Futures, Carry, and Basis

Futures are where a huge share of quant trading actually happens — index, rates, energy, FX — and their pricing runs on one idea already established in this curriculum: replication by carry. This lesson extends it to term structures, roll yield, and the basis trades desks live on.

Cost of carry, completed

The no-arbitrage forward from the options course, with the full carry ledger:

$$F = S \cdot e^{(r + u - q - y)T}$$

— financing $r$ and storage $u$ push the forward up; dividends/coupons $q$ and convenience yield $y$ (the value of holding the physical thing — an oil refiner can't run on futures) pull it down. Each market emphasizes different terms: equity index futures ≈ financing minus dividends (tight arbitrage, the basis trades within basis points); commodities carry storage and convenience (wide, economically meaningful deviations); FX forwards are pure interest differential — covered interest parity, the cleanest arbitrage relation in finance (and its small persistent post-2008 violations, driven by balance-sheet costs, are a famous "free lunch that isn't" — worth one knowing sentence).

Contango, backwardation, and roll yield

Contango: futures curve upward-sloping (far > near) — normal when carry costs dominate. Backwardation: downward — scarcity now, high convenience yield. The P&L consequence is roll yield: a long futures position held in contango repeatedly sells the cheap expiring contract and buys the richer next one, bleeding even if spot goes nowhere — the mechanism behind the long-run decay of long-commodity ETPs, and the exact same curve mathematics as the VIX-futures decay in the variance lesson (short-vol ETPs are a contango-roll story). In backwardation the roll pays you. The general statement interviewers want: a futures position's return = spot return + roll yield, and ignoring the second term misreads entire asset classes.

Basis and basis risk

The basis — futures minus (fair-value) spot — is a mean-reverting spread (a stat-arb lesson object) bounded by arbitrage costs. Trading it is a core desk activity: index arbitrage (futures vs the basket), cash-and-carry in commodities, and the giant Treasury cash-futures basis trade (bond vs future, levered through repo — whose crowded unwind in March 2020 is the standard modern example of a "riskless" spread becoming a liquidity event; the crises lesson will pick this thread up). Basis risk is the hedger's residual: hedging jet fuel with crude futures, or a stock portfolio with index futures, leaves the spread between your exposure and your instrument — smaller than outright risk, but concentrated and correlation-dependent (the portfolio lesson's warning again: those correlations widen under stress).

Operational facts that mark familiarity: futures are exchange-cleared, margined daily (mark-to-market cash flows, so P&L timing differs from forwards), quoted in ticks with contract multipliers, and most positions roll before delivery — nobody wants the barrels.

The interview version

"Index future trades at 5020, spot 5000, rates 5%, dividend yield 1%, 3 months to expiry — anything to do?" — Fair forward: $5000 \times e^{(0.04)(0.25)} \approx 5050$; the future is ~30 points cheap: buy futures, short the basket, collect the convergence (mind financing and transaction costs before declaring free money). "Why do long-oil ETFs lose money in flat markets?" — contango roll, two sentences. "You hedge a corporate bond portfolio with Treasury futures — what's left?" — credit-spread risk plus basis risk, named separately. Carry ledger, curve shape, residual risk: three habits, every futures question.