Learn about basis and cost of carry

5.1.6 · Stock-Market › Futures

Overview

Basis aur cost of carry spot aur futures prices ko no-arbitrage logic ke through link karte hain. Inhe samajhna pricing, hedging, aur arbitrage opportunities dhundhne ke liye zaroori hai.

Basis: The Core Concept

WHY this definition? Hum measure karna chahte hain ki futures current reality se kitna deviate kar raha hai. Basis ek mispricing signal hai arbitrage ke liye.

HOW it evolves: Jaise-jaise expiration paas aata hai, basis → 0 kyunki futures aur spot ko converge karna hi hoga ("deliver now" aur "deliver in zero time" mein koi farq nahi). Is convergence ko basis convergence kehte hain.

Cost of Carry: The Fair-Value Formula

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

jahan $r$ = risk-free rate, $T$ = time to expiration, $y$ = dividend yield ya convenience yield hai.

Derivation: Futures Price from First Principles

GOAL: Aisa $F_0$ dhundho ki koi arbitrage na ho.

Step 1: Do equivalent strategies set up karo

Strategy A: Futures contract kharido (aaj $0$ cost, expiry par $F_0$ pay karo, asset lo). Strategy B: Paisa udhaaro, abhi spot asset $S_0$ par kharido, expiry tak hold karo, storage aur financing pay karo, dividends collect karo.

Expiry par dono strategies same asset deliver karti hain. No-arbitrage ke hisaab se, dono aaj same cost ki honi chahiye.

Step 2: Strategy B ki cost

• $S_0$ rate $r$ par borrow karo → $S_0{e}^{rT}$ wapas do
• Storage cost $U\cdot S_0\cdot T$ (proportional model)
• Dividends worth $D$ lo (ya yield $y\cdot S_0\cdot T$)

Expiry par net outflow: $S_0{e}^{rT}+U\cdot S_0\cdot T-D$

Step 3: Futures se equate karo

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

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

WHY exponential? Continuous compounding ki wajah se. Chhote $T$ ke liye approximate karo: $F_0\approx S_0\left[1+(r+U-y)T\right]$

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

Worked Examples

Find: Fair futures price.

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

Yeh step kyun?

• Hum dividend yield subtract karte hain kyunki spot asset hold karne par dividends milte hain, jo net carry cost ko kam karta hai.
• Exponential isliye kyunki hum continuous compounding assume karte hain.

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

Negative basis: futures > spot kyunki financing cost dividend income se zyada hai.

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

Storage kyun add karte hain? Physical commodities ko store karna costly hota hai. Yeh carry cost badhata hai, futures ko spot se upar push karta hai.

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

Arbitrage strategy (Cash-and-Carry):

1. Aaj: ₹18,000 ko 6% par borrow karo, Nifty spot 18,000 par kharido, futures 18,500 par becho.
2. 3 mahine hold karo: Dividends worth $18,000\cdot 0.015\dot{2}5=67.5$ collect karo.
3. Expiry par: Futures ke against spot deliver karo, 18,500 lo. Loan $18,000\cdot e^{0.06\cdot 0.25}=18,271$ wapas karo.

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

Yeh kyun kaam karta hai? Market futures price fair value se zyada hai. Sasta (spot) kharido, mehnga (futures) becho, spread lock in karo.

Common Mistakes

Sahi lagta kyun hai: Simple interest compounding jaisa dikhta hai.

Fix: Stocks dividends dete hain. Spot hold karna income deta hai, isliye futures price pure financing cost se kam hoti hai. Hamesha $y$ subtract karo: $F_0=S_0{e}^{(r-y)T}$

Sahi lagta kyun hai: "Positive" sunne mein "higher" lagta hai.

Fix: $\text{Basis}=S_0-F_0$ Positive basis: spot zyada hota hai (backwardation). Negative basis: futures zyada hote hain (contango). Yaad rakho: basis spot-centric hai.

Sahi lagta kyun hai: Static thinking.

Fix: Basis expiry par zero par converge karta hai. Convergence ki speed changing $r$, $y$, time decay par depend karti hai. Basis risk isliye arise karta hai kyunki convergence linear nahi hoti.

Active Recall Checks

Kyunki agar aaj kharido, toh ₹80,000 tie up ho jaata hai (bank mein interest earn ho sakta tha). Plus, Apple ko tumhare liye phone store karna padega (unhe paisa lagta hai). Toh 3-mahine wali price zyada hoti hai cost of waiting ke barabar: jo interest tum lose karte ho + storage cost. Yahi "cost of carry" hai.

Agar Apple ₹79,000 mein future delivery offer kare, toh sab log apne iPhones aaj bech denge aur sasta future wala khareed lenge—arbitrage! Market future price ko aaj ki price plus carry cost ke barabar force kar deta hai.

"Back-ward-ation = Back to reality" → Spot higher (market expect karta hai ki prices girenge, ya abhi scarcity hai).

Connections

• Introduction-to-futures-contracts – Basis hi reason hai ki futures ≠ spot
• Futures-pricing-vs-spot-pricing – Cost of carry bridge ka kaam karta hai
• Hedging-with-futures – Basis risk hedges ko imperfect banata hai
• Arbitrage-strategies-cash-and-carry – Mispriced basis ko exploit karta hai
• Backwardation-and-contango – Basis sign market structure define karta hai
• Marking-to-market – Daily settlement carry returns ko affect karta hai
• Commodity-futures – Storage aur convenience yield dominate karte hain

#flashcards/stock-market

Basis ka formula kya hai? :: Basis = Spot Price - Futures Price = $S_0-F_0$

Dividend yield $y$ wale stock ke liye fair futures price derive karo :: $F_0=S_0{e}^{(r-y)T}$ — spot kharido (cost $S_0$), $r$ par finance karo, dividends $y$ earn karo, no-arbitrage futures price ke barabar karta hai

Concept Map