Show that the trinomial model

is arbitrage free.

And suppose inin(P(ST = u), P(ST = m), P(ST = d)) > 0. Then the risk neutral probability Q needs to put positive weight on all these outcomes. So let PQ(St = u) = q1, PQ(ST = erT) = q2, PQ(Sr = d) = 1-q1-q2. We then require EQ(e-rtSr) = e-rTS0(uq1 + erTq2 + d(1 - q1 - q2)) = S0. That is qi1u - d) + q2(erT- d) = ert-d. There are several choices for q1, q2. is arbitrage free