Pricing of Stock and Index Futures: Cost of Carry Method

Understand Pricing of Stock and Index Futures

Key rules to remember

Fair price, simple interest, no dividend

F = S × (1 + r × t)

r is the annual rate, t is the time in years (e.g. 3 months = 0.25). Use when the question says simple interest or gives no compounding instruction.

Fair price, continuous compounding, no dividend

F = S × e^(rT)

Use when the question says continuous compounding. The e^x value is normally given in the question.

Fair price with known dividends

F = S × (1 + r × t) − FV of dividends

Compound each dividend from its receipt date to the expiry date at r. A dividend received after expiry is ignored.

Fair price with dividend yield, continuous

F = S × e^((r − q)T)

q is the continuous dividend yield. Typical for index futures.

Cost of carry

Cost of carry = Financing cost − Dividend income

Fair futures price = Spot + Cost of carry. Basis = Spot − Futures; it is zero at expiry.

Arbitrage profit

Profit = |Market futures price − Fair futures price| × lot size

Sell futures if market > fair. Buy futures if market < fair.

How to solve Pricing of Stock and Index Futures questions

1. 1Write down spot price, interest rate, time to expiry in years, dividends with their dates, and the quoted futures price.
2. 2Choose the compounding method from the question: simple, annual or continuous.
3. 3Compute the future value of spot: S × (1 + r × t) or S × e^(rT).
4. 4List the dividends that fall on or before expiry. Compound each one forward to expiry and add them up. Ignore any dividend after expiry.
5. 5Fair futures price = Step 3 minus Step 4. For an index with a yield, use S × e^((r − q)T).
6. 6Compare with the market price. Higher means overpriced, so sell futures and buy spot. Lower means underpriced, so buy futures and short spot.
7. 7Compute the arbitrage profit per unit and multiply by lot size. Show the cash flows today and at expiry if asked.
8. 8State the conclusion in one line: fair price, mispricing and action.

Quickest way: Fair price in three lines

Common mistakes in Pricing of Stock and Index Futures

• Using the annual rate without converting the time period

  Students see 8% and apply it directly when the contract is for 3 months.

  Fix: Always convert months to years first: t = months ÷ 12. Write t on the page before calculating.

• Forgetting to subtract dividends

  The basic formula has no dividend term, so it gets dropped.

  Fix: Read the question for any dividend or yield. Subtract its future value from the carried-forward spot price.

• Including a dividend that falls after expiry

  Students subtract every dividend that is listed.

  Fix: Check the date of each dividend. Only dividends received before or at expiry reduce the futures price.

• Taking the wrong side of the arbitrage

  Students mix up overpriced and underpriced.

  Fix: Remember: overpriced futures, sell them and buy the share. Underpriced futures, buy them and sell the share.

• Treating the index like a share and ignoring lot size

  The per-unit gap is found but the contract value is not.

  Fix: Multiply the per-unit price gap by the lot size or the index multiplier to get the rupee profit.

• Mixing compounding methods

  Using e^(rT) for spot but simple interest for dividends.

  Fix: Use one method throughout, as the question states.

Worked examples