Cash and carry arbitrage

When a futures contract is too expensive compared with what it costs to buy the asset today and hold it, you can do both at once: buy the cheap thing (the asset in the cash market) and sell the expensive thing (the futures).

The workbook's definition: cash and carry arbitrage refers to a long position in the cash or underlying market and a short position in futures market.

You then "carry" the asset to expiry. At expiry the futures price converges to the spot price, so the two legs offset each other whatever the market did, and what remains is the original overpricing — minus the cost of carrying the asset.

This is also the mechanism that corrects the overpricing. As arbitrageurs buy in the cash market and sell futures, the spot price rises and the futures price falls, until the gap closes.

1. Compute the fair futures price: spot plus cost of carry (less any dividend).
2. Compare with the traded futures price. If traded > fair, cash and carry is profitable.
3. Buy the underlying, sell the futures, usually financing the purchase with borrowed money.
4. At expiry, deliver or square off both legs. The amount received on the futures leg repays the financier.

The arbitrage profit is traded futures price − fair futures price. It survives only if it exceeds transaction costs, brokerage, taxes and margin costs — the frictions that create no-arbitrage bounds around the fair price.

Fair price F* = S × e^(r × T)
Arbitrage profit per share = Traded futures price − F*
Triggered when Traded futures price > F*

The workbook's stock A (Chapter 5, section 5.1):

Fair price     = 1,500 × e^(0.09 × 3/12) = 1,534.13
Overpricing    = 1,550 − 1,534.13        = 15.87 per share
Arbitrage gain = 15.87 × 100             = Rs 1,587

Buy 100 shares at 1,500, sell 1 futures contract at 1,550. Financing cost = 34.13 × 100 = Rs 3,413.

Case I — stock ends at Rs 1,580:

Profit on shares  = (1,580 − 1,500) × 100 = +8,000
Loss on futures   = (1,580 − 1,550) × 100 = −3,000
Gain before cost  = 5,000
Net gain          = 5,000 − 3,413 = Rs 1,587

Case II — stock ends at Rs 1,480:

Loss on shares    = (1,500 − 1,480) × 100 = −2,000
Profit on futures = (1,550 − 1,480) × 100 = +7,000
Gain before cost  = 5,000
Net gain          = 5,000 − 3,413 = Rs 1,587

The answer is Rs 1,587 in both cases — exactly the overpricing. Brokerage and taxes are not deducted in the workbook's example.

Chapter 3, section 3.7, introduces cash and carry arbitrage through the gold example and section 3.9 lists it among the three arbitrage types. Chapter 5, section 5.1, works the stock A example above. The paper tests both the direction (which leg is long, which is short) and the arithmetic (fair price, profit per lot).

• Futures overpriced → buy cash, sell futures. Reversing the legs is the most common wrong answer.
• The profit does not depend on the expiry price. Both cases give Rs 1,587.
• Deduct the financing cost. The Rs 5,000 gross gain is not the arbitrage profit.
• Convergence makes it work: at expiry futures settle at the underlying's price, so the two legs offset.
• It is profitable only outside the no-arbitrage bounds once transaction costs are counted.