= Solution
For $f=kS^\alpha$, $Sf'=\alpha f$ and $S^2f''=\alpha(\alpha-1)f$. Thus $\lambda=\lambda_\alpha=\sigma^2\alpha(\alpha-1)/2+(r-q)\alpha$, and all values with this fixed spatial factor are
$$
\boxed{V(S,t)=A S^\alpha e^{(r-\lambda_\alpha)(t-T)}.}
$$
This <Power payoff in the Black-Scholes model> calculation remains valid for any real $\alpha$ on $S>0$; the specified nonzero exponent excludes the constant-payoff subpart. It also supplies the moment term in <power put-call parity> above.
Back to article page
