The model is about exponential growth. In finance exponential growth is most likely to occur during financial bubbles. The essence of exponential growth: the rate of growth is proportional to the quantity itself.
At first glance, Tesla stock fits nicely in this pattern.
The second pillar — nothing can grow exponentially forever, therefore there must exist a point in time that this pattern breaks. Let’s call it tc — critical time.
Now, let’s perform the most beloved trick in econometrics and finance — linearize the time series by taking logarithm of stock price.
The logarithm of price is not quite linear: it has ups and downs and we need to somehow change the regression to account for it. Logical choice is to use trigonometric functions, like cosines. And also we should remember about critical point in time when this pattern breaks.
Let’s construct a function that can account for both effects:
It’s a first approximation to LPPLS model. We have a linear part and cosines part. Logarithm under cosines helps to model variability: it is approaching infinity as the time gets closer to the end. Moreover, ω helps to model the number of waves, and C - their sizes. If we fit this equation using Excel solver, results should be similar to those below
This equation predicts several outcomes:
- firstly, the closer to the critical time, the more volatile the behavior of stock (more uncertainty embedded),
- secondly, waves become more short-lived.
Several improvements can be made to come to the canonical LPPLS model. At first, instead of linear part we can insert power part, secondly, to connect the behavior of linear and cosines part — multiply them.
Power part helps to model super-exponential growth, i.e. around the critical time the stock rises above any expectations (“mania” period).
A few words about restrictions to facilitate the optimization process:
- B < 0
- 0<β<1 (according to practitioners, it can be further restricted to 0,2<β<0,8)
- t is measured as a fractions of year (t = 1/365). For instance, 1.3 means 1.3 years from the start of the data.
Both models predict 2–2,5x increase around New Year: 800–100 USD per share. Maybe, it as an inflection point. This model also predicts 4000 USD next March (10x greater), which 99,9999% will not happen.
LPPLS model is a simple tool that can be used to measure the possibilities of future growth in fast growing assets, like gold, Tesla, Amazon, Zoom etc. The main difficulty is to calibrate coefficients, in Excel it can be made by running solver from different starting sets and narrowing coefficients restrictions.
A book with theory and examples: Why Stock Markets Crash: Critical Events in Complex Financial Systems