## Basis

In futures trading, the basis is the relative difference between the price of the future contract and the spot price. $${futurePrice - spot} \over spot$$

This quantity is usually presented as an absolute price difference, but in its relative form can be annualised by dividing by the time to expiry \(\tau\) of the contract expressed in years:

$$ basis =\left( \frac{futurePrice}{spot} - 1 \right) \times \frac{1}{\tau} $$

## Rolling Basis

Let us consider a hypothetical contract starting today with a given tenor of 1 month (1M) or 3 months (3M). Such a contract does not necessarily trade in the market and thus has no observable price to calculate the basis. The rolling basis is the basis between this ’bespoke’ contract and the spot.

This quantity is inferred from existing contracts’ basis and the calculation is accomplished in two steps:

### 1. Forward yield

First, we select two futures contracts with expiries \(T_{1}\) and \(T_{2}\) surrounding the hypothetical contract’s expiry \( T_{target} \, (\, T_{1} \leq T_{target} \leq T_{2})\, \). The forward yield is calculated as:

$$ yield ^{fwd} = {{basis_{2} \times \tau_{2} - basis_{1} \times \tau_{1}} \over {\tau_{2} - \tau_{1}}}$$

Where:

- \( \tau_{i} \): time to maturity for contract i, expressed as a year fraction
- \(basis_{i} \): annualised basis for contract i

### 2. Weighted Basis Average

The rolling basis is calculated as a weighted average between the first contract’s basis and the forward yield:

$$ basis^{rolling} = {{basis_{1} \times \tau_{1} + yield^{fwd} \times ( \tau_{target} - \tau_{1}})\, \over \tau_{target}} $$