1/H is the right anwser
replicate a risk free portfolio and derive a SDE exacty the same as Black Scholes
1/2*S^2*Vol^2*dV^2/dS^2+dV/dt-r*V+r*S*dV/dS=0
since r=0, V is independent of time so dV/dt=0
this leads to 1/2*S^2*Vol^2*dV^2/dS^2=0 which is a heat equation
dV^2/dS^2=0 with bountry condition V=1 when S=H
the solution is V is a constant 1/H
