Arbitrage, Hedging, and the Law of One Price
- Nov 4, 2005
Identifying Arbitrage Opportunities
Arbitrage Situations
Arbitrage opportunities exist when an investor either invests nothing and yet still expects a positive payoff in the future or receives an initial net inflow on an investment and still expects a positive or zero payoff in the future.^{18}
This appeals to the commonsense expectation that money must be invested to result in a positive payoff. Furthermore, if you receive money upfront, you expect at the least to pay it back and certainly do not expect the investment to produce positive payoffs in the future. It is also reasonable to expect the value of a portfolio of assets to properly reflect the prices of the underlying components of that portfolio. Thus, the situations described in this chapter indicate arbitrage opportunities in which deviations from the Law of One Price can potentially be exploited. Any one of these conditions is sufficient for the presence of an arbitrage opportunity. Consider the following examples, which indicate the presence of an arbitrage opportunity.
Arbitrage When "Whole" Portfolios Do Not Equal the Sum of Their "Parts"^{19}
What if the price of a portfolio is not equal to the sum of the prices of the assets when purchased separately and combined into an equivalent portfolio? This summons the earlier image of a basket of fruit selling for a price different from the cost of buying all its contents individually. More specifically, if fruit basket prices are too high, people will buy individual fruit and sell baskets of fruit. They would consequently "play both ends against the middle" to make a profit.
This situation could occur when commodities or securities are sold both separately and as a "packaged" bundle. For example, the Standard & Poor's 500 Composite Index (S&P 500) is a portfolio consisting of 500 U.S. stocks that can be traded as a package using an SPDR.^{20} Of course, the stocks can also be traded individually. Thus, an arbitrage opportunity would exist if the S&P 500-based SPDR sold at a price different from the cost of separately buying the 500 stocks comprising the index.
Consider what happens if this condition is not satisfied for a two-stock portfolio consisting of one share of Merck (MRK) selling at $31.46 and one share of Yahoo (YHOO) selling at $34.02. If the price of the equal-weighted portfolio differs from $31.46 + $34.02 = $65.48, an investor could profit without assuming any risk.
The two possible imbalances are as follows:
Price portfolio (MRK + YHOO) > $31.46 + $34.02
or
Price portfolio (MRK + YHOO) < $31.46 + $34.02
In the first case, the portfolio is overpriced relative to its two underlying components. In the second case, the portfolio is underpriced relative to its components. More specifically, assume in the first case that the portfolio sells for $75.00 and in the second case that the portfolio sells for $55.00. We expect that the sum of the prices of MRK and YHOO will equal the price of the portfolio at some time in the future. However, in light of the earlier discussion of convergence, we must admit that because there was mispricing to begin with, there is no certainty that the relevant prices will equalize in the future. We assume that such convergence will occur eventually.
If the price of the portfolio is $75.00 and therefore exceeds the costs of buying MRK and YHOO individually, the strategy is to buy a share of MRK for $31.46 and a share of YHOO for $34.02 separately because they are cheap relative to the price of the portfolio. To finance the purchases, it is necessary to sell short the portfolio for $75.00 at the same time. Because the price of the portfolio exceeds the cost of buying each of its members separately, selling the portfolio short generates sufficient money to purchase the stocks individually. The strategy consequently is self-financing. It generates a net initial cash inflow of $75.00 – $65.48 = $9.52.
Yet what will the net long and short positions yield in the future? You will have to return the portfolio at some time in the future to cover the short position, which involves a cash outflow to buy the portfolio. However, you already own the shares that constitute that portfolio. Thus, subsequent moves in the prices of MRK and YHOO are neutralized by the offsetting changes in the value of the portfolio consisting of the same two stocks. Thus, the net cash flow in the future is zero.
What does this mean? It means that you could generate an initial cash inflow of $9.52—that is like getting a loan you never have to repay! This cannot last, because everyone would pursue this strategy. Indeed, investors would pursue this with as much money as possible! Ultimately the increased demand to buy MRK and YHOO would put upward pressure on their prices, and the demand to sell short the portfolio would put downward pressure on its price. Consequently, an arbitrage-free position will ultimately be reached in which the price of the portfolio equals the sum of the prices of the assets when purchased separately.
To reinforce this result, consider the other imbalance, in which the price of the portfolio is only $55.00, which is less than the costs of buying MRK and YHOO individually for a total of $65.48. The strategy is to sell short a share of MRK for $31.46 and to sell short a share of YHOO for $34.02 separately because they are expensive relative to the price of the portfolio at $55.00. Similarly, you would buy the portfolio for $55.00 because it is cheap relative to its underlying components.
It is obvious that selling short the two stocks individually generates more cash inflow than the cash outflow required to purchase the portfolio. Thus, the investment generates an initial positive net cash inflow of $65.48 – $55.00 = $10.48. As in the case just evaluated, it is important to consider the cash flow at termination of the investment positions in the future. Some time in the future you will have to return the shares of MRK and YHOO to cover the short sale of each stock, which involves the cash outflow to buy each of the two stocks. However, you already own the portfolio, which consists of a share each of MRK and YHOO. Thus, subsequent moves in the prices of the long positions in MRK and YHOO are neutralized by the equivalent, mirroring price moves of the same stocks within the short portfolio. Consequently, the net cash flow in the future is zero. As observed with the other imbalance, you can effectively borrow money that never has to be paid back! This indicates an arbitrage opportunity and shows why only arbitrage-free asset and portfolio prices persist.
Arbitrage When Investing at Zero or Negative Upfront Cost with a Zero or Positive Future Payoff
An arbitrage opportunity can be identified based on the relationship between the initial and future cash flows of a portfolio formed by an investor who buys and sells the component assets separately. Consider the case in which putting together a portfolio of individual assets generates either a zero net cash flow or a cash inflow initially and yet that portfolio produces a positive or zero cash inflow in the future. This situation produces an arbitrage opportunity because everyone would want to replicate the portfolio at no cost or even receive money up-front and also receive money or not have to pay it back in the future.
Consider three individual assets that can be purchased separately and as a portfolio. Table 1.2 portrays the cash flows to be paid by each of the three assets and the portfolio at the end of the period as well as their prices at the beginning of the period. The future cash flow payoffs are also presented.
Table 1.2 Example Identifying an Arbitrage Opportunity
Asset |
Current Price |
Cash Flow Next Period |
1 |
$1/1.08 =~ $0.926 |
$1 |
2 |
$900 |
$972 |
3 |
$1,800 |
$2,200 |
Portfolio (1,080 units of asset 1) |
$900 |
$1,080 |
Table 1.2 shows that an arbitrage opportunity exists. Remember that an arbitrage opportunity is present if the price of a portfolio differs from the cost of putting together an equivalent group of securities purchased separately. In this example, the portfolio of 1,080 units of asset 1 can be purchased more cheaply than if 1,080 units of asset 1 are purchased separately. Specifically, it would cost $1,000 or 1,080 (0.926) to buy 1,080 units of asset 1 individually, while a portfolio of 1,080 units of asset 1 is priced at only $900. Thus, the "whole" portfolio is not equal to the sum of its "parts."
The arbitrage strategy is to sell short 1,080 units of asset 1 for $1,000 now to finance the purchase of one (undervalued) portfolio that contains 1,080 units of asset 1 for only $900. The resulting current cash inflow is $1,000 – $900 = $100. No cash inflow or outflow would occur at the end of the period, because you would hold a portfolio of asset 1 that is worth $1,080, which is the same value you must return to cover the short position in 1,080 individual units of asset 1. Thus, $100 is generated upfront, and nothing must be returned. This is either a dream come true or an arbitrage opportunity—one and the same. Obviously, investors would pursue this opportunity on the largest possible scale.
Another arbitrage condition is satisfied using assets 2 and 3. The current value of the portfolio formed by buying and selling these two assets separately is nonpositive, which means that either there is no initial cash flow or there is an initial cash inflow. Thus, the portfolio either is costless or produces a positive cash inflow when established and yet still generates cash at the end of the period. Using the data in Table 1.2, the arbitrage portfolio is formed by selling short two units of asset 2 and buying one unit of asset 3. The initial outlay would be –2($900) + 1($1,800) = 0. Notwithstanding the zero cost of forming the portfolio, at the end of the period the cash flow is expected to be –2($1,800) + 1($2,200) = $400. Thus, an arbitrage opportunity exists because the strategy is costless but still produces a future positive cash inflow. The portfolio consequently is a proverbial money machine that investors would exploit on the greatest scale available to them.
Market Implications of Arbitrage-Free Prices
The conditions required for the presence of an arbitrage opportunity imply that their absence also places a structure on asset prices. As noted, prices are at rest when they preclude arbitrage. Specifically, arbitrage-free prices imply two properties. First, asset prices are linearly related to cash flows. Known as the value additivity property, this implies that the value of the whole portfolio is simply the added values of its parts. Thus, the value of an asset should be independent of whether it is purchased or sold individually or as a member of a portfolio. Second, any asset or portfolio that has positive cash flows in the future must necessarily have a positive current price. This is often referred to as the dominance criterion. Thus, the absence of arbitrage opportunities places a structure on asset prices.