# Can we proof the boundary condition for the Black Scholes derived from a replicating Portfolio?

So for Black Scholes we know that the PDE is the follwing: $${\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}=rV-rS{\frac {\partial V}{\partial S}}$$. The boundary conditions for that equations then are the following:

\begin{aligned}C(0,t)&=0{\text{ for all }}t\\C(S,t)&\rightarrow S{\text{ as }}S\rightarrow \infty \\C(S,T)&=\max\{S-K,0\}\end{aligned}

How do we proof the boundary condition exactly? Can we do it by induction (e.g. that the value of the replicating portfolio should be equal to the call option?) or by some thing else ? Thank you in advance for helping out!

• 1) The first condition follows from the fact that a stock at level $S=0$ implies, effectively, that the underlying (company) is bankrupt and hence will not recover from that level. Another way to see this is that, at $S=0$, $dS=S\times(...)$ is zero as well. 2) Should follow from logical reasoning: If the stock goes to $\infty$, the present value of a claim on the stock above and beyond the strike is effectively as valuable as a claim on the stock itself. 3) The last statement is stipulated by the option contract itself. HTH? These conditions are not proven, but (sensibly) assumed. Mar 1, 2021 at 8:29
• Cool, that is the perfect answer. Thank You! Mar 1, 2021 at 11:36