Unbiasedness of Garman–Klass and Rogers–Satchell

Let the intraday log price be $\sigma W_s$, $s\in[0,1]$, with $W$ standard Brownian motion, and use the notation (9.4). (a) Show that the Garman–Klass estimator (9.7) is unbiased. (b) Using the reflection $W\mapsto-W$, show $E[u(u-c)]=E[d(d-c)]$. (c) Using Lévy's theorem that $\max_{s\le1}W_s-W_1\overset{d}{=}|W_1|$, show $E[uc]=\sigma^2/2$ and hence that the Rogers–Satchell estimator (9.8) is unbiased when the drift is zero.