 Solution

For $f=k{S}^{\alpha }$, $S{f}^{\prime }=\alpha f$ and $S^2{f}^{\prime \prime }=\alpha (\alpha -1)f$. Thus $\lambda ={\lambda }_{\alpha }={\sigma }^2\alpha (\alpha -1)/2+(r-q)\alpha$, and all values with this fixed spatial factor areThis 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.