Today is assumed to be 2021/1/15, and you are asked to analyze the following trading strategies constructed based on S&P 500 index options (listed on CBOE) and E-Mini S&P 500 futures (listed on CME) after the markets are closed. Both S&P 500 index options and E-Mini S&P 500 futures are assumed to be settled in cash on 15th of the contract month. For S&P 500 index options (E-Mini S&P 500 futures), one index point is worth $100 ($50). Suppose that there is no margin requirement for trading S&P 500 index options and E-Mini S&P 500 futures. In addition, the settlement price of the E-Mini S&P 500 futures on the delivery date is assumed to converge perfectly to the closing price of the S&P 500 index on the same day, i.e., ignoring the basis risk. Last, transaction costs are not considered except the bid and ask prices for trading S&P 500 index options. The current market information is summarized as follows. The closing prices of the S&P 500 index and the E-Mini S&P 500 March index futures are 3768.25 and 3762.25, respectively, on 2021/1/15. In addition, Tables 1 and 2 show the quoted closing prices of S&P 500 index options. Note that an individual trader like you always buys options at the ask price and writes options at the bid price. For the following two months, the risk-free interest rate is assumed to be a constant of 0.08% per annum, the dividend yield for the S&P 500 index is assumed to be a constant of 1.55% per annum, and the annualized volatility (o) of log returns of S&P 500 index is assumed to be 19.45%. For simplicity, the E-Mini S&P 500 futures and S&P index call or put options are hereafter mentioned as futures and calls or puts, respectively. The strike price is denoted as K. The S&P 500 index levels today and at maturity are denoted as So and ST, respectively. 1. (8%) Answer these questions based on the formulas in Chapters 5 and 13. Note that the time to maturity is approximated as 2/12 years. a. (2%) What is the theoretical price (in points) for the E-Mini S&P 500 March index futures on 2021/1/15? b. (6%) What are the Black-Scholes-Merton theoretical values (in points) for the S&P 500 March index call and put given the strike price K = 3800 on 2021/1/15? Ans: Note that to take the dividend yield into account, European options can be priced by simply reducing the current stock price to Soe-ar in the Black-Scholes-Merton formula (formally introduced on Slide 13.18): c = Soe TN (d) - Ke-N(d), p = Ke TN(-d) - Soe-9T N(-d), where d = In(Soe-9T/K)+(r+o/2)T In(So/K)+(r-q+o/2)T OVT and d = OVT In(Soe-T/K)+(r-o/2)T_In(So/K)+(r-q-o/2)T OVT = -=doVT. To calculate N(d), the cumulative OVT distribution function of the standard normal distribution, one can employ the function of NORMSDIST() in EXCEL. Today is assumed to be 2021/1/15, and you are asked to analyze the following trading strategies constructed based on S&P 500 index options (listed on CBOE) and E-Mini S&P 500 futures (listed on CME) after the markets are closed. Both S&P 500 index options and E-Mini S&P 500 futures are assumed to be settled in cash on 15th of the contract month. For S&P 500 index options (E-Mini S&P 500 futures), one index point is worth $100 ($50). Suppose that there is no margin requirement for trading S&P 500 index options and E-Mini S&P 500 futures. In addition, the settlement price of the E-Mini S&P 500 futures on the delivery date is assumed to converge perfectly to the closing price of the S&P 500 index on the same day, i.e., ignoring the basis risk. Last, transaction costs are not considered except the bid and ask prices for trading S&P 500 index options. The current market information is summarized as follows. The closing prices of the S&P 500 index and the E-Mini S&P 500 March index futures are 3768.25 and 3762.25, respectively, on 2021/1/15. In addition, Tables 1 and 2 show the quoted closing prices of S&P 500 index options. Note that an individual trader like you always buys options at the ask price and writes options at the bid price. For the following two months, the risk-free interest rate is assumed to be a constant of 0.08% per annum, the dividend yield for the S&P 500 index is assumed to be a constant of 1.55% per annum, and the annualized volatility (o) of log returns of S&P 500 index is assumed to be 19.45%. For simplicity, the E-Mini S&P 500 futures and S&P index call or put options are hereafter mentioned as futures and calls or puts, respectively. The strike price is denoted as K. The S&P 500 index levels today and at maturity are denoted as So and ST, respectively. 1. (8%) Answer these questions based on the formulas in Chapters 5 and 13. Note that the time to maturity is approximated as 2/12 years. a. (2%) What is the theoretical price (in points) for the E-Mini S&P 500 March index futures on 2021/1/15? b. (6%) What are the Black-Scholes-Merton theoretical values (in points) for the S&P 500 March index call and put given the strike price K = 3800 on 2021/1/15? Ans: Note that to take the dividend yield into account, European options can be priced by simply reducing the current stock price to Soe-ar in the Black-Scholes-Merton formula (formally introduced on Slide 13.18): c = Soe TN (d) - Ke-N(d), p = Ke TN(-d) - Soe-9T N(-d), where d = In(Soe-9T/K)+(r+o/2)T In(So/K)+(r-q+o/2)T OVT and d = OVT In(Soe-T/K)+(r-o/2)T_In(So/K)+(r-q-o/2)T OVT = -=doVT. To calculate N(d), the cumulative OVT distribution function of the standard normal distribution, one can employ the function of NORMSDIST() in EXCEL