Let's break down the problem step by step: **Given:** - Current stock price, \( S_0 = \$60 \) - Strike price, \( E = \$55 \) - Up factor, \( u = 0.10 \Rightarrow S_u = S_0 \times (1 + u) = 60 \times 1.10 = \$66 \) - Down factor, \( d = -0.15 \Rightarrow S_d = S_0 \times (1 + d) = 60 \times 0.85 = \$51 \) - Risk-free rate, \( r = 8\% \Rightarrow R_f = 1.08 \) - One period (single period binomial model) - Call price to be calculated: \( C \) - Consider call priced at \( \$15.00 \) and \( \$6.00 \) for arbitrage. --- ### **Step 1: Calculate Call Payoff at Expiration** - **Up:** \( S_u = \$66 \Rightarrow \text{Call payoff } = \max(66 - 55, 0) = 11 \) - **Down:** \( S_d = \$51 \Rightarrow \text{Call payoff } = \max(51 - 55, 0) = 0 \) --- ### **Step 2: Risk-neutral Probability** The risk-neutral probability \( p \) is: \[ p = \frac{R_f - d}{u - d} \] Plug in the values: \[ R_f = 1.08, \quad u = 0.10, \quad d = -0.15 \] But for the binomial model, \( u \) and \( d \) are *multiplicative factors*, so typically: - \( u = 1.10 \) - \( d = 0.85 \) So: \[ p = \frac{1.08 - 0.85}{1.10 - 0.85} = \frac{0.23}{0.25} = 0.92 \] --- ### **Step 3: Calculate Hedge Ratio (\( \Delta \))** The hedge ratio \( \Delta \) is: \[ \Delta = \frac{C_u - C_d}{S_u - S_d} \] - \( C_u = 11 \), \( C_d = 0 \) - \( S_u = 66 \), \( S_d = 51 \) \[ \Delta = \frac{11 - 0}{66 - 51} = \frac{11}{15} = 0.733 \] --- ### **Step 4: Call Price (\( C \))** The risk-neutral value of the call is: \[ C = \frac{pC_u + (1-p)C_d}{R_f} \] \[ C = \frac{0.92 \times 11 + 0.08 \times 0}{1.08} \] \[ C = \frac{10.12}{1.08} = 9.37 \] --- ### **Step 5: Replicating Portfolio** Construct a portfolio of \( \Delta \) shares and borrow \( B \): \[ B = \frac{uC_d - dC_u}{u - d} \] But let's use the constructed replicating portfolio: At time 0: \[ C_0 = \Delta S_0 + B \] \[ 9.37 = 0.733 \times 60 + B \] \[ 9.37 = 43.98 + B \] \[ B = 9.37 - 43.98 = -34.61 \] So, **borrow** \$34.61 at the risk-free rate. At \( t = 1 \): - **If up:** Portfolio is \( 0.733 \times 66 - 34.61 \times 1.08 = 48.378 - 37.379 = 11 \) - **If down:** \( 0.733 \times 51 - 34.61 \times 1.08 = 37.383 - 37.379 = 0.004 \approx 0 \) This matches the call payoffs. --- ### **Step 6: Arbitrage at C = \$15.00 and C = \$6.00** #### **If \( C = \$15.00 \) (Overpriced):** - **Short the call at \$15.00** - **Buy the replicating portfolio for \$9.37** - **Profit:** \( 15.00 - 9.37 = \$5.63 \) risk-free #### **If \( C = \$6.00 \) (Underpriced):** - **Buy the call at \$6.00** - **Sell the replicating portfolio at \$9.37** - **Profit:** \( 9.37 - 6.00 = \$3.37 \) risk-free --- ## **Summary Table** | Item | Value | |---------------------|-----------| | Hedge ratio (\( \Delta \)) | 0.733 | | Call price (\( C \)) | \$9.37 | | p (risk-neutral prob.) | 0.92 | | Replicating portfolio cost | \$9.37 | | Arbitrage if \( C = 15 \) | +\$5.63 | | Arbitrage if \( C = 6 \) | +\$3.37 | --- ## **Conclusion** 1. **Hedge ratio (\( \Delta \))**: **0.733** 2. **Call price**: **\$9.37** 3. **Risk-neutral portfolio**: Buy 0.733 shares, borrow \$34.61, matches call payoff. 4. **If call is overpriced at \$15**, arbitrage profit: **\$5.63** per option (short call, buy hedge). 5. **If call is underpriced at \$6**, arbitrage profit: **\$3.37** per option (buy call, sell hedge). **This demonstrates risk-neutral pricing and the absence of arbitrage in equilibrium.**