Laplace Transform Derivation
Starting Point: Compound Interest
The journey begins with the most basic exponential process in finance — compound interest.
If you invest principal \( P_0 \) at continuous rate \( r \), the value at time \( t \) is:
\[ P(t) = P_0 \cdot e^{rt} \]
The Laplace Transform Definition
The Laplace transform takes a function of time \( f(t) \) and converts it into a function of a complex variable \( s \):
\[ \mathcal{L}\{f(t)\}(s) = \int_{0}^{\infty} f(t) e^{-st} \, dt \]
The Key Equivalence: s = r
Now substitute the compound interest growth function into the Laplace transform.
Assume \( P_0 = 1 \) for simplicity. The growth function is \( e^{rt} \).
When we set \( s = r \) in the transform kernel, we get:
\[ e^{st} = e^{rt} \quad \Rightarrow \quad s = r \]
This shows that the exponential kernel of compound interest is identical to the kernel of the Laplace transform. They are mathematically the same function.
Reversing Dependent and Independent Variables
In standard analysis we treat price as a function of time: \( P = f(t) \).
Financial Interferometry reverses this perspective. By working in the s-domain, we effectively treat price (or log returns) as the independent variable and extract information about time, direction, and trend strength.
This reversal is the conceptual leap that allows us to move from “what will the price be at time t?” to “what is the probabilistic steering signal at the current price level?”
Taylor Series Connection
The exponential function that links both concepts expands as a power series:
\[ e^{st} = \sum_{n=0}^{\infty} \frac{(st)^n}{n!} \]
This series underpins the entire framework and shows why the Laplace transform is analytically tractable for market data.
From Theory to the Real-Time Delta
Once we operate on the s-plane with log returns, the real part of the transform reveals exponential growth or decay rates. Combining magnitude and phase information yields the real-time delta — a value between -1 and +1 that indicates instantaneous trend direction and strength.
This delta is the practical output of the Laplace perspective and forms the basis for the Stepped Log Contour system.
Last updated: April 2026