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Fertik's Law

Michael Fertik
December 17, 2001

Fertik’s Law states:

Increasing entropy geometrically increases arbitrage opportunities.

In other words: More products and more relationships among products result directly in more arbitrage possibilities, and they do so at a geometric rate, faster than a simple arithmetic increase.

It is best to put this principle, dubbed “Fertik’s Law”, in terms of one of today’s major industry debates. Leo Melamed stated in his November speech at the Tokyo conference that it is “axiomatic” that screen-based electronic trading will replace traditional pit-based trading in the coming years. But will the electronification of trading exchanges spell the end of arbitrageurs, especially pit arbitrageurs?

Put it another way, with an example taken from a real exchange: right now, when the July contract for Orange Juice (just take any contract as an example, irrespective of how or where it’s traded) ticks, the calendar spread between the July and September Orange does not follow automatically and instantaneously. So there’s a chance to arb, and some guys make money keeping the underlying contracts and the spreads in line.

Technology, that Specter of the Trading Pit, can and will soon be doing the job of this type of arbitrageur. The new generation of electronic trading platforms enable the exchanges to “connect” or otherwise create relationships among products, so that when the monthly contract changes, the calendar spread updates in sub-second speed. The market is therefore, in this way and in this instance, efficient at just about all times.

So won’t this new technology kill arbitrage opportunities? Won’t it take away the need for arbitrageurs?

No. The answer is no. In fact the opposite is true. That is the meaning of Fertik’s Law.

First, it is important to recognize that “arbitrage”, both as an idea and as a practice that is potentially lucrative, includes a range of buying, selling, and matching activities with respect to tradable products. More properly, arbitrage includes a wide range of products and relationships among products. This means that arbitrage can and does take place not just between two products from the same class, such as July and September Orange contracts, but among all kinds of products whose prices are interconnected.

An example may help illustrate: Trader John from Major Financial notices that the price of stock options on one market follows the price of single stock futures on another market, which itself follows the price of equities on a third market. The lags in between each of these markets represents a potentially very lucrative arbitrage opportunity, and John can take positions to take advantage of the follow-on ticks. These types of inter-market connections are likely, it is worth noting, not going to be updated automatically by computers, though the technology to do so will be available soon, because they are inter-market in nature.

Another example may help seal the deal: Trader Jane from Major Financial observes that every time weather derivatives for March change X%, volatility of softs increases Y%, and prices on certain March soft options change Z%. The reverse is also true, though in a slightly different pattern: if the March soft options prices tick, so will the March weathers. So she buys low and sells high as appropriate, filling the market demands that she knows will occur as soon as something changes. In this way, Jane is aribitraging the same way as the guy in the Orange market.

Second, arbitrage is based on lack of information, or on a mismatch in the flow of information among various parties interested in trading. Arbitrageurs take advantage of this lack of information in order to make money, and, in doing so, they “return” the market to an efficient state.

Electronic trading will enable the trading of many more products. Today’s electronic trading platforms permit the launch of new products on-the-fly, and with little financial or time cost to the exchange. Electronic trading allows market participants to take and monitor positions in many more products simultaneously. In sum, electronic trading will increase the amount of trading in a larger number of products.

When there are more products traded, more information is produced to be consumed by different market participants. It is axiomatic that when more information is produced in an environment (in our case, an exchange or market), there will be a higher lack of information among some portion of the market participants with respect to some portion of the market. Individual market participants will not have an easier time following 10 to 50 times the amount of information tomorrow than they do today.

In this way, increased trading activity of an increased number of products will produce more arbitrage opportunities.

Third, we must acknowledge that relationships among tradable products, and not just the raw volume of products, create arbitrage opportunities. To return to the format of our Trader Jane example, it is clear that different trading houses, and different brokers within different trading houses, will have different views of the proper price relationships among weather derivatives and soft contracts. But the point is that they will have opinions, and that they trade based on their sense of formulaic relationships among those products.

Of course, not all products can rightly be related in this way. For example, the price of T-Bills and the price of Oranges are not likely going to correlate in any substantially firm way over the course of a year. But the price of IBM Single Stock Futures and the price of T-Bills might be tied. And the price of gold and the price of Brent crude might even be tied in some environmental circumstances, such as a wartime footing by the United States. And certainly the price of IBM Single Stock and IBM options will be connected. Moreover, those relationships, as they are essentially speculative in nature-because they are not functions of the exact same product in the exact same market-will and can never be performed “automatically” by an electronic trading engine.

Are these last few arbitrage opportunities the same as closing the gap between July Oranges and the July-September spread? Not all of them. The industry must wait till Single Stock Futures start trading before we can tell how close the SSF/Equities/Options connection will be. I believe the relationship among those three will be quite close, and that they will provide very “pure” arbitrage opportunities. Other examples of arbitrage, as described above, will be more speculative in nature, since not everyone is destined to agree on the price relationships among some non-fungible products. Nonetheless, arbitrage is arbitrage, and someone will be able to take home the bacon for closing the efficiency gaps when they open.

So what is all this about “entropy”? How does physics find its way into the world of derivatives trading?

Rather directly, in fact. Entropy is the measure of disorder, randomness, or chaos in a system. As a system gets more disordered, more random, and more chaotic, its entropy increases.

Here’s the point: 1) Electronic trading enables more products to be traded, and more relationships to be set up by market participants among the tradable products; 2) More products, and more relationships among them, result in more activity in the market; 3) more activity in the market creates more information, which leads directly to increased lack of and lag in equal information distribution in the market; 4) this lack of and lag in information creates arbitrage opportunities all over the market.

In other words, if you increase the friction points in an exchange by creating a LOT more moving parts (i.e. tradable instruments and, by necessary consequence, relationships among those instruments), you increase the number of places where a fire might start. You create chaos. You increase entropy.

That’s Part 1 of Fertik’s Law: Increasing entropy…increases arbitrage opportunities. In terms of today’s industry debate, electronic trading will increase arbitrage opportunities.

Part II of Fertik’s Law is in the word “geometrically.” I posit that increasing chaos in a market will increase arbitrage opportunities at a faster rate than the chaos itself.

Demonstrating the basic distinction I’m trying to draw will help illustrate the general principle:

Under the mathematical rule called an “Arithmetic Progression,” numbers rise like this:

2, 4, 6, 8, 10


5, 10, 15, 20

The sequence changes according to an increase or decrease of “equal differences.” In the first example, the sequence rises by adding 2 to every previous number. In the second sequence, we add 5.

By contrast, a “Geometric Progression” looks like this:

2, 4, 8, 16, 32

5, 25, 125, 625, 3125

In a Geometric Progression, the sequence progresses by increase or decrease by “equal ratio.” Multiplication or division is applied to each previous number at a standard rate. In the first example, each previous number is multiplied by 2. In the second, each previous number is multiplied by 5.

Fertik’s Law states that when an exchange increases the number of products and relationships among products in its market, it will increase the number of arbitrage opportunities at a Geometric, or multiplicative, rate. In other words, adding four more tradable instruments will create not just four more arbitrage opportunities, but a multiple of four more arbitrage opportunities.

Clearly, geometric rates will (almost always) produce higher numbers than arithmetic ones. I posit that increasing entropy in a market, by increasing the number of tradable products and the relationships among them, will result in this accelerated rate of growth in arbitrage opportunities.

It may be noted that I have not included a fixed geometric rate in the Law. That is because each market will experience a growth in the number of arbitrage opportunities at a different rate. The growth will be a function of both the actual number of tradable instruments added to the total, as well as the exchange rules, if any, governing the relationships among products. If the exchange sanctions certainly relationships expressly, it will stimulate market participants such as FCMs to use them and to create their own. The bottom line is, however, that arbitrage opportunities will rise in an asymptotic fashion, so that as you keep adding more instruments and relationships, you get an ever-increasing return in the number of arbitrage opportunities that are created.

Leo Melamed believes that it is “axiomatic” that screen-based electronic trading will supplant traditional pit-based trading in the near future. But much of the industry will remain the same: centralized exchanges, or virtually centralized exchanges, will remain dominant for several legitimate reasons. Moreover, as I have outlined here in this essay, the role of traditional arbitrageurs will only become stronger in the electronified markets. As electronic trading enables the proliferation of trading many products, arbitrage opportunities will only increase. Today’s savvy arbitrageurs, and tomorrow’s new breed, will have more, not fewer opportunities, to make money and make efficient markets.