1. Show that the discounted price ( ˜ Sn) is a martingale under P if and only if E(Tn+1|Fn)=1 + r, 8n 2 {0, 1, . . . ,N − 1}.

The equality E( ˜ Sn+1|Fn)= ˜ Sn is equivalent to E( ˜ Sn+1/ ˜ Sn|Fn)=1, since ˜ Sn is Fn-measurable and this last equality is actually equivalent to E(Tn+1|Fn)=

1 + r.