7.3 Basis and convergence · M07 Forwards and futures

Motivation

A hedger never gets the spot price or the futures price; she gets the futures price she locked in, adjusted by the basis on the day she lifts the hedge. Basis is also the cleanest real-time gauge of funding stress in the index market.

7.3.1Convergence and the hedger's price#

Proposition 7.7convergence and basis

(a) At expiry $F_T = S_T$ , so $b_T = 0$ . (b) Under the cost-of-carry model,

b_t = S_t\left(e^{(r-q)\tau} - 1\right) = S_t(r-q)\tau + O(\tau^2), \tag{7.9}

which shrinks roughly linearly to zero as $\tau \to 0$ . (c) A hedger who opens a futures hedge at price $F_1$ at time $t_1$ and closes it at $t_2$ , while transacting in the spot market at $t_2$ , realises an effective price of

F_1 - b_2 = F_1 + (S_2 - F_2), \tag{7.10}

whether she is a short hedger selling the asset or a long hedger buying it.

Proof

(a) For a physically settled contract in its delivery instant, if $F_T > S_T$ one buys spot, sells futures and delivers at once for a riskless profit $F_T - S_T$ ; if $F_T < S_T$ one buys futures, takes delivery and sells spot. Hence $F_T = S_T$ up to transaction costs. For a cash-settled index future, $F_T$ is defined as the final settlement price, which is the index close on expiry day, so $b_T = 0$ by contract. (b) Substitute (7.1) at time $t$ and expand the exponential. (c) Short hedger: sells the asset at $S_2$ and gains $F_1 - F_2$ on the short futures; total $S_2 + F_1 - F_2 = F_1 - b_2$ . Long hedger: buys at $S_2$ and gains $F_2 - F_1$ on the long futures; net cost $S_2 - (F_2 - F_1) = F_1 - b_2$ .∎

7.3.2Basis risk#

In Hull's convention ( $b = S - F$ ) equation (7.10) reads $F_1 + b_2$ . The economic content is the same: $F_1$ is known at $t_1$ , and the only uncertainty left is the basis at close-out. This uncertainty is basis risk. It is zero if the hedge is lifted at expiry and the asset hedged is exactly the futures' underlying. It is positive whenever the hedge horizon differs from the expiry, or the asset differs from the underlying (a cross hedge, e.g. a stock portfolio hedged with Nifty futures, or aviation turbine fuel hedged with crude).

7.3.3Reading the Nifty basis#

Worked example 7.7term structure of the Nifty basis, 5 Nov 2026

From the table in 7.1.6, the basis per day to expiry is $51.40/18 = 2.86$ , $182.50/54 = 3.38$ and $298.20/81 = 3.68$ points per day. Under the cost-of-carry model with constant $r-q$ , the three ratios would be nearly equal ( $S(r-q)$ , about 3.0–3.4 points a day for $r - q$ between 4.5% and 5%). The rising pattern says the market's implied carry rises with maturity, from 4.24% to 5.44% a year. One reading is a liquidity effect: the near month is where hedgers and arbitrageurs concentrate, so its premium is competed down hardest.

Practitioner notes

EOD basis is noisy. The futures settlement price is a half-hour VWAP of the futures, and the index close is computed from the constituents' own closing prices. On a volatile day the two describe slightly different moments, so a "basis" from EOD files can differ from any basis that was tradable.
A calendar spread (long one expiry, short another) is a pure bet on the change in carry between two dates. Its P&L is a basis P&L, not an index P&L.
On expiry day, the basis of the expiring contract must go to zero by the close. Anything that moves the closing index, legitimately or otherwise, moves the final settlement of every expiring contract. That is why expiry-day closing mechanics are heavily regulated (M01; M08 · 8.12 on the 2025 SEBI interim order).