Thanks -- I guess the smooth paste is best seen by the continuity of delta.
You're welcome. I took FDM to get delta which is OK since it is applied to an exact solution (it is _not_ a divided difference of a divided difference).
Another CSE approach (as in Matt Robinson's thesis 2019) is to differentiate the ODE wrt [$]S[$] to get an ODE for [$]\Delta[$]
\[0 + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 \Delta}{\partial S^2} + (\sigma^2 + r )S \frac{\partial \Delta}{\partial S} = 0, \]
We can probably solve this
analytically as before. Of course, we need 2 extra conditions at the free boundaries.
I suspect [$]\Delta[$] is not [$]C^1[$] continuous because we differentiate it at the boundaries wrt payoff (gamma, thus).
Or .. we can just differentiate [$]V[$] all over the place to see where it becomes discontinuous, if at all.