\beta_1 >\beta_2$, a short position in \[N=(\beta_1-\beta_2)\frac{V_a}{V_f}\] contracts required. When $\beta_1 <\beta_2$, a long trade in \[N=(\beta_2-\beta_1)\frac{V_a}{V_f}\] contracts is needed.

### Conclusion

While this blog post showed the basics of the hedging using futures, it is important to say that futures hedging can be useful not only in the long-term investments but also speculative options trading. It also builds fundamentals used for hedging of other risk factors like volatility and via more complex derivatives. Hope you found this post useful!