Price options and bonds via binomial no-arbitrage

Use binomial no-arbitrage to price options and bonds. Part A (equity/options): Stock S0=100; two periods; per-period up factor u=1.25 and down factor d=0.8; risk-free rate per period r=5% (simple); strike K=100. 1) Compute the risk-neutral probability q and the European call price at t=0 by backward induction. 2) Price the American put and identify any node(s) where early exercise is optimal. 3) Prove whether early exercise of a non-dividend-paying American call can ever be optimal in this setup and explain. Part B (rates/bonds): A two-period short-rate tree with r0=5%; next period r_u=7%, r_d=3%; final period from r_u → {9%, 5%}, from r_d → {4%, 2%}. Assume risk-neutral q=0.5 at each node and discount factors (1+r_node)^{-1} per period. 4) Price a zero-coupon bond maturing at t=2 with face 100. 5) Price a two-period coupon bond with face 100 and coupon 5 paid at t=1 and t=2; report its yield to maturity and explain the rational (no-arbitrage) pricing principle you used.