Binomial No-Arbitrage Pricing: Options and Bonds

This problem involves pricing financial derivatives (options and bonds) using a two-period binomial model with simple compounding.

Setup and Assumptions

• The model spans two periods.
• Compounding is simple per period.
• The discount factor per period is (1 + r_node)^-1.
• The equity does not pay dividends in Part A.

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

• Initial stock price (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 price (K) = 100.

Tasks:

1. Compute the risk-neutral probability q and the European call price C0 using 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:

1. Price a zero-coupon bond maturing at t=2 with a face value of 100.
2. Price a two-period coupon bond with a face value of 100 and a coupon of 5 paid at t=1 and t=2. Report its yield to maturity (per period) and explain the no-arbitrage pricing principle used.