## Code Outline and Intuition

The art of good trading is to be one step behind the market and two steps ahead of everybody else.

The Application of the Laplace Transform to Financial Markets Intuitional Steps.

• The price axis is in log Pi as opposed to base 10.
• This shows that all the booms and busts of the last century are related to each other by Pi and the square roots of Pi.

• This is the last thing you would expect a random walk to do.
• It is however the first thing you would expect a wave equation to do.

• Money circulates therefore its internal dynamics are circular in nature.
• They are polar functions and are measured in radians per unit time.
• This makes numeric analysis on the complex plane possible.

• This is what W.D. Gann’s was doing with his various wheels.
• There is however a far easier way to do it, by rearranging the question and using differential calculus to solve it.

• To do this you need to know the general form of the underlying integral equation of the market.
• The compound interest rate function is at the heart of all monetary systems.
• Which you integrate and convolute with the market to integrate out time from price altogether.

• To reveal the directions and magnitudes of the underlying wave functions of the long term carrier waves / swings in the market.
• Which you can then solve algebraically for a trading solution using an exponential step controller to regulate a discrete trading signal.
• This returns a process control solution that switches when the major changes in real time.

The Process Controller.

• To regulate the output of the system you have employ an escapement mechanism that prevents the function from constantly over running the market and creating false signals.
• It is all but identical to the mechanism that stops clock spring from releasing all its energy at once.
• To capture the energy of an irregular out of balance oscillation with the objective of damping out the short term oscillations / swings and capturing the energy of the long term swings.

• To reveal the true state of play in the major trend as it happens in real time without the phase delay and false signals of moving averages
• So when the market goes sideways so do integral contour functions of your signal whereas moving averages converge to the price and generate false signals and lots of them.
• If you keep the change natural log of the price as a as a cumulative time series

• You can then do your signal processing using linear algebra.
• With a tri state logic output to solve  for the direction of underlying trend in real time.

Function Outline: S is constant typically 10^-1 < S < 10^.

• Es(0) = Ln(Close);
• Es(1) = Es(0) + Ln(Close1) – Ln(Close0);
• Es(2) = Es(1) + Ln(Close2) – Ln(Close1);

• Fs(t) = if(|dEs(t)| > S) Fs(t) += Fs(t+1) + jS;
• j = Ln(Close1) – Fs(n-1) / | Ln(Close1) – Fs(t+1) |;

• Js(t) = Fs(t) – Fs(t-1) / | Fs(t) – Fs(t-1) |;
• Hs(n) = Js(t) – Js(t-1) / | Js(t) – Js(t-1) |;

Coding Outline:

• Function Inputs
• Time
• The last market price
• The value of S.
• The order of magnitude S.
• The Z axis Resolution Step
• Data Structures.

• The incoming market data
• The integrations of the market prices.
• The exponential unit step controller.
• The outputs are.

• The market chart contour.
• The Fringe magnitudes
• The Poisson distribution of the signal
• The market signal the center line of the Poisson distribution
• The filtered / rectified return market signal.

Contour Processing.

• The method is to create a time series of the differences between a known differential equation / contour and the market you are tracking as a complex number.
• You can do this by using an exponential step controller that ignores the passage of time and simply looks at the unit step change in price.
• If the distance is greater that a specific value it take one unit step in up or down depending on the direction of the market.
• When a change in trend is detected steps in the other direction.
• Each contour is tracking the one before it.
• Signal Processing is done by taking the sum of the directions of all the contours and plotting it as a Poisson / Bernoulli count distribution.
• This returns the magnitude of the underlying trend as a stepped ramp function and a tri state logic signal -1 0 1.