Markets are places where people go and exchange things, called assets. People go to a market in order to buy something they need or sell something they no longer need.
Markets can be used in order to buy and sell mostly anything, but for the sake of our discussion, let's focus on 3 things:
- Foreign exchange, where currencies are exchanged for one another (e.g. euros vs. dollars).
- Stocks, where money is exchanged for ownership in a company (e.g. dollars vs Google stocks).
- Commodities, where money is exchanged for a specific good (e.g. dollars vs. oil).
Although buying and selling can be done directly, as in a barter system format (I give you 1 euro in exchange for 2 dollars), this process is typically done by using an universal exchange medium, such as money (I give you 1 Google stock in exchange for 10$ and I use this 10$ in order to buy 1 barrel of oil).
Every asset has a price, which is the amount you have to pay if you want to buy that good or the amount you will obtain if you sell it. In the world of free markets, prices are dynamic, that is, they change over time, depending on the supply and demand. For example, if a resource is very scarce (low supply), but highly desirable (high demand), such as gold, then it is only normal that it has a very high price. The other way around, a high supply, but low demand means a low price.
The study of the way prices evolve over time has an obvious appeal to it. For example, if the oil price today is 100$ and I can predict with a very high certainty that tomorrow it will be 150$ dollars, then it makes sense to buy oil today at a lower price and sell it tomorrow at a higher price in order to make a profit.
One way people study and predict prices is by looking at historical values. They try to learn certain patterns of past price behavior and hope that under similar conditions these patterns will repeat themselves, and thus make a profit by betting in the right direction.
The most interesting aspect to look at are price changes, since this is how you make a profit. You buy today at a low price, hoping that the price will go up (have a positive change) and sell tomorrow in order to make a profit.
An example of a historical hourly price record can be seen below:
This record depicts the euro vs. U.S. dollar exchange rate, as it was in 1st of April, 2011, at 16:00. In that moment, the opening price (the price at the beginning of the hour 16:00) was 1.40635, the closing price (the price at the end of the hour 16:59) was 1.41460, the highest price achieved within this interval was 1.41487, the lowest was 1.40625, and the total volume of trading (how much euros and dollars where exchanged) was 19561.
Based on the above record, we can make some additional observations. The closing price (1.41460) is higher than the opening price (1.40635), thus we can say that within that trading hour the exchange rate went up (positive evolution). The absolute amount with which it went up (let's call this absolute difference), calculated as the absolute value of the difference between the closing price and the opening price was 0.00825 units, and the relative difference (as expressed in percentages from the opening price, that is (absolute difference) / (opening price) * 100) was 0.586%.
In other words, if I would have bought an amount of say 10 000 euros at 16:00 and sold that amount at 16:59, I would have made a relative profit of 0.586%, or in absolute terms, 58.6 euros (not bad for a single hour).
Note that in the example above I have depicted an hourly price change, but we can look at price records at other time granularities: changes within a second, a minute, an hour, a day, a week, trimester, semester, year, decade, century, millennium and so on (well actually it doesn't make sense to look at centuries and millennia, simply because the markets haven't been around for that long, but if the current market mechanism continues, this might be applicable in the future).
Let's recap the most interesting parameters we have discovered so far:
- Time granularity - the length of the period we look at a price difference.
- Binary evolution - if the price went up or down.
- Absolute difference - the difference between closing and opening price.
- Relative difference (magnitude) - the absolute difference expressed as a percentage from the opening price.
Looking at a single historical price record doesn't really reveal much useful information (except how much money we could have made or lost, if we traded in a certain direction or another) so in order to give laws referring to price changes, we have to study more of them.
But I believe we have enough information in order to propose the first law:
1. For every asset (euro vs dollar, oil, stocks, etc.), when looking at every possible time granularity (seconds, minutes, hours, days, etc.), within any time interval of a reasonable size, there is an almost equal amount of positive and negative price changes.
I've first made this observation by looking at the hourly price changes in the euro/dollar exchange rate, starting from the 1st of January 2010 until the 20th of April 2013. Within this time period, there are 20972 active trading hours (hours during weekends or holidays do not count, since the markets are closed). Out of these 20972 records, there are 10495 when the price went up (positive evolution) and 10477 when the prices went down (negative evolution). Expressed as percentages, there are 50.049% positive price changes and 49.951% negative price changes.
I wanted to see if this observation holds for every other asset, so I've downloaded the entire history of daily, weekly and monthly price changes referring to stocks for all the companies listed in the Standard and Poor's 500 index. For each of the downloaded time series that had more than 300 records I have computed the percentage of positive price changes and plotted all of these percentages as a histogram. You can see the results below:
The series formed by all the positive percentages taken together has a mean of 50.614, standard deviation of 1.606, a minimum value of 43.40 and a maximum value of 55.816. This distribution is depicted in the picture above with a blue line. My intuition was that positive percentages follow a normal distribution, so in order to test this I've generated random points from a normal distribution with the same mean and standard deviation and plotted their histogram as well (depicted with green above). As you can see, the lines match pretty good, which confirms my intuition.
What this rule actually shows is that in every price time series, no matter the granularity, there are an equal amount of good and bad trading intervals.
The first rule only looks at the sign of the price change. Even though this is a good starting point, it would be much more interesting to also study the magnitude of price changes. So the second law proposes the following:
2. For every asset, when looking at every possible time granularity within any time interval of a reasonable size, the magnitudes (relative differences) of price changes follow a fat-tailed distribution.
This can be seen in the chart below:
The chart was generated by computing all of the magnitudes for the hourly price changes for the euro vs. dollar exchange rate from the 1st of January 2010 until the 20th of April 2013. This series has a mean of -0.000336 and a standard deviation of 0.132782, with a maximum of 1.625 and a minimum of -0.962 and it is depicted with the blue line.
By comparison, with the green line I have depicted a normal distribution with the same mean and the same standard deviation, and as you can see, the 2 distributions do not really fit.
I don't want to go into the details of what a fat-tailed distribution or a normal distribution are, but the basic difference is that in a normal distribution, extreme values are highly unlikely (99.7% of the data falls within 3 standard deviations of the mean) , while in a fat-tailed distribution, extreme values are more likely.
This difference is very important, since extreme price changes have to be taken into account, as they can bring huge profits or high losses. Even more important is that until recently, most of the predictive mathematical models made the assumption that price changes follow a normal distribution, which is false. The most dangerous part is that this normal distribution assumption is also made by official institutions. Fortunately, this view has been slowly changing in the recent years, mainly thanks to the works of Benoit Mandlebrot and Nassim Nicholas Taleb.
And slowly we get to the 3rd and final law:
3. Future price changes cannot be predicted based on past price changes, except in some limited special cases.
In other words, no matter how you look at past price data, the best you can do regarding the future is to toss a coin.
Although at the moment I cannot give an empirical or theoretical proof regarding this law, I can offer an intuition as why this is the case. Let's consider for example the Apple Company. Apple owns much of its commercial success (which is translated into high stock valuations) to the iPhone device, which it launched in 2007. This can be also seen on the chart depicting the evolution of the Apple stock price. After 2007, with a couple of exceptions related to the financial crisis of 2008-2009 and the death of Steve Jobs in 2011, the stock price of the company has been continuously rising.
So if this high success (high stock valuation) is due to the success of the iPhone device, my question is how could you have predicted in say 2006 (before the launch of iPhone) that such a device will be launched and it will be so successful, just by looking at past Apple stock prices? Obviously, you couldn't have. Or, another example, under the assumption that the exchange rate of a currency reflects the efficiency of a country's government, how can you predict this efficiency just by looking at the past movements of the exchange rate?
The special cases, where one could make a prediction, relate to the notion of a self-fulfilling prophecy. If enough traders hold the same belief, that some random past pattern can be used in order to predict the future, then when that pattern activates, all of them will make the same decision and this will be reflected in the stock price. For example, in the world of FOREX prediction, people talk about psychological levels of quotation, when the exchange rate of a currency is close to some round number. Some traders believe that if a currency can break through this level, then it will aggressively continue its rising or falling course. This may or may not be the case in reality, but the fact is that if enough traders believe that this strategy works, they will buy (or sell) that currency after it goes through this psychological barrier. As a result, the price will change according to their beliefs, and so the prophecy will come true.
In the next post, I will try to give some advice regarding as to how to trade, based on the laws stated above.