Let's proceed **step by step** as requested, with explanations at each stage and all calculations shown clearly. Since you need detail and accuracy, I will walk through the **initial riskless portfolio**, **probabilities of terminal prices**, and **option values** using the **binomial model**. --- ### **Step 1: Setting Up the Initial Riskless Portfolio (Delta Hedge)** **A. Price Tree:** - The price tree is a **two-period binomial tree**: - **Initial Price (Now):** $32.00 - **First Period:** $35.20 (UP), $28.80 (DOWN) - **Second Period:** - If UP, then $38.72 (UP again), $31.68 (DOWN after UP) - If DOWN, then $25.92 (DOWN again), $31.68 (UP after DOWN) **B. Call Option Details:** - **Exercise Price (K):** $30 - **Type:** European Call - **Maturity:** 1 year (2 periods) - **Risk-Free Rate (per period):** 4% (so, per period compounding) **C. Hedge Portfolio (Delta Hedge):** To form a **riskless hedge**, you combine **stock (Delta)** and **short call positions** such that the portfolio has the same payoff regardless of the up/down move in the next period. #### **Delta Calculation (First Step):** Let's get the call values at the terminal nodes (t=2): - At $38.72: Call payoff = max($38.72 - $30, 0) = $8.72 - At $31.68: Call payoff = max($31.68 - $30, 0) = $1.68 - At $25.92: Call payoff = max($25.92 - $30, 0) = $0 At **t=1**, if price is $35.20: - Up to $38.72 (payoff $8.72), Down to $31.68 ($1.68) - At $28.80: - Up to $31.68 ($1.68), Down to $25.92 ($0) **Delta at t=0** (initial hedge; use $35.20 and $28.80 branches): Let C_u = Value of call if stock goes up to $35.20 Let C_d = Value of call if stock goes down to $28.80 But those are not directly known; we can compute them using risk-neutral valuation (see next step). **However,** to set up the initial hedge, we need the delta at t=0: \[ \Delta_0 = \frac{C_{u} - C_{d}}{S_{u} - S_{d}} \] Where: - S_u = $35.20, S_d = $28.80 **First, calculate risk-neutral probability (p):** \[ u = \frac{35.20}{32.00} = 1.10 \\ d = \frac{28.80}{32.00} = 0.90 \\ R = 1 + 0.04 = 1.04 \] \[ p = \frac{R - d}{u - d} = \frac{1.04 - 0.90}{1.10 - 0.90} = \frac{0.14}{0.20} = 0.70 \] **Now, calculate call values at t=1 nodes:** - At $35.20: - Up to $38.72: payoff $8.72 - Down to $31.68: payoff $1.68 - Value at $35.20: \[ C_{35.20} = \frac{1}{1.04} [0.70 \cdot 8.72 + 0.30 \cdot 1.68] = \frac{1}{1.04}[6.104 + 0.504] = \frac{6.608}{1.04} = 6.35 \] - At $28.80: - Up to $31.68: $1.68 - Down to $25.92: $0 - Value at $28.80: \[ C_{28.80} = \frac{1}{1.04}[0.70 \cdot 1.68 + 0.30 \cdot 0] = \frac{1}{1.04}[1.176 + 0] = \frac{1.176}{1.04} = 1.13 \] **Delta at t=0:** \[ \Delta_0 = \frac{C_{35.20} - C_{28.80}}{35.20 - 28.80} = \frac{6.35 - 1.13}{6.40} = \frac{5.22}{6.40} = 0.8156 \] Round: **0.82** **Initial riskless portfolio:** - **1 share of stock** - **Short 0.82 call options** (to hedge) --- **Explanation:** At the start, to form a riskless portfolio, you need to **buy 1 share of stock and short 0.82 call options**. The Delta represents the sensitivity of the option value to the stock price, so by shorting 0.82 calls per stock, your portfolio is hedged for small moves in the underlying. --- ### **Step 2: Hedge Ratio After Up Move and Probabilities of Terminal Prices** #### **A. Hedge Ratio After Up Move** After first up move, stock is at $35.20. At this node, possible next moves are: - Up to $38.72: Call payoff $8.72 - Down to $31.68: Call payoff $1.68 The new delta here: \[ \Delta_1 = \frac{8.72 - 1.68}{38.72 - 31.68} = \frac{7.04}{7.04} = 1.00 \] **So, after an initial up move to $35.20, the hedge ratio is 1.00.** --- **Explanation:** If the stock rises to $35.20 after the first period, you would need to adjust your hedge so that you are **long 1 share and short 1 call option** to keep the portfolio riskless. --- #### **B. Probabilities of Terminal Prices** We have **p = 0.70, (q = 0.30)**. There are three possible terminal prices after two steps: 1. **$38.72** (UP, UP): - Probability: \(p^2 = 0.70 \times 0.70 = 0.49\) 2. **$31.68** (UP, DOWN or DOWN, UP): - There are **two paths**: Up then Down, or Down then Up. - Probability: \(2 \times (p \times q) = 2 \times (0.70 \times 0.30) = 2 \times 0.21 = 0.42\) 3. **$25.92** (DOWN, DOWN): - Probability: \(q^2 = 0.30 \times 0.30 = 0.09\) **Summary Table:** | Ending Price | Probability | |--------------|-------------| | $38.72 | 0.490 | | $31.68 | 0.420 | | $25.92 | 0.090 | --- **Explanation:** The binomial model gives us the probabilities for each terminal price by multiplying the probability of up and down moves over two periods, and taking into account the number of ways to reach each terminal price. --- ### **Step 3: Present Value of Call and Put Option (Binomial Model)** #### **A. Present Value of Call Option** Recall terminal payoffs for the call ($K = 30$): - $38.72: 8.72$ - $31.68: 1.68$ - $25.92: 0$ \[ C_0 = \frac{1}{(1.04)^2} [0.49 \times 8.72 + 0.42 \times 1.68 + 0.09 \times 0] \] \[ (1.04)^2 = 1.0816 \] \[ 0.49 \times 8.72 = 4.2728 \\ 0.42 \times 1.68 = 0.7056 \\ \text{Sum: } 4.2728 + 0.7056 = 4.9784 \] \[ C_0 = \frac{4.9784}{1.0816} = 4.604 \] **Rounded to nearest cent: $4.60** --- #### **B. Present Value of Put Option** Put payoff at each terminal price ($K = 30$): - $38.72: max(30 - 38.72, 0) = 0$ - $31.68: max(30 - 31.68, 0) = 0$ - $25.92: max(30 - 25.92, 0) = 4.08$ So only the lowest node pays out. \[ P_0 = \frac{1}{1.0816} [0.49 \times 0 + 0.42 \times 0 + 0.09 \times 4.08 ] \] \[ 0.09 \times 4.08 = 0.3672 \] \[ P_0 = \frac{0.3672}{1.0816} = 0.3396 \] **Rounded to nearest cent: $0.34** --- **Explanation:** The present values of the call and put options are computed using the risk-neutral probabilities and discounting the expected payoff at the risk-free rate over two periods. --- ## **Final Answers Summary** 1. **The riskless portfolio will initially contain**: **1 share of stock and short 0.82 call options.** 2. **After an initial up move to $35.20, the hedge ratio is**: **1.00** 3. **Probabilities of terminal stock prices:** | Ending Price | Probability | |--------------|-------------| | $38.72 | 0.490 | | $31.68 | 0.420 | | $25.92 | 0.090 | 4. **Present value of the call option:** **$4.60** 5. **Value of a one-year put option (same strike):** **$0.34** --- ## **Summary Table for Entering Answers:** | Question Part | Value | |------------------------------------------------------|----------| | Number of call options to short initially | 0.82 | | Hedge ratio after up move to $35.20 | 1.00 | | Probability for $38.72 (ending price) | 0.490 | | Probability for $31.68 (ending price) | 0.420 | | Probability for $25.92 (ending price) | 0.090 | | Value of call option (to nearest cent) | $4.60 | | Value of put option (to nearest cent) | $0.34 | **All calculations shown and double-checked for accuracy.**