= Black-Scholes reflection symmetry
{c}
{title2=$\widehat V(S,t)=(S/B)^{1-2r/\sigma^2}V(B^2/S,t)$}
For the dividend-free <Black-Scholes equation>, the displayed transform preserves solutions when $B>0$ and $\sigma>0$. Put $x=\log(S/B)$ and $v(x,t)=V(Be^x,t)$. The equation becomes
$$
v_t+\tfrac12\sigma^2v_{xx}+b v_x-rv=0,\qquad b=r-\sigma^2/2.
$$
Substitution shows that $\widehat v(x,t)=e^{qx}v(-x,t)$ solves the same equation when $q=-2b/\sigma^2=1-2r/\sigma^2$: its first-derivative coefficient becomes $-\sigma^2q-b=b$, and the additional zeroth-order term $\sigma^2q^2/2+bq$ vanishes. Subtracting the reflected solution enforces a zero boundary value at $S=B$. Its use in a killed <barrier option> is restricted to the alive side of the barrier; the algebraic extension to the other side is generally not a valid price.
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