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Outline the strategy arbitrageurs follow and compute an arbitrage payoff.

FRM® Part I · Financial Markets and Products · Introduction to Derivatives · FRM1.FM.4.h

The idea

Arbitrage means locking in a riskless profit with no net investment by exploiting a price inconsistency. For an asset with no income or carrying costs before maturity, the no-arbitrage forward price is: F = S × (1 + r)^T. Here S is the spot price, r the annual risk-free rate with annual compounding and T the time in years. For an asset with income, such as a dividend, subtract the future value of that income from this price, and add the future value of any storage or other carrying costs. If the quoted forward is above this value, do cash and carry: borrow, buy the asset, sell the forward, deliver at maturity and repay the loan. If it is below, do reverse cash and carry: short the asset, lend the proceeds and buy forward. The profit at maturity is the difference between the quoted and the fair forward price, taken as a positive amount. The usual mistakes are ignoring the financing cost, forgetting income such as dividends, doing the strategy in the wrong direction, and ignoring transaction costs, which can remove an apparent opportunity.

Formulas

No-arbitrage forward price of an asset with no income, annual compounding

F₀ = S₀ × (1 + r)ᵗ

S₀
the spot price today
r
the risk-free rate
t
the years to delivery
F₀
the price agreed today for delivery then

No income or carrying costs before maturity. Income lowers the fair forward and carrying costs raise it; if the quote is above, do cash and carry, if below, reverse cash and carry.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which description best fits an arbitrage opportunity?

  1. A strategy with a high expected return but some probability of lossWhy not: Exposure to loss means risk, so this is speculation, not arbitrage.
  2. A strategy that earns the risk-free rate on the capital investedWhy not: A risk-free return on capital is fair compensation, not a profit from mispricing.
  3. A strategy that reduces exposure to a known risk at a modest costWhy not: Describes a hedge, which may cost money and does not aim at a riskless profit.
  4. A strategy needing no net investment that has no risk of loss and a positive profit in some outcomes - correct

Arbitrage means a riskless profit with no net investment, which is made possible by an inconsistency between prices. Its existence is incompatible with equilibrium.

A harder one

A non-dividend-paying stock trades at 40. The one-year forward price is 44 and the annual risk-free rate (annual compounding) is 6 percent. An arbitrageur borrows to buy the stock and sells the forward. The arbitrage profit at maturity per share is:

  1. 4.00Why not: Ignores the cost of financing: 44 - 40.
  2. 2.40Why not: Takes the interest cost (40 x 6%) as if it were the profit.
  3. 1.60 - correct
  4. 6.40Why not: Adds the interest to the forward price instead of deducting it from the proceeds (44 + 2.40 - 40).

The no-arbitrage forward is 40 x 1.06 = 42.40. The arbitrageur delivers the stock for 44, repays 42.40 and keeps 44 - 42.40 = 1.60.

The hardest tier

The spot rate is 1.1000 USD per EUR. One-year interest rates are 5 percent in USD and 3 percent in EUR (annual compounding). The one-year forward is quoted at 1.1300 USD per EUR. An arbitrageur borrows USD 11 million, converts it to EUR 10 million, deposits the euros, and sells the forward on the maturity value. The arbitrage profit at maturity is closest to:

  1. USD 300,000Why not: Compares the forward and spot rates directly, ignoring both interest rates.
  2. USD 80,000Why not: Uses a simple rate differential (5% - 3%) for the fair forward instead of (1.05 / 1.03).
  3. USD 639,000Why not: Repays the USD loan without interest, ignoring the 5 percent cost of borrowing.
  4. USD 89,000 - correct

EUR deposit grows to 10,300,000; sold forward at 1.1300 gives USD 11,639,000. The USD loan costs 11,000,000 x 1.05 = 11,550,000. Profit = 89,000. Covered interest parity fair forward is 1.1000 x 1.05 / 1.03 = 1.1214, well below 1.1300.

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