Certainly! Let's break down the problem and each sub-question step by step. --- ## **Given Data** - **SET 50 Index (Spot Price, S)**: 840 points - **850 Call price (C)**: 8 points - **850 Put price (P)**: 3 points - **Strike Price (K)**: 850 points - **Risk-Free Rate (r)**: 6% per year - **Time to Maturity (T)**: 90 days = .25 years --- ## **(i) Put-Call Parity and Mispricing** **Put-Call Parity Formula:** \[ C - P = S - K e^{-rT} \] Let’s plug in the values: - \( S = 840 \) - \( K = 850 \) - \( C = 8 \) - \( P = 3 \) - \( r = .06 \) - \( T = .25 \) First, calculate \( K e^{-rT} \): \[ K e^{-rT} = 850 \times e^{-.06 \times .25} \approx 850 \times e^{-.015} \approx 850 \times .9851 \approx 837.335 \] Now, plug into the parity: \[ C - P = S - K e^{-rT} \] \[ 8 - 3 = 840 - 837.335 \] \[ 5 = 2.665 \] **Nature of Mispricing:** - The left side (actual option prices): 5 - The right side (theoretical value from parity): 2.665 **Conclusion:** There is a mispricing. The options are overpriced by \(5 - 2.665 = 2.335\) points compared to the put-call parity. --- ## **(ii) Arbitrage Strategy and Arbitrage Profit** Since \( C - P > S - K e^{-rT} \), you should: 1. **Sell the call (receive 8)** 2. **Buy the put (pay 3)** 3. **Buy the index (pay 840)** 4. **Borrow \( K e^{-rT} = 837.335 \) at risk-free rate (receive 837.335)** ### **Cash Flows Today:** - Sell call: \( +8 \) - Buy put: \( -3 \) - Buy index: \( -840 \) - Borrow: \( +837.335 \) **Net Cost:** \[ = 8 - 3 - 840 + 837.335 = 2.335 \] So, you **receive 2.335 points today**. ### **At Expiry:** - **Index matures at \( S_T \)** - **Call and put payoff depend on \( S_T \)** - **Repay loan: \( K = 850 \) (since you borrowed \( 837.335 \) at risk-free rate for .25 years)** Let's check the final position at expiry for both cases: #### **Case 1: \( S_T > 850 \)** - Call is exercised: you must deliver the index at 850 (but you own it) - Put expires worthless - Sell index at 850 (via call) - Repay loan: 850 #### **Case 2: \( S_T < 850 \)** - Put is exercised: you sell index at 850 (though it is worth less) - Call expires worthless - Repay loan: 850 **In both cases, your net payoff from the options, index, and loan is zero.** **You keep the initial 2.335 points as arbitrage profit.** --- ## **(iii) Graphing the Arbitrage Strategy and Overall Position** Graph the net payoff at expiry: - **X-axis:** \( S_T \) (Index at expiry) - **Y-axis:** Payoff Since you have no exposure at expiry (payoff always zero), your graph is a flat line at zero for all \( S_T \). --- ## **(iv) Proving the Strategy is Riskless** - Your final payoff at expiry is zero, regardless of where the index ends. - You already locked in a gain of 2.335 points at inception. - All positions (long index, short call, long put, borrowed money) cancel out at expiry. **Conclusion:** **Yes, the arbitrage is riskless because your net payoff is independent of the index's final value.** --- ### **Summary Table** | Action | Today (points) | At Expiry (\( S_T > 850 \)) | At Expiry (\( S_T < 850 \)) | |---------------|---------------|----------------------------|----------------------------| | Sell Call | +8 | \(- (S_T - 850)\) | | | Buy Put | -3 | | \(850 - S_T\) | | Buy Index | -840 | \(S_T\) | \(S_T\) | | Borrow | +837.335 | -850 | -850 | | **Total** | **+2.335** | | | --- If you have questions about any step or need the graph visualized, let me know!

# Explanation of Arbitrage Strategy Using Put-Call Parity ## Given Data - **SET 50 Index (Spot Price, S)**: 840 points - **850 Call Price (C)**: 8 points - **850 Put Price (P)**: 3 points - **Strike Price (K)**: 850 points - **Risk-Free Rate (r)**: 6% per year - **Time to Maturity (T)**: 90 days (0.25 years) ## (i) Put-Call Parity and Mispricing ### Put-Call Parity Formula \[ C - P = S - K e^{-rT} \] ### Calculating \( K e^{-rT} \) 1. Calculate \( e^{-rT} \): \[ e^{-0.06 \times 0.25} \approx e^{-0.015} \approx 0.9851 \] 2. Calculate \( K e^{-rT} \): \[ K e^{-rT} = 850 \times 0.9851 \approx 837.335 \] ### Plugging Values into the Parity Equation \[ C - P = S - K e^{-rT} \] \[ 8 - 3 = 840 - 837.335 \] \[ 5 = 2.665 \] ### Nature of Mispricing - **Left Side (Actual Prices)**: 5 - **Right Side (Theoretical Value)**: 2.665 **Conclusion**: There is a mispricing of approximately 2.335 points, indicating that the options are overpriced. ## (ii) Arbitrage Strategy and Profit ### Strategy Outline 1. **Sell the Call**: Receive 8 points 2. **Buy the Put**: Pay 3 points 3. **Buy the Index**: Pay 840 points 4. **Borrow at Risk-Free Rate**: Receive \( K e^{-rT} = 837.335 \) ### Cash Flows Today - Sell Call: \( +8 \) - Buy Put: \( -3 \) - Buy Index: \( -840 \) - Borrow: \( +837.335 \) ### Net Cost Calculation \[ \text{Net Cost} = 8 - 3 - 840 + 837.335 = 2.335 \] You **receive 2.335 points today**. ### Payoff at Expiry #### Case 1: \( S_T > 850 \) - Call is exercised (must deliver index). - Put expires worthless. - Sell index at 850. - Repay loan of 850. #### Case 2: \( S_T < 850 \) - Put is exercised (sell index at 850). - Call expires worthless. - Repay loan of 850. **Final Position**: In both cases, net payoff is zero. Arbitrage profit remains at 2.335 points. ## (iii) Graphing the Arbitrage Strategy ### Graph Description - **X-axis**: \( S_T \) (Index at expiry) - **Y-axis**: Payoff The graph is a flat line at zero for all \( S_T \) values, indicating no exposure at expiry. ## (iv) Riskless Nature of the Strategy - Final payoff at expiry is zero, independent of \( S_T \). - Initial gain of 2.335 points is locked in. - All positions offset each other at expiry. **Conclusion**: The arbitrage strategy is riskless due to a guaranteed net payoff of 2.335 points, unaffected by the index's final value. --- If you need further clarification or additional details, feel free to ask!

# Summary of Arbitrage Strategy Using Put-Call Parity ## Given Data - Spot Price of SET 50 Index: 840 points - 850 Call Price: 8 points - 850 Put Price: 3 points - Strike Price: 850 points - Risk-Free Rate: 6% per year - Time to Maturity: 90 days (0.25 years) ## (i) Mispricing Analysis Using the put-call parity formula \( C - P = S - K e^{-rT} \), we calculated: 1. \( K e^{-rT} \approx 837.335 \) 2. The left side of the equation (actual prices) yields 5, while the right side (theoretical value) is approximately 2.665. This indicates a mispricing of about 2.335 points, suggesting the options are overpriced. ## (ii) Arbitrage Strategy To capitalize on this mispricing: 1. Sell the call and receive 8 points. 2. Buy the put and pay 3 points. 3. Buy the index and pay 840 points. 4. Borrow approximately 837.335 points at the risk-free rate. ### Cash Flows Today The net cash flow at inception is a receipt of 2.335 points. ### Payoff at Expiry - If the index price is above 850, the call is exercised. - If the index price is below 850, the put is exercised. In both scenarios, the final payoff is zero, ensuring the initial arbitrage profit remains intact. ## (iii) Graphing the Arbitrage Strategy The graph depicts a flat line at zero payoffs across all possible index prices at expiry, demonstrating no risk exposure. ## (iv) Riskless Nature of the Strategy The strategy guarantees a net profit of 2.335 points, independent of the index's final value, confirming its riskless nature.