Question 6 (9 marks) You would like to set up a portfolio with the payoff structure at time I depicted in the figure below. The underlying stock is a non-dividend paying stock, and its current stock price is $23. The prevailing continuous compound risk-free rate is 5% per annum, and T = 1 year. Figure: Payoff Structure (in %) 10 20 30 40 O ST -10 Payoff Suppose that except the shares of the underlying stock and the risk-free bonds, there are also six call options with strike prices X = $20, $21. ..., $25 (all integer values from $20 to $25), and five put options with strike prices X = $26. $27, .... $30 (all integer values from $26 to $30) available in the market. All the option contracts are European style options which mature at time T and are written on the same underlying stock. Assume the time value of options are non-negative, and any of the securities can be bought or (short) sold. a). Clarify how you would set up the portfolio today to ensure the payoff structure at time T in the above figure, assuming that put-call parity holds for the options. (3 marks) b). Suppose that the cost for setting up the portfolio in Part a) is $6.80. The negative cost indicates that you receive, rather than pay out, $6.80 when forming the portfolio. Explain why the payoff structure could limit the downside risk at time T, and calculate the maximum (possible) loss of the portfolio at time T. (2 marks) c). Suppose for the six call options and five put options, you observe their current option prices in the two tables below: Table: Call option premium with different strike prices Call Option Premium $4.26 $3.58 $2.96 $2.40 $2.96 $1.45 Strike Price X $20 $21 $22 $23 $24 $25 Table: Put option premium with different strike prices Put Option Premium $4.05 $4.69 Strike Price. X $26 $27 $5.38 $28 $5.90 $29 $7.08 $30 Assume among all the option contracts above, there are one call option and one put option that are being mispriced, respectively. That is, their prices deviate from the theoretical value. Explain which call and which put are the mostly likely being mispriced and why. (Your answer should be no longer than 150 words) (4 marks) Question 6 (9 marks) You would like to set up a portfolio with the payoff structure at time I depicted in the figure below. The underlying stock is a non-dividend paying stock, and its current stock price is $23. The prevailing continuous compound risk-free rate is 5% per annum, and T = 1 year. Figure: Payoff Structure (in %) 10 20 30 40 O ST -10 Payoff Suppose that except the shares of the underlying stock and the risk-free bonds, there are also six call options with strike prices X = $20, $21. ..., $25 (all integer values from $20 to $25), and five put options with strike prices X = $26. $27, .... $30 (all integer values from $26 to $30) available in the market. All the option contracts are European style options which mature at time T and are written on the same underlying stock. Assume the time value of options are non-negative, and any of the securities can be bought or (short) sold. a). Clarify how you would set up the portfolio today to ensure the payoff structure at time T in the above figure, assuming that put-call parity holds for the options. (3 marks) b). Suppose that the cost for setting up the portfolio in Part a) is $6.80. The negative cost indicates that you receive, rather than pay out, $6.80 when forming the portfolio. Explain why the payoff structure could limit the downside risk at time T, and calculate the maximum (possible) loss of the portfolio at time T. (2 marks) c). Suppose for the six call options and five put options, you observe their current option prices in the two tables below: Table: Call option premium with different strike prices Call Option Premium $4.26 $3.58 $2.96 $2.40 $2.96 $1.45 Strike Price X $20 $21 $22 $23 $24 $25 Table: Put option premium with different strike prices Put Option Premium $4.05 $4.69 Strike Price. X $26 $27 $5.38 $28 $5.90 $29 $7.08 $30 Assume among all the option contracts above, there are one call option and one put option that are being mispriced, respectively. That is, their prices deviate from the theoretical value. Explain which call and which put are the mostly likely being mispriced and why. (Your answer should be no longer than 150 words) (4 marks)
