Binomial No-Arbitrage Pricing: Options and Bonds

Setup and Assumptions

• Two-period models; per-period simple compounding.
• Discount per period: (1 + r_node)^{-1}.
• No dividends on the equity in Part A.

Part A: Equity and Options (Two-Period Binomial Stock Tree)

• Stock: S0 = 100
• Per-period up factor u = 1.25; down factor d = 0.8
• Risk-free rate per period r = 5% (simple), so 1 + r = 1.05
• Strike K = 100

Tasks:

1. Compute the risk-neutral probability q and the European call price C0 by backward induction.
2. Price the American put P0 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: Short-Rate Tree (Two Periods)

• Short rates:
  • t=0: r0 = 5%
  • t=1: up node ru = 7%, down node rd = 3%
  • t=2: from ru → {9%, 5%}, from rd → {4%, 2%}
• Risk-neutral probability at each node q = 0.5
• Discount per period using the node’s short rate: (1 + r_node)^{-1}

Tasks: 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 (per period) and explain the no-arbitrage pricing principle used.