Learn about basis and cost of carry

Overview

The basis and cost of carry link spot and futures prices through no-arbitrage logic. Understanding them is essential for pricing, hedging, and spoting arbitrage opportunities.

Basis: The Core Concept

WHY this definition? We want to measure how much futures deviate from current reality. Basis is the mispricing signal for arbitrage.

HOW it evolves: As expiration approaches, basis → 0 because futures and spot must converge (no difference between "deliver now" and "deliver in zero time"). This convergence is called basis convergence.

Cost of Carry: The Fair-Value Formula

$\text{Cost of Carry} = (r + U - y) \cdot S_0 \cdot T$

where $r$ = risk-free rate, $T$ = time to expiration, $y$ = dividend yield or convenience yield.

Derivation: Futures Price from First Principles

GOAL: Find $F_0$ such that no arbitrage exists.

Step 1: Set up two equivalent strategies

Strategy A: Buy the futures contract (costs $0$ today, pay $F_0$ at expiry, receive asset). Strategy B: Borrow money, buy spot asset now at $S_0$, hold until expiry, pay storage and financing, collect dividends.

At expiry, both strategies deliver the same asset. By no-arbitrage, they must cost the same today.

Step 2: Cost of Strategy B

• Borrow $S_0$ at rate $r$ → repay $S_0 e^{rT}$
• Storage cost $U \cdot S_0 \cdot T$ (proportional model)
• Receive dividends worth $D$ (or yield $y \cdot S_0 \cdot T$)

Net outflow at expiry: $S_0 e^{rT} + U \cdot S_0 \cdot T - D$

Step 3: Equate to futures

Futures price $F_0$ is the forward commitment price. No-arbitrage: $F_0 = S_0 e^{rT} + U \cdot S_0 \cdot T - D$

Simplify (continuous compounding, yield model): $F_0 = S_0 e^{(r + U - y)T}$

WHY exponential? Continuous compounding. For small $T$, approximate: $F_0 \approx S_0 \left[1 + (r + U - y)T\right]$

$F_0 = S_0 e^{(r + u - y)T}$ (for commodities: $u$ = storage cost rate, $y$ = convenience yield)

Worked Examples

Find: Fair futures price.

Solution: $F_0 = S_0 e^{(r - y)T} = 18{,}000 \cdot e^{(0.06 - 0.015) \cdot 0.25}$ $= 18{,}000 \cdot e^{0.0125} = 18{,}000 \cdot 1.0113 = 18{,}203$

Why this step?

• We subtract dividend yield because holding the spot asset earns dividends, reducing net carry cost.
• Exponential because we assume continuous compounding.

Basis: $\text{Basis} = 18{,}000 - 18{,}203 = -203\quad \text{(contango)}$

Negative basis: futures > spot because cost of financing exceds dividend income.

Solution: $F_0 = S_0 e^{(r + u)T} = 60{,}000 \cdot e^{(0.05 + 0.02) \cdot 0.5}$ $= 60{,}000 \cdot e^{0.035} = 60{,}000 \cdot 1.0356 = 62{,}136$

Why add storage? Physical commodities cost money to store. This increases carry cost, pushing futures higher than spot.

Basis: $\text{Basis} = 60{,}000 - 62{,}136 = -2{,}136 \quad \text{(contango)}$

Arbitrage strategy (Cash-and-Carry):

1. Today: Borrow ₹18,000 at 6%, buy Nifty spot at 18,000, sell futures at 18,500.
2. Hold 3 months: Collect dividends worth $18{,}000 \cdot 0.015 \.25 = 67.5$.
3. At expiry: Deliver spot against futures, receive18,500. Repay loan $18{,}000 \cdot e^{0.06 \cdot 0.25} = 18{,}271$.

Profit: $18{,}500 + 67.5 - 18{,}271 = 296.5 \quad \text{(risk-free)}$

Why this works? Market futures price exceds fair value. Buy cheap (spot), sell expensive (futures), lock in spread.

Common Mistakes

Why it feels right: Looks like simple interest compounding.

The fix: Stocks pay dividends. Holding spot earns you income, so futures price is lower than pure financing cost suggests. Always subtract $y$: $F_0 = S_0 e^{(r - y)T}$

Why it feels right: "Positive" sounds like "higher."

The fix: $\text{Basis} = S_0 - F_0$ Positive basis: spot is higher (backwardation). Negative basis: futures are higher (contango). Remember: basis is spot-centric.

Why it feels right: Static thinking.

The fix: Basis converges to zero at expiry. The rate of convergence depends on changing $r$, $y$, time decay. Basis risk arises because convergence is not linear.

Active Recall Checks

Because if you buy today, you tie up ₹80,000 (could have earned interest in a bank). Plus, Apple has to store the phone for you (costs them money). So the 3-month price is higher by the cost of waiting: interest you lose + storage cost. That's the "cost of carry."

If Apple offered₹79,000 for future delivery, everyone would sell their iPhones today and buy the cheaper future one—arbitrage! The market forces the future price to equal today's price plus carry cost.

"Back-ward-ation = Back to reality" → Spot higher (market expects prices to fall, or scarcity now).

Connections

• Introduction-to-futures-contracts – Basis is why futures ≠ spot
• Futures-pricing-vs-spot-pricing – Cost of carry is the bridge
• Hedging-with-futures – Basis risk makes hedges imperfect
• Arbitrage-strategies-cash-and-carry – Exploits mispriced basis
• Backwardation-and-contango – Basis sign defines market structure
• Marking-to-market – Daily settlement affects carry returns
• Commodity-futures – Storage and convenience yield dominate

#flashcards/stock-market

What is the formula for basis? :: Basis = Spot Price - Futures Price = $S_0 - F_0$

Derive the fair futures price for a stock with dividend yield $y$ :: $F_0 = S_0 e^{(r - y)T}$ — buy spot (cost $S_0$), finance at $r$, earn dividends $y$, no-arbitrage equates to futures price

Concept Map

Hinglish (regional understanding)