Im currently working on option pricing using characteristic funcitons, anf have come across a problem.
BACKGROUND:
I have implemented an algorithm that generates the characteristic function giving the pdf in blue:
These oscillations prove very difficult to remove due to the cutoff. Because of the nature of the options I am pricing, what the pdf does left of the cutoff does not matter.
Hence, I had the idea of mirroring the right hand side of the pdf to the left side, creating somewhat of a bell curve in this situation. This would take away the cutoff and thus allow me to use spectral filters (used to remove this oscillating 'noise') more effectively.
THE PROBLEM:
I have a characteristic function for some unknown $f(y)$, the definition of a characteristic funtion is $$ \phi(u) = \int_{-\infty}^{\infty} e^{iuy} f(y)dy $$
Can I transform this into the characteristic function for a function $g(y)$ which is defined as:
$$ g(y) = \begin{cases} f(-(y+a))&\text{ if }y<a, \hspace{1cm}\\ f(y)&\text{ if }y\geq a\\ \end{cases} $$ where $a$ is the axis i want to mirror from.
You're defenition of g can't be a density function.
Lets say $X\sim U(0,1)$ so $f_X=1_{(0,1)}$ And if we choose $a=0$. We'll try to define Y with a density function $g(t)=1_{(-1,1)}$
But now $\lim _{t\rightarrow \infty}F_Y(t)=\int_{-\infty}^t g(t)dt=2$, so g cannot be a density function.