= Separated solutions of the Black-Scholes equation
{title2=$V(S,t)=S^\alpha e^{(r-\lambda_\alpha)t},\quad\lambda_\alpha=\frac12\sigma^2\alpha(\alpha-1)+(r-q)\alpha$}
In the <Black-Scholes equation with continuous stock dividends>, <separation of variables> gives $g'=(r-\lambda)g$ and $\sigma^2S^2f''/2+(r-q)Sf'-\lambda f=0$. Substituting $f=S^\alpha$ gives the displayed eigenvalue. A stationary solution uses the two roots of $\sigma^2\alpha(\alpha-1)/2+(r-q)\alpha-r=0$. For positive volatility and $r>0$, their product is negative, so they are distinct real roots.
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