Volumes of symmetric random polytopes

We consider the moments of the volume of the symmetric convex hull of independent random points in an $n$-dimensional symmetric convex body. We calculate explicitly the second and fourth moments for $n$ points when the given body is $B_q^n$ (and all of the moments for the case $q = 2$), and derive from these the asymptotic behavior of the expected volume of a random simplex in those bodies.