Laplace and Compound Interest
“Mathematics is the art of giving the same name to different things.” Henri Poincare
SP 500 1996 – 2017 As a Complex Harmonic Function.
Compound Interest and The Laplace Transform
Contrary to what the conventional wisdom would have you believe that it is impossible to apply physics to finance the above proof would suggest otherwise, it is indeed possible to apply the principles of calculus to markets, after all market fluctuate and calculus is all about fluxions. It is just a matter of figuring out how to apply this which will inevitably mean finding the true names of the variables.
ie: Price = Voltage , Volume = Current and so on, and then figure out a the differential equation that filters out all the noise and high frequency content of the market action and returns low frequency carrier wave ie the trend. Just as you would do if it was a telecommunications signal, the Laplace transform is the tool of choice for this problem in physics so why not finance?
The only difference is the Laplace transform has an integration wrapped around it. The difference is the kernel of the compound interest rate function and the Laplace transform use a different letter for the exponent, that is s instead of r.
They are therefore the same formula and thus the same thing that is to say the Laplace transform is an integration of the compound interest rate function.
This also indicates that compound interest is by definition a complex number and thus can be translated to the complex plane and analyzed there for it wave components.
e^st = e^rt
st = rt
s = r
So any impulse (transaction) will create point source interference that will propagate a wave front through the medium.
If many impulses that is to say trades are applied to the medium (market) they will form interference patterns as each wave front decays over time.
Where the interference is generated by the constant impulse responses from transactions that is taking an open position and closing it out at some future time.
This will look very much like a chaotic system but don’t be fooled the reason markets form linear trend lines is because they are linear systems with a chaotic component as there inputs however the way the market reacts is what matters and that is a function of the elasticity of the market in question which remains constant over time.
More to the point markets are sum total of the supposition of the wave that form them. These patterns are not apparent to due to calibration problems inherent in the base ten number system when looking at a logarithmic system which can be easily solved on the complex plane for the component parts of the interfering waves. Base ten pricing is the problem here, we know the markets are inherently exponential in there behavior which does show itself in standard base ten charts.
The Laplace transform makes it possible to measure the change in the major trend in real time with out the phase delay in the signal caused by moving averages.
SP 500 1996 – 2015 Weekly Parabolic Distribution Signal.
SP 500 2014 – 2020 Weekly Laplace Signal.
SP 500 2019 – 2020 Daily Laplace Signal.
The filter in the above charts uses a an inverse Laplace transform coupled with a limit step function that follows the market to market the change in the natural log of price to integrate out time. It is essentially a step filter that removes the high frequency fluctuations from the data letting the low frequency’s pass.
It then filters the output through a Poisson distribution counter to remove out the short term noise and swings to reveal the long term trend signal, without all the phase delay involved with moving averages that in many ways make them worse than useless.
Markets are telecommunications networks, the only difference between a telephone exchange and a financial exchange is the spelling. They both perform identical functions they match callers with receivers or buyers with sellers in real time at the speed of light.
We just use different names for same functions, to put it another way money and light are the same thing, money is information traveling at the speed of light and therefore obeys the same laws of physics as the wave propagation of electromagnetic energy.
Specifically markets when viewed on the Cartesian plane are price interference patterns that can be analyzed on the complex plane for the vector co ordinates of the component waves that form the primary trends as they change in real time.
The trading inputs of a market may seem to random from day to day but over time they accumulate and distribute in one direction or another and create stable trends that can last for decades which can be analyzed, as a time independent linear systems that have constant behaviors independent of time.
The way the markets trend and oscillate and crash has not changed over the centuries the laws of mathematics are the same as they were and must be obeyed as Richard Feynman said in his report on the space shuttle Challenger accident.
“For a successful technology, reality must take precedence over public relations, for Nature cannot be fooled.”