## Dangerous Markets and Didier’s Formula

**Dangerous Markets and Didier’s Formula**

In 1923 an earthquake devastated Tokyo and became known as the Great Kanto earthquake. This mega earthquake was preceded by several years of increased incidents of intermediate sized seismic events. Similarly, large earthquakes in San Francisco were also preceded by a period of increased intermediate seismic activity. In fact, evidence would suggest that large earthquakes are often preceded by these patterns of activity. Furthermore, mathematically similar precursor behaviour is a known feature of many catastrophic transitions that occur in nature. Other examples are metal fatigue, vortices in certain weather systems and biologically, for example in epileptic fits and heart attacks.

**The Golden ratio of bubbles…**

Just as the famous “Golden Ratio” appears over and over throughout nature, catastrophic behaviour also seemingly exhibits a precursor signature.

But I am supposed to be writing about something finance related, right?

Well, I am. I want to say something about crashes and specifically about a subject that has been in the press recently, which relates to a model that I wrote about now and again years ago as a sell side strategist. The connection to the above natural phenomena is that similar precursor signatures have been observed in the run up to catastrophic events in financial markets, i.e., crashes. And the specific type of pattern that links all of these catastrophes is so called log periodic oscillations, neatly captured in a formula derived by Didier Sornette, which when applied to markets would represent the log of the price of the asset as it approaches a crash at some critical time (tc). In the months, or even years, leading to that critical time, prices would oscillate with diminishing amplitude and increasing frequency:

**…Didier’s Formula**

p(t) ≈ p(c) + A + B*((1-t/tc)^beta)*(1+C*cosine(w*ln(1-t/tc)+Phi))

Now, before you all start screaming that markets are noisy and complicated and can’t be captured by a simple formula….I know that, I too have wasted years and years of my life trading and will probably never be free of this debilitating addiction… the way to think about this formula is in the way you would about, say, a moving average, or a graph of RSI etc. Think of it as the Sornette Study.

Didier Sornette studied the behaviour of a lattice of agents exhibiting imitative and herding behaviour with nonlinear feedback. Such systems in nature are often found to exhibit sudden phase transitions from one state to another, much like a tectonic rupture, catastrophic failure in a metal or crystal subjected to stress, in magnetic substances…or a stock market crash. The theory is far more complex than this formula suggests, this is just a special case, an approximation in a specific situation as a critical point is approached…let’s say an approximation to the

**…death throes of a bubble**….

Taking illustrative parameters (A =8, B=0.86, C=0.1, tc=25, w=20, beta = 0.35) prices (not log actual) look like this as that critical day is approached:

Note that I chose parameters roughly to look a bit like the S&P from early 2012 to the present, with a frequency to be close to monthly. A statistically sophisticated (watch this space) using maximum likelihood methods, or simulated annealing, or just plain old least squares will result in a better fit and somewhat predictive end point. But I am more interested in recognising the pattern “sort of fits” and just keeping that in the back of my mind, much as I would be aware of technical levels, or levels of indicators such as RSI or moving averages when thinking about entry or exit points in a regular trading strategy…all part of the picture.

It isn’t just that the ascending prices with ups and down look a little like the S&P. The ups and downs have fallen in amplitude and increased in frequency, in log periodic fashion. In other words, we’ve seen the trend up get stronger, the dips shallower, but instead of every 3 months, we got them every month and then every week and so on .We can’t expect an exact fit of course, but there are remarkable similarities.

**Fear order not disorder**

We could argue that a healthy market is a disordered one in which there are a mixture of speculative buyers and speculative sellers. An unhealthy market is one that is ordered, like an upward trend with everyone wanting to be a buyer.

A sudden transition from one state of unhealthy order to another state of unhealthy order would be the rapid flip between everyone wanting to buy to everyone wanting to sell. That transition could be a crash but its onset is not really instantaneous as there may be a pattern, that of log periodic oscillation, that leads to it. You must all have noticed that traders copy other traders and friends copy friends or people are led by media stories…of course we have all noticed that…like a Mexican wave of markets. If you ever worked in a bank trading floor you may have noticed how neighbouring traders often share similar positions and so on. There is very little originality out there and in the absence of originality all that remains is imitation and trend following. And as for price feedback, there is no better generator of bullish trading ideas than a rising market. The price leads, random justifications to buy follow. Again, rarely an original or contrarian thought.

The fit has been good to crashing markets such as in 1987, the dotcom bubble, more than one currency crash and the crash of the Nikkei, to name just a few that have been analysed. There are no particular financial parameters or assumptions; it’s more about calibration and pattern recognition.

This certainly isn’t the only model helpful in understanding crashes, but it is one you may have heard of and has been mentioned in the press recently, but often without any kind of quantitative description of what it really is. Markets are like that, words are often thrown around for affect.

What we have been seeing recently in the S&P 500 is smaller dips as buyers are confident the dip is always a buy. We have also been seeing less pronounced oscillatory behaviour as the more rapid ups and downs seem to have been replaced by a seeming endless upward wave. Just look at the SNP over the past year or so and you should see what I mean, its been a long time since any kind of purgatory correction. It is this kind of trading behaviour that suggests the complacency of a “bubble” and quantitatively a chart of the S&P arguably shows log periodic oscillations.

**Crashing uncertainty, crashing volatility**

A corollary to this log periodic type behaviour is also decreasing realised volatility, a low in volatility being another coincident factor in the start of most market crashes and corrections. The longer this log periodic behaviour persists, with traders gaining confidence with every dip that is then followed by higher prices, the more complacent the trading community becomes. There are of course sceptics, but what we care about is what people do, not what they say. Just because everyone feels deep down that the market has become one big bubble doesn’t mean the bubble won’t burst. After all, insane or not, few can resist the temptation to be part of this bubble and few can afford the alternative of leaving their cash to rot away in the bank. Of course, once the price rises slow and price falls take over then the opposite will apply.

**The good times never last forever**

So as the trend continues and as confidence builds, the log periodic fit is another sign that we may be approaching a critical point. To stress, I am not saying this is about to happen, we may be months away, or we may just happily trend up forever, to infinity, I am just saying its another indication that all is not well.