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Commodity Futures Storage And Carry Cost Modeling

commodity-futures-storage-and-carry-cost-modelingsource

Use when pricing physical commodity futures or designing roll strategies, linking spot to futures through financing, storage and insurance, extracting implied convenience yield and classifying contango against backwardation.

Domain
Multi-Asset Derivatives
Version
1.1.0
Reading
5 min
Hands off to
5
Handed off from
3
License
Apache-2.0
CoversGeneric Derivatives Pricing

When to Use

Use this skill when pricing physical commodity futures (Crude Oil CL, Natural Gas NG, Gold GC, Agriculture ZC) or designing term-structure roll strategies. The Cost of Carry model links spot prices ($S_0$) to futures prices ($F_T$) using financing costs ($r$), physical storage/insurance costs ($c$ proportional and/or $U$ per unit), and implied convenience yield ($y$). High convenience yield causes Backwardation ($F_T < S_0$), signaling physical inventory scarcity, whereas high storage costs relative to convenience yield lead to Contango ($F_T > S_0$).

When NOT to Use

  • As an arbitrage signal generator on the cheap side of the curve. For a consumption commodity the cost-of-carry relation is an inequality, $F_0 \le (S_0 + U)e^{(r+c)T}$, not an equality. Only the rich side is enforceable. A futures price below your fair-value estimate is a view about convenience yield, not a riskless trade, because you cannot generally borrow and sell short a physical commodity.
  • For non-storable commodities (electricity, most weather and freight underlyings). Storage arbitrage is the entire basis of this model; without storability the futures price is a risk-neutral expectation, not a carry relation. See weather-derivatives-and-niche-instrument-handling.
  • When spot and futures are not the same deliverable. A refiner's local crude assessment is not the contract-grade deliverable at the delivery point. A basis difference between grades or locations shows up here as a spurious convenience yield or a spurious arbitrage.
  • At sub-daily maturities. The implied-yield inversion divides by $T$, so within a day or two of expiry ordinary quote noise annualises into implausible yields.

Prerequisites

  • Spot price $S_0$ and futures contract price $F_{market}$ sampled at the same timestamp, for the contract-deliverable grade and delivery point.
  • Time to maturity $T$ in years, on a stated day-count basis.
  • A continuously compounded annual financing rate $r$ on that same day-count basis. Money-market quotes (e.g. SOFR, ACT/360, simple) must be converted before use.
  • Storage cost as a proportional annual rate $c$, a fixed currency amount per unit per year $U$, or both.

Workflow

  1. Full-Carry Price (the no-arbitrage bound):
    • $F_{full} = (S_0 + U_{PV}) \cdot e^{(r + c) T}$, where $U_{PV}$ is the present value of the fixed per-unit storage charge accruing over $[0, T]$.
    • This is the upper bound on the futures price, and equals the price at $y = 0$.
  2. Theoretical Futures Price at an Assumed Yield:
    • $F_{theoretical} = F_{full} \cdot e^{-yT}$. Because $y$ is unobservable, this is a view, not a fair value that arbitrage enforces.
  3. Implied Convenience Yield Extraction:
    • $y = \frac{1}{T}\ln\left(\frac{F_{full}}{F_{market}}\right)$.
    • If $y < 0$ the bound is violated. Before treating it as profit, re-check timestamp synchronisation, grade/location deliverability, and whether storage and financing costs are understated — those explain the great majority of apparent violations.
  4. Regime Identification:
    • $F_{market} > S_0$ is CONTANGO; $F_{market} < S_0$ is BACKWARDATION; equality is FLAT. Do not fold the equality case into either regime — a flat curve is a distinct, informative state.
  5. Arbitrage Audit (Cash-and-Carry only):
    • Raise a CASH_AND_CARRY signal only when $F_{market} > F_{full} \cdot (1 + \text{round-trip costs})$: buy spot, pay financing and storage, sell futures, deliver. This leg is executable by anyone with capital and storage capacity.
    • Do not raise an arbitrage on the other side. A cheap futures price is surfaced separately as a reverse-carry candidate, actionable only by an existing inventory holder (who is really monetising their own convenience yield) or in a commodity with a genuine lease/borrow market such as gold.

Full procedure: see references/workflows.md. Standards reference: see references/standards.md. Printable pre-flight checklist: see assets/checklist.md.

Common Pitfalls

  • Treating backwardation as a reverse cash-and-carry arbitrage: Crude oil routinely trades at implied convenience yields of tens of percent annualised during tight-inventory periods. A model that compares the market price to a fixed "baseline" convenience yield and flags every deviation will fire a false arbitrage on essentially every backwardated market, because the short-physical leg needed to capture it does not exist.
  • Ignoring Convenience Yield ($y$): Assuming futures prices are purely driven by $r + c$. In tight physical markets, convenience yield surges, causing deep backwardation that pure storage models fail to explain.
  • Fixed vs. Proportional Storage Costs: Exchanges regulate physical storage as a fixed charge per unit per day, not as a percentage of spot — CBOT caps grain storage in fractions of a cent per bushel per day and adjusts that cap through the Variable Storage Rate mechanism. Modelling a fixed charge as a percentage of spot silently makes storage cheap when the commodity is cheap, which is exactly backwards.
  • Day-Count / Compounding Misalignment: $T$ and $r$ must share a day-count basis, and $r$ must be continuously compounded. Pairing an ACT/360 simple money-market quote with an ACT/365 $T$ biases the implied yield by roughly 1.4% of the rate before any market signal is present.
  • Silent NaN propagation: nan <= 0 is False, so a naive positivity check passes NaN straight through to math.log and returns a NaN price alongside a confidently wrong regime string. Validate for finiteness, not just sign.
  • Non-synchronous quotes: A settlement-price futures quote against a live spot tick manufactures basis out of nothing. At short maturities the $1/T$ factor amplifies it into a headline-grade convenience yield.

Verification

  • Instantiate CommodityCarryCostModel(risk_free_rate=0.05, storage_cost_rate=0.02). With $S_0 = 100$, $T = 1.0$, $y = 0.01$, verify the theoretical price is $100 \cdot e^{0.06} \approx 106.1837$; with $y = 0.10$, verify $100 \cdot e^{-0.03} \approx 97.0446$. The full-carry price ($y = 0$) must be $100 \cdot e^{0.07} \approx 107.2508$.
  • Feed a WTI-like backwardated curve ($S_0 = 80$, $F = 76$, $T = 0.5$). Confirm the regime is BACKWARDATION, the implied convenience yield is roughly 17%, and is_arbitrage_opportunity is False — this is a normal market, not a trade.
  • Feed $F_{market} = 115$ against $S_0 = 100$, $T = 1.0$. Confirm CASH_AND_CARRY, a negative implied yield, and convenience_yield_bound_violated.
  • Feed $F_{market} = S_0$ and confirm the regime is FLAT, not BACKWARDATION.
  • Run python -m unittest discover -s skills/commodity-futures-storage-and-carry-cost-modeling/scripts and confirm 100% pass rate.

Verify it, from the repository root

python -m unittest discover -s skills/commodity-futures-storage-and-carry-cost-modeling/scripts

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