Crossposted at Quantitative Finance SE
I have seen lot of discussions in this Math. S.E. platform about 'degenerate partial differential equations'. But I still unclear about the 'technical difficulty' with the degeneracy present in the Black-Scholes model governing option pricing. While dealing the generalized Black-Scholes equation
$$\frac{\partial{V}}{\partial{t}}+\frac{1}{2}\sigma^2(S,t)S^2\frac{\partial^2{V}}{\partial{S}^2}+(r(S,t)-D(S,t))S \frac{\partial{V}}{\partial{S}}-r(S,t)V=0,~~~~S\in (0,\infty),~~~t\in(0,T),$$
with the terminal condition $V (S,T)=\max(S-E,0)$, most of the numerical analysts (Page no: 1767, 2230 ) attempt transforming the price variable $S$ to a dimensionless variable $x=\ln S$, and claim thereby tackled the degeneracy of the model. I feel some trouble should be there near $S=0$ affecting the nature of the PDE. Further, I got the exact definition from the discussion. Is it a matter of the possible domain class of the B-S operator?
Regardless, I am interested to know about the analytical difficulty while dealing with such degenerate PDEs. I appreciate any help you can provide.
It's not really a problem per se. If $S = 0$ then the dynamics of the risky asset vanishes and the solution is just the present value of the payoff.
If anything, it may cause a problem since it changes the nature of the PDE. With the transformation you proposed, the coefficients of the derivatives won't vanish and then the nature won't change either.