I wonder if someone could help with an approximation to an "invert and scale" transformation that occurs in a (simple) financial maths problem. It has aspects that looks as through they may be familiar from other applications (renormalization methods in physics?) and I suspect may part of a more general theory.
The function in question is this
$$ C(y,z)=\frac{1}{2}+\frac{1}{2}\textrm{erf}\left(\frac{z+y^{2}/2}{\sqrt{2}y}\right) -\frac{e^{-z}}{2}-\frac{e^{-z}}{2}\textrm{erf}\left(\frac{z-y^{2}/2}{\sqrt{2}y}\right) $$
The range of $C(y;z)$ is $[(1-e^{-z}):\infty]$. [Observant readers will see $C$ as the price of a (scaled) call option as a function of $y$, the variance and $z$, the moneyness. The lower limit of the range is the intrinsic value.] Let $C^{-1}$ be the inverse of $C$ with respect to $y$ i.e. $C^{-1}(C(y;z))=y$.
What I am looking at is the quantity $$ V_{sp}(V; z, k)= C[k C^{-1}(V);z] $$ where $k$ is a parameter.
The derivatives of $C$ is easy to obtain: $\frac{\partial C}{\partial y}= \frac{1}{\sqrt{2\pi}} e^{-z/2} e^{-(z^{2}+y^{4}/4)/(2y^{2})}$ and $\frac{\partial^{2} C}{\partial y^{2}}= -\frac{1}{4}y\left( 1-\frac{4z^{2}}{y^{4}} \right) \frac{\partial C}{\partial y} $, so there is an inflection at $y=\sqrt{2 |z|}$, $C$ is convex below this point and concave above.
Intuitively I expect a power law kind of behaviour $V_{sp}(V) \simeq V^{f(k,z)}$. For $|z| \gg 0$ (further out of the money options) I would imagine the exponent is bigger, however I struggle to show this.
For lower values of $y$ a plot of $C(y)$, one could argue for an approximate exponential form
small $y$: $C(y)\simeq e^{y}$ and hence $V_{sp}(V)=V^{k}$
For larger values of $y$ one could argue for an log form
large $y$: $C(y)\simeq \ln y$ and hence $V_{sp}(V)=V + \ln k$
For arbitrary sizes of $k$, which can project variables from one region to another, I feel there might be some blend of this dependency. Does anyone recognize this kind of problem from other applications or elsewhere?