Cost of arbitrage

An arbitrage trade rarely delivers its whole "gross" profit as clean cash in hand. Money has to be tied up to run it, and tying up money has a cost.

In a cash-and-carry arbitrage, the arbitrageur buys the stock and sells a futures contract against it. Whatever the stock price does before expiry, the futures leg exactly offsets the cash leg, and a fixed gross gain results. But that gain still has to be reduced by the cost of arbitrage — the financing cost baked into the fair futures price at the start — to arrive at the arbitrageur's true net profit.

The workbook's worked illustration (Chapter 5) makes the cost concrete. Stock A trades at a cash price of Rs 1,500 and a 3-month futures price of Rs 1,550, with a cost of carry of 9% a year (about 0.75% a month). The fair 3-month futures price works out to Rs 1,534.13 — so the traded price of Rs 1,550 is overvalued, and the mispricing is Rs 15.87 per share (1550 − 1534.13).

An arbitrageur buys 100 shares and sells one futures contract (contract multiplier 100 shares). Whatever the stock does by expiry, the underlying position and the futures loss/gain offset to a fixed gross gain of Rs 5,000. But the cost of arbitrage — the financing built into the 3-month carry, Rs 34.13 per share × 100 shares = Rs 3,413 — is deducted from that gross gain to leave the net gain of Rs 1,587, which exactly equals the original mispricing of Rs 15.87 × 100 shares.

Case: the stock rises to Rs 1,580 by expiry.

Profit on underlying (100 shares)   = (1580 - 1500) x 100 = Rs 8,000
Loss on futures (100 shares)        = (1580 - 1550) x 100 = Rs 3,000
Gain on arbitrage (gross)           = Rs 5,000
Cost of arbitrage (financing)       = Rs 34.13 x 100        = Rs 3,413
Net gain out of arbitrage           = 5,000 - 3,413 = Rs 1,587

If instead the stock falls to Rs 1,480, the underlying shows a loss of Rs 2,000 and the futures a profit of Rs 7,000 — still a gross gain of Rs 5,000, still a cost of arbitrage of Rs 3,413, and still a net gain of Rs 1,587. The net figure does not depend on which way the stock moves, because the cash-and-carry position is hedged either way.

Chapter 5, section 5.2 (Using futures for arbitrage), works this exact stock-A illustration and computes the cost of arbitrage explicitly at both a rising and a falling expiry price. Expect a numeric question that gives spot price, futures price and cost of carry, and asks for the net arbitrage gain after deducting the cost of arbitrage.

• The net gain is the same whether the stock rises or falls — because the position is fully hedged, the cost of arbitrage and the gross gain both stay fixed regardless of the expiry price.
• The net gain equals the original mispricing per share, times the number of shares — Rs 15.87 × 100 = Rs 1,587, the same figure reached by subtracting the cost of arbitrage from the gross gain.
• Do not confuse the cost of arbitrage (the financing cost) with brokerage or transaction costs — the workbook's illustration is purely about the carry cost embedded in the fair futures price.