Analyzing tick data of currency markets we have discovered 12 new scaling laws in the foreign exchange markets that complement the two scaling laws that we uncovered in the 90s. A scaling law exists when two quantities maintain the same proportions over a certain range. For the scaling laws that we have discovered the range of proportionality is big, a factor of 1000, from the purely intraday domain to inter-day and longer. Scaling laws play an increasingly important role in natural sciences from biology to physics and complex systems for the calibration of models. In economics, this has so far not happened. I believe that this could change in view of the many new scaling laws.
I clearly remember when we discovered the first scaling law in the late 80s (for a full account see our book). At the time, the research tools were primitive and graphical visualization could not be taken for granted. When Ueli Müller, a researcher in our team, made the first print out of the original scaling law, I thought that this was a test of the graphics program to draw a straight line, so good was the fit. The first scaling law states that there is a fixed relationship between the average price move and the time interval over which the price move is measured. The most striking feature of this scaling law is that ‘intraday’ proportionality is the same as that of longer-term data of days, weeks, months and even years. This self-similarity indicates that short- and long-term price moves are more closely related than we are inclined to believe.
A few years later, we discovered another scaling law of the number of directional price changes per time period. This scaling law, which belongs to the same class of scaling laws that we have just discovered, did not receive sufficient attention at the time and was accidentally not included in the book ‘Introduction to High Frequency Finance’, published in 2001. In 2005, we started to systematically search for additional scaling laws. We did not only look for scaling laws in relation to physical time but for relative and specific events, e.g., turning points.
We define an event as having occurred, when the price has moved down by more than 0.5% from its last high or the reverse; the next event has happened when the price has moved up by more than 0.5% from its last low. We then investigate the behavior of the price curve in the intervals between these events. We observe the following: whenever the price changes its direction with a 0.5% threshold, the average overshoot before it reaches its next low or high respectively is an average 0.5%. This relationship holds for ultra-small thresholds of 0.01% to 5 percent or even more, in other words the average overshoot is equal to the threshold. If a price has started a new trend with a threshold of 0.5% then the first 0.5% overshoot represents the expected average overshoot. Only when the market has exaggerated the move and has gone beyond this point to say 1%, 2%, 3% or more of overshoot, is it considered a tail event with an increased likelihood for a rebound.
Another important new scaling law deals with the length of the coastline of the sum of the entire price move; both up and down. If we consider price moves of 0.05% and discard any smaller price moves, then the average sum of total price move during the course of a year is an astounding 2500 percent for exchange rates, such as EUR_USD. If we only consider price moves of 0.1%, then the total coastline reduces to 1250%. For thresholds of 0.2%, the total coastline length is 670%, for 0.4% it is 340%, 0.8% thresholds 175% and for 1.6% only 90%.
How can we leverage these scaling laws for model building?
Scaling laws are efficient at condensing a lot of data, they are computed as follows: for every x quantity, we observe quantity y and then average y. The scaling law tells us, how average y changes with x. The intellectual beauty of scaling laws is that by considering all possible thresholds of x, we automatically include data of all y. The scaling law consists of an interception to the y-axis and the slope; so only two numbers ‘intercept’ and ‘slope’ summarize the whole time series with all its data, very elegant. Scaling laws only provide information about averages and do not make a statement about the distribution.
A remarkable feature of the scaling laws is that only little data is required to arrive at accurate estimates of the scaling parameters. I currently teach at the Essex University, where one of my students had accidentally only received a short data sample of one month but his estimates of the scaling parameters were astonishingly accurate. This is no small feat in economics where everything seems to be afloat.
In economics and finance it is standard practice to come up with the following type of model: variable x is a function of variable y, w and v. Where variable y, w, and v, may originate either from completely different time series (some of them collected with daily, weekly or monthly data) or from the same data series but sampled at different frequencies. Typically, there is the unspoken understanding that the sampling frequency of the data is only a minor issue; a consequence of the availability of the data, in actual fact its impact is more profound. The sampling frequency modulates the observations, just remember the length of the coastline. If an indicator based on monthly data is compared with another indicator that is updated with every 0.05% price move, then the first indicator relates to a coastline of a length of 20% to 30%, whereas the second relates to a coastline of 2500%. It goes without saying that the statistical properties of these indicators are different and the results may not mesh easily. To use an analogy, to disregard a scaling law is like planning a road trip with different road maps that have different yardsticks, but assuming that they are all the same.
Scaling laws have an important role as a yardstick for measurements and more importantly carry a significant message for the design of economic and financial models. Traditionally, economic models have assumed that markets have an equilibrium price level, where prices match fundamentals and forces of demand and supply are in balance. Increasingly, economists are becoming aware that there is no such thing as a fundamental price and that there is a need to find a substitute for the role of fundamental price, which is a kind of anchor, where market prices are expected to gravitate towards. I conjecture that scaling laws are a substitute and are an indicator for the dynamic equilibrium where financial markets and the economy at large gravitate, not in terms of absolute price levels but rates of change.
Why is there no fundamental price, this concept seems so intuitive? Just consider the question, of what is the fundamental price of gold? Is it the current market price quoted on the over the counter market or on the CME, the Chicago Board of Trade? To be pedantic, is it the bid or ask price that is quoted at this very second on the market, or is it the price that I read in my newspaper that publishes daily closing prices, which may be two or more percent away from the current price? Or is the fundamental price, the end of month price or for that matter the end of quarter or end of year price? Is the price the same for small or big quantities of gold? Anyway market prices are subject to the randomness of market oscillations and investor exuberance and a more objective measure for fundamental prices may be actual production costs. Where do we measure production costs in South Africa or in Australia and which gold mine should be used as the benchmark, or more involved questions, what interest rate assumptions should be made to compute the production costs, or how do we input costs for energy or over how many years are the production facilities amortized? However attractive the concept of fundamental prices seems at first glance it hinges on too many assumptions and is thus not a robust reference point.
Scaling laws are an indicator of the dynamic equilibrium. In context of the overshoot scaling law: if the market price starts a new trend, of say a 1% downward move computed from its previous peak, there is an average overshoot of another 1%, which is the dynamic equilibrium. If the price overshoots by 2% or more, the market is clearly off equilibrium. The scaling law is a metric to determine, how far the market has diverged from its equilibrium. With the scale of market quake (SMQ) service, we do exactly this: we measure the extent in which the market price has diverged from its dynamic equilibrium.
The research of scaling laws in financial markets has just started. Even though, we have discovered 12 new scaling laws, there are many other scaling laws still to be discovered. The more scaling laws are known, the easier it is to understand, where the financial markets and economy general has its dynamic equilibrium. Other important questions have not been answered: how do the scaling laws originate and why are they so pervasive and hold for so many orders of magnitude? Is it because financial markets are a half open system, where there are a large number of participants, where nobody is the ‘master’ and everyone competes with the other participants that have different time horizons and position sizes?
Because scaling laws hold true for so many orders of magnitude, we can really leverage the usage of high frequency data: we can calibrate the models with tick by tick data and then apply the resultant model to lower frequency data, where data is sparse. This gives financial and economic model builders a substitute for the lack of long-term data. Scaling law research in economics is in its infancy and a lot of exciting research remains to be done.
Richard Olsen is founder and chief executive of Olsen Ltd and the Chairman of OANDA, a leading foreign exchange broker and market maker.
Olsen Ltd is a research and development company and investment manager based in Zurich, Switzerland. Olsen has yielded practical applications and managed accounts and third-party products, investing in currencies as a separate asset class or as an overlay to an existing currency exposure.
Author: Richard B. Olsen, Founder and CEO of Olsen Ltd
October 27th, 2009 | High frequency finance, News | RSS feed