In this Article I faced with Asymptotic Expansion method for pricing American option. the price $P(S,t)$ of this option satisfies the partial differential equation (PDE): $${{P}_{t}}+(r-\delta )S{{P}_{S}}+\frac{1}{2}{{\sigma }^{2}}{{S}^{2}}{{P}_{SS}}-rP=0$$ with boundary conditions:
$\begin{align} & P(\infty ,t)=0,\,\,\,\,\,\,\,\,\,P(S,T)=\max \left( K-S,0 \right) \\ & P\left( \bar{S}\left( T-t \right),t \right)=\max \left( K-\bar{S}\left( T-t \right),0 \right),\,\,\,\,\,\,\,\,{{P}_{S}}\left( \bar{S}\left( T-t \right),t \right)=-1 \\ \end{align}$
Here, $\bar S(T-t)$ is the early exercise price, which depends on the option time-to-maturity $\tau =T-t$. then the author mention that To derive proper asymptotic expansions, We rewrite PDE in terms of $(\theta , \tau )$ instead of $(S,t)$. Using the definition of $\theta =\frac{\ln \left( K/S \right)}{\sigma \sqrt{\tau }}$ and setting $P\left( \theta ,\tau \right)=P\left( K{{e}^{-\sigma \theta \sqrt{\tau }}},T-\tau \right)$ ,then We obtain: $$\theta {{P}_{\theta }}+{{P}_{\theta \theta }}+\frac{1}{\sigma }[{{\sigma }^{2}}+2(\delta -r)]{{P}_{\theta }}\sqrt{\tau }-2({{P}_{\tau }}+rP)\tau =0$$
and satisfying boundary conditions in the form
$\begin{align} & P(-\infty ,\tau )=0 \\ & P\left( y,\tau \right)=K\max \left( 1-{{e}^{-\sigma y\sqrt{\tau }}},0 \right)=K\left( 1-{{e}^{-\sigma y\sqrt{\tau }}} \right) \\ \end{align}$
and which has regular asymptotics near maturity of the form
$$P\left( \theta ,\tau \right)=\sum\limits_{n=1}^{\infty }{{{P}_{n}}\left( \theta \right){{\tau }^{n/2}}}$$
Now I have 2 questions that any recommendations/suggestions would be greatly appreciated.
First: Why Change of variables makes the PDE proper than it was ?
Second: How We can obtain the form of regular asymptotics near maturity ?