Understand spot vs futures pricing in commodities — Stock-Market notes

Overview

When trading commodities like oil, gold, or wheat, you encounter two fundamental prices: the spot price (buy/sell RIGHT NOW) and the futures price (locked-in price for delivery LATER). Understanding their relationship unlocks arbitrage opportunities, hedging strategies, and market sentiment reading.

Figure — Understand spot vs futures pricing in commodities

Core Concepts

The Cost-of-Carry Model (Derivation from Scratch)

Starting Point: The No-Arbitrage Principle

Core idea: If futures price deviates too far from spot + carry costs, arbitrageurs will force prices back in line by simultaneously buying cheap and selling expensive.

Let's derive the fair futures price step-by-step:

Formula Cost-of-Carry Formula (Derived)

Given:

$S_0$ = Spot price today
$F$ = Futures price for delivery at time $T$
$r$ = Risk-free interest rate (annual, continuously compounded)
$c$ = Storage cost per unit per year (as % of spot, continuously compounded)
$y$ = Convenience yield (benefit of holding physical, as % per year, continuously compounded)
$T$ = Time to maturity (in years) Derivation (all costs treated as continuous rates for consistency):

Strategy A (Synthetic Futures): Buy spot commodity now + hold until $T$
- Cost today: $S_0$
- Finance $S_0$ at continuous rate $r$ → the debt grows to $S_0 e^{rT}$ at time $T$
- Storage is charged as a continuous rate $c$ on the value, so paying it via financing turns the total into $S_0 e^{(r+c)T}$ (interest and storage compound together the same way)
- Total cost at time $T$ : $S_0 e^{(r+c)T}$
Why treat storage as a continuous rate? For consistency with the financing term. If instead you model storage as a discrete lump-sum $U$ paid today, the formula becomes $F = (S_0 + U)e^{rT}$ . Both are valid; we pick the continuous-rate convention throughout this note so every term ( $r$ , $c$ , $y$ ) lives in the same exponent. The earlier " $S_0 \cdot c \cdot T$ " is only the first-order Taylor approximation of $S_0(e^{cT}-1)$ for small $cT$ —we do not mix it with the exponential form.
Strategy B (Direct Futures): Buy futures contract at $F$ , pay at delivery
- Cost at time $T$ : $F$
No-Arbitrage Condition: If $F > S_0 e^{(r+c)T}$ , everyone would do Strategy A (buy spot, store, deliver against futures → riskless profit). If $F < S_0 e^{(r+c)T}$ , holders would sell spot, invest proceeds, buy futures back.

Including Convenience Yield (benefit of holding physical, like emergency access):

$F = S_0 e^{(r + c - y)T}$

Why this step? Convenience yield $y$ REDUCES the futures price because physical holders get a benefit (optionality to use immediately, avoid stockouts) that futures holders don't. Think of $y$ as a "dividend" from holding the commodity.

Simplified for small $T$ (linear approximation when $T < 0.5$ years): $F \approx S_0 \left(1 + (r + c - y)T\right)$

Why Each Term Matters

$r$ (interest rate): Money to buy spot could earn risk-free return → futures must compensate
$c$ (storage): Physical commodities need warehouses, insurance → futures must cover this
$y$ (convenience yield): Oil refiners pay a premium to have oil on hand NOW vs. a futures contract → reduces futures price

Contango vs Backwardation

Worked Examples

Worked example Example 1: Calculating Fair Futures Price for Crude Oil

Given:

Spot WTI crude: $S_0 = 75$ USD/barrel
Annual risk-free rate: $r = 4\% = 0.04$
Annual storage cost: $c = 2\% = 0.02$ of spot value
Convenience yield: $y = 1\% = 0.01$
Time to delivery: $T = 0.5$ years (6 months) Find: Fair 6-month futures price.

Solution: $F = S_0 e^{(r + c - y)T} = 75 \cdot e^{(0.04 + 0.02 - 0.01) \cdot 0.5}$ $F = 75 \cdot e^{0.025} = 75 \cdot 1.0253 = 76.90 \text{ USD/barrel}$

Why this step? We compound the net carry cost (5% annual = 4% interest + 2% storage - 1% convenience) over half a year. The exponential accounts for continuous compounding.

Interpretation: If market quotes 6-month futures at $78, it's overpriced → arbitrage opportunity (buy spot + store, sell futures).

Worked example Example 3: Arbitrage Calculation

Market anomaly (using the continuous-rate convention throughout):

Gold spot: $2000/oz
1-year gold futures: $2100/oz
Risk-free rate: $r = 3\%$ (continuous)
Storage cost: $c = 0.5\%$ per year (continuous rate)
Gold has zero convenience yield ( $y = 0$ )

Fair futures price: $F = 2000 \cdot e^{(0.03 + 0.005) \cdot 1} = 2000 \cdot e^{0.035} = 2071.23$

Observed futures: $2100 (overpriced by$ 2100 - 2071.23 = 28.77$!)

Arbitrage strategy (fully consistent — finance BOTH the gold and the storage at $r$ ):

Today: Borrow $2000 at 3\% (continuous), buy 1 oz spot gold, sell 1-year futures at$ 2100. Storage is charged continuously at $c$ on the position and financed at $r$ alongside the gold.
In 1 year: Deliver gold against futures → receive $2100
Repay all financing (gold + storage compounded together): $2000 \cdot e^{(0.03 + 0.005) \cdot 1} = 2071.23$
Profit: $2100 - 2071.23 = **$ 28.77 riskless profit**

Why this step? By financing the storage cost at the same continuous rate $r$ inside the exponent, the total repayment is exactly the fair futures price $S_0 e^{(r+c)T}$ . The arbitrage profit is therefore precisely observed futures − fair futures = $28.77$ , with no leftover discrepancy from mixing discrete and continuous conventions.

Common Mistakes & Steel-Manning

Active Recall Practice

Recall Feynman: Explain to a 12-year-old

Imagine you want to buy a PlayStation 5, but the store says they're sold out right now and you can only get one in 3 months. They offer you two deals:

Wait until they restock and pay whatever the price is then (spot price in 3 months)
**Pay $520 TODAY to lock in your PS5 for 3-month delivery**, even though today's cash price is$ 500 (futures price)

Why would deal 2 cost $520 when PS5s are$ 500 today? Because the store has to keep your PS5 in a warehouse (storage cost) and they could've sold it and invested that $500 (interest cost). This is contango—the locked-in future price is ABOVE today's spot.

Now flip it. Imagine a crazy holiday shortage where everybody needs a PS5 RIGHT NOW. The spot price shoots up to $600 today. But everyone also knows a huge new shipment lands in 3 months, so the store will only promise you$ 560 for 3-month delivery. This is backwardation—the future price ( $560) is BELOW today's spot ($ 600), because having the console immediately is worth a premium and future supply will be plentiful.

Commodities like oil and gold work the same way. Spot = right now. Futures = locked-in future price. Contango = future price higher; backwardation = future price lower.

Memory Aids

Connections

Commodity Trading Strategies - How to exploit spot-futures spreads
Hedging with Futures - Producers/consumers use futures to lock in prices
Cost of Carry Arbitrage - Riskless profit from mispricing
Convenience Yield in Oil Markets - Why backwardation dominates energy
Contango Bleed in Commodity ETFs - Rolling futures destroys returns
Basis Risk - Spot-futures spread doesn't always converge perfectly
Normal Backwardation Theory - Keynes' view on futures risk premium
Gold vs Oil Futures Curves - Storage costs determine curve shape

Flashcards

#flashcards/stock-market

What is the spot price in commodities? :: The current market price for immediate delivery (within 2 business days) of a physical commodity. It reflects real-time supply-demand.

What is a futures price in commodities?

A standardized contract price agreed upon today for delivery of a commodity on a specific future date. You lock the price now, settle later.

Define contango

When futures price > spot price. The futures curve slopes upward. Typical for cheaply-storable commodities where carry costs exceed convenience yield.

Define backwardation

When futures price < spot price. The futures curve slopes downward. Occurs frequently (not rarely) when immediate demand is high or convenience yield dominates carry costs.

Cost-of-carry formula for futures pricing

F = S_0 e^{(r + c - y)T}

where

r

= risk-free rate,

c

= storage cost (as continuous rate),

y

= convenience yield,

T

= time to maturity.

Why does the risk-free rate

r

increase futures price?

Capital used to buy spot commodity could earn risk-free return. Futures must compensate for this opportunity cost, pushing futures price above spot.

What is convenience yield

y

The benefit of holding a physical commodity (immediate access, avoiding stockouts, operational flexibility). It REDUCES futures price because futures holders don't get this benefit.

If gold spot =

2000, 1-year futures =

2080, risk-free rate = 3%, storage = 1%, what is implied convenience yield?

F = S_0 e^{(r+c-y)T}

→

2080 = 2000 e^{(0.03 + 0.01 - y)}

→

1.04 = e^{0.04- y}

→

0.04 - y = \ln(1.04) = 0.0392

→

y = 0.08\%

(almost zero, typical for gold).

Why is agricultural commodity backwardation common before harvest?

Current supply is tight (high spot), but harvest will flood supply soon (lower futures). Immediate demand > expected future demand.

What is roll yield?

The profit/loss from closing an expiring futures contract and opening the next month's contract. Negative in contango (buy higher), positive in backwardation (buy lower).

When is arbitrage possible between spot and futures?

When

F \neq S_0 e^{(r+c-y)T}

. If futures are too high, buy spot + store + sell futures. If too low, sell spot + invest + buy futures.

Does futures price predict future spot price?

No. Futures price = spot + cost of carry - convenience yield. It's a no-arbitrage price, not a forecast. Use fundamentals to predict spot.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, yahan pe do prices ki baat ho rahi hai jo har commodity trade karte time samne aati hai. Spot price matlab abhi ka price - agar tumhe oil ya gold aaj hi chahiye, aaj hi delivery, toh jo rate pay karoge woh spot hai. Futures price matlab tum aaj hi ek future date ke liye price lock kar dete ho, delivery baad me hogi. Ab sawaal yeh aata hai ki futures price zyada kyun hota hai spot se? Iska simple reason hai "cost of carry" - agar tum aaj oil kharid ke 6 mahine store karte ho, toh tumhe storage, insurance, aur apne paise ka interest bhi lagta hai. Yeh saara kharcha spot price me add ho jata hai, isliye futures usually thoda upar hota hai.

Ab isme ek interesting cheez hai jise backwardation kehte hain - jab futures price spot se kam ho jaata hai. Yeh tab hota hai jab market ko abhi turant commodity chahiye, future me nahi. Jaise crude oil ya natural gas me jab immediate demand tight ho jaati hai, toh yeh situation kaafi common hai. Iske peeche jo main logic kaam karta hai woh hai no-arbitrage principle - matlab agar futures price apni fair value se bahut door chala jaaye, toh arbitrageurs foran cheap wali cheez khareed ke expensive wali bech denge, aur prices wapas line me aa jayengi. Isi se cost-of-carry formula banta hai: $F = S_0 e^{(r+c-y)T}$ , jisme $r$ interest rate, $c$ storage cost, aur $y$ convenience yield hai.

Yeh samajhna important isliye hai kyunki iske through tum market ka sentiment padh sakte ho - contango (futures upar) matlab market future ke liye relaxed hai, aur backwardation matlab abhi shortage ki tension hai. Saath hi yeh hedging aur arbitrage strategies ka foundation hai. Ek chhoti si tip: is note me saare costs continuous rate ( $e$ wale form) me liye gaye hain taaki consistency bani rahe - storage ko lump-sum me mat mix karo exponential ke saath, warna galti ho jayegi. Yeh conceptual clarity aage jaake derivatives aur real trading dono me kaam aayegi.

Test yourself — Commodities, Forex & Crypto

Connections