# Motivation of the singular perturbation solution formulation for local volatility model

I am puzzled by the motivation of the particular choice of the (singular) perturbation method used in Equivalent Black Volatilities. Equation (A.6a) sets $$\epsilon:= A(K)\ll 1.$$ What is the motivation for this setting? I find it surprising that $A(K)$ is to be infinitesimal. However, at later expansion in Equation (A.9a), $A(K)$ seems to be treated independently from $\epsilon$ which is $A(K)$ itself. Moreover, Equation (A.9b) seems to assume $A'(K)$ and $A''(K)$ to be infinitesimal as well, if $\nu_1$ and $\nu_2$ are to be finite. This setting seems to be rather contrived.

What is going on?

This local volatility analysis is referenced in the paper Managing Smile Risk on the SABR model.

• Maybe the M.Sc. thesis of de Jong "Option Pricing with Perturbation Methods", available at repository.tudelft.nl, is helpful. If I remember correctly, she works through all the details of the derivation in the original Hagen et al. paper. – LocalVolatility Mar 11 '18 at 20:39
• @LocalVolatility: Thank you for the reference. That master thesis deals with SABR model per se. I am fine with the usage there. I should be clearer (I have rephrased my question to that effect), that my concern is with the perturbation in the local volatility model deal with in the Equivalent Black Volatilities paper. It appears pretty odd to me. Care to look into the details? – Hans Mar 12 '18 at 6:51
• @LocalVolatility: I have resolved the issue as described in my answer below. – Hans Mar 21 '18 at 6:02

In fact, this is a confusion caused by a sloppy notation. The rigorous version of the setup should be $$A(K)\rightarrow \epsilon A(K).$$ Then we let $x:=\frac{f-K}\epsilon$. The rest is the usual singular perturbation operation.