Chapter 3 of 8

Basis, cost of carry and convergence

Understand the relationship between spot and futures, financing, income, storage and the risk of assuming an automatic arbitrage.

3 min read · Original educational chapter

In this chapter

State the sign convention
A simplified theoretical relationship
Convergence is contract-dependent
Decision lab: Convergence does not remove carrying uncertainty
Worked example
Common mistakes
Check your understanding

State the sign convention #

One common definition of basis is futures price minus spot price. Some discussions reverse that sign, so always state the convention. A positive basis under this definition means futures trade above spot; it does not by itself predict that spot must rise.

The relationship reflects financing, expected income, storage or availability, contract terms and supply–demand conditions. A dividend-paying equity exposure and a physical metal have different carrying components.

A simplified theoretical relationship #

For an idealized continuously financed asset with a continuous income yield, a common model is F = S × exp((r − q) × T). S is spot, r financing rate, q income yield and T years. Commodity models may also incorporate storage and convenience yield. Real execution, funding restrictions and taxes can create deviations.

A quoted premium over spot is not automatically a free profit. A cash-and-carry transaction may require buying the asset, financing it, selling the future and managing custody, corporate actions, settlement and all costs.

Convergence is contract-dependent #

As a contract approaches its relevant settlement point, its relationship to the deliverable or settlement reference tightens through the settlement mechanism. This is not a guarantee that two screens with different timestamps or qualities must show identical prices.

A study must use matching units, timestamps, delivery locations, quality and currency. A synthetic continuous futures series can add roll adjustments that are not directly comparable to a spot quote. Basis risk is central when the exposure being hedged differs from the contract.

Decision lab: Convergence does not remove carrying uncertainty #

A basis trade involves financing, contract terms and the ability to carry both legs. A favourable-looking spot/futures difference may be consumed by funding, storage, borrow constraints, taxes or execution. A theoretical convergence relation is not automatically an executable arbitrage for every participant.

Build an all-in cost bridge and specify delivery or exit mechanics. Stress the funding rate and a failure to obtain the intended underlying. Separate theoretical fair-value inputs from prices at which both legs can actually trade at the same time.

Hypothetical worked example

Carry is not a directional forecast

With fictional spot 1,000, financing rate 8%, income yield 2% and half a year, the simplified model gives 1,000 × exp(0.06 × 0.5) ≈ 1,030.45.

The ₹30.45 premium results from the model’s assumptions. It does not say that spot will be 1,030.45 in six months. Changing financing, income or delivery assumptions changes the comparison.

For a physical commodity, adding storage and changing convenience yield can alter the relationship further. A real arbitrage analysis must include both legs’ executable prices and the ability to complete settlement.

Common mistakes

Reading a futures premium as an unconditional price forecast.
Comparing unmatched timestamps, purity or delivery locations.

Check your understanding

Under the stated convention, basis equals what?

Make this chapter useful

Can you defend the reasoning?

Before moving on, answer the self-check in your own words and write why each alternative fails. Use the worked example to test that explanation. Then apply it in the module research task; keep the assumptions and the evidence that could change your answer.

Sources and further reading

Further-reading references from the curriculum source list. Contract sizes, expiry calendars, settlement references, margin parameters and tax treatment can change; verify the current operative documents for the actual product. Worked figures are illustrative, not current quotes.

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General financial education. No securities recommendation, assured outcome or personal advice. Read the risk disclosures.