Index futures basis: cheap or just illiquid?
A dealer on the institutional derivatives desk is reviewing a roll request from a corporate treasury client. The client holds 400 long Nifty 50 futures contracts, lot size 25, and wants to roll them to the three-month contract. Market data at 11:05 a.m.: Nifty 50 spot 24,650; near-month futures 24,720; three-month futures 24,980; annualised risk-free rate 6.75%; expected dividend yield 1.10%. The dealer's cost assumptions: brokerage 0.02% of notional per leg, exchange transaction charge 0.002% per leg, STT 0.0125% on the sell side of futures, bid-ask impact 0.05 index points per leg, and initial margin 12% of notional. Excess cash collateral earns 5.50% while the desk funds at 6.75%. The client asks whether the three-month futures are cheap and whether the desk can arbitrage the difference. The desk only executes arbitrage if expected net edge exceeds ₹250 per lot after all costs and financing.
1. Compute the fair three-month futures price and quantify the gross mispricing. Is the contract cheap in a tradeable sense?
Read the worked answer
Use the cost-of-carry relationship for an index future:
F = S × e^(r − q)T
where S = 24,650, r = 6.75%, q = 1.10%, and T = 0.25 years.
r − q = 5.65% = 0.0565 Exponent = 0.0565 × 0.25 = 0.014125 e^0.014125 ≈ 1.014225 Fair futures = 24,650 × 1.014225 ≈ 25,000.6
The observed three-month futures price is 24,980, so it is about 20.6 index points below fair value. That is roughly 0.08% of spot. Per lot, the gross gap is 20.6 × 25 ≈ ₹515.
So the contract is 'cheap' only on a gross basis. The gap is small relative to the bid-ask, brokerage, STT and financing costs. Also, the continuous dividend yield is an approximation: if the index has lumpy dividend flows in the next three months, the fair value itself could shift by more than 20 points. The conclusion is that the contract is marginally below model value, not necessarily mispriced in an exploitable way.
2. Itemise the costs of the reverse cash-and-carry (buy futures, sell the index basket) for one lot and decide whether the desk should trade.
Read the worked answer
The reverse cash-and-carry would buy one futures lot and sell the equivalent index basket. For one lot (25 units):
- Spot-leg brokerage: 0.02% × 24,650 × 25 = ₹123.25
- Futures-leg brokerage: 0.02% × 24,980 × 25 = ₹124.90
- Exchange charges: 0.002% × (24,650 + 24,980) × 25 = ₹24.82
- STT on futures sale: 0.0125% × 24,980 × 25 = ₹78.06
- Bid-ask impact: 0.05 × 25 × 2 legs = ₹2.50
- Margin financing: 12% × 24,980 × 25 = ₹74,940. The desk funds at 6.75% but collateral earns 5.50%, a 1.25% spread. For 0.25 years: 1.25% × 0.25 × 74,940 = ₹234.19
Total identified costs ≈ ₹587.72. Gross edge ≈ ₹515.60. Net edge ≈ −₹72.12 per lot, before any securities-borrowing fee, dividends owed on the short basket, or SLB scarcity premium.
The desk's threshold is ₹250 per lot. The trade fails on cost even before the practical problem: shorting the index basket requires borrowing securities, which may be unavailable or expensive. The correct decision is no arbitrage; the desk should execute the client's roll as an agency order and not put on a proprietary reverse cash-and-carry.
3. Why might the three-month futures trade below the calculated fair value even when the desk's arithmetic is correct? Identify three frictions and explain how each affects the desk's decision.
Read the worked answer
Three plausible frictions are:
Dividend timing and measurement. The fair value uses a continuous 1.10% dividend yield. If actual dividends are lumpy, or if the market expects higher dividends than the yield implies, the fair value falls. A 20-point gap is small enough to be explained by dividend timing alone, so the desk should not treat the model output as an arbitrage signal unless it has the dividend calendar.
Short-selling and SLB constraints. The reverse cash-and-carry requires borrowing the index basket. If the basket is hard to borrow, the SLB fee and recall risk can exceed the ₹515 gross edge. If borrowing is unavailable, the arbitrage is not executable at all, so the mispricing can persist.
Balance-sheet and financing costs. The desk funds margin at 6.75% while excess cash earns 5.50%. Higher funding costs or a higher margin requirement widen the no-arbitrage band. A small basis below fair value is therefore consistent with the desk's own cost of capital.
A fourth practical friction is the client's own roll flow: the corporate treasury is selling near-month and buying three-month futures, which can move the far contract. The desk should separate client execution from proprietary arbitrage and document why it did not trade the basis.