3. Mimi is an intern working in Goldman Sachs. One day, his boss asked him to form the following portfolio: long 1 European put with strike = $30 long 1 European put with strike = $40 short 2 European puts with strike = $35 All the calls in the portfolio are European options expiring in 8 months written on the same non-dividend paying stock. The underlying stock is currently trading at $35 per share. Assume a constant cash (i.e., risk-free) rate of 1%, and that the stock price follows a gBm with a constant volatility of 10%. (a) Draw the portfolio's payoff at maturity as a function of the spot price at maturity. (b) What is the cost of the porfolio? (e) Compute the "delta of the portfolio. How could Mimi hedge against the risk of this portfolio at this moment? (d) It turns out that only European calls instead of puts are available in the option market at the moment. To avoid being sacked on the spot, please help Mimi form another portfolio consisting of only European calls but yielding identical payoff as the above portfolio at maturity. (Hint: Use the put-call parity.) 4. (Ex: 7.1 of Tsay (2010, 3rd ed)) Consider the daily returns of GE stock from January 2, 1998, to December 31, 2008. The data can be obtained from CRSP or the file d-ge9808.txt. Convert the simple returns into log returns. Suppose that you hold a long position on the stock valued at $1 million. Use the tail probability 0.01. Compute the value at risk of your position for 1-day horizon and 15-day horizon using the following methods: (a) The Risk Metrics method. (b) A Gaussian ARMA-GARCH model. (c) An ARMA-GARCH model with a Student-t distribution. You should also estimate the degrees of freedom. (d) The traditional extreme value theory with subperiod length n= 21. 3. Mimi is an intern working in Goldman Sachs. One day, his boss asked him to form the following portfolio: long 1 European put with strike = $30 long 1 European put with strike = $40 short 2 European puts with strike = $35 All the calls in the portfolio are European options expiring in 8 months written on the same non-dividend paying stock. The underlying stock is currently trading at $35 per share. Assume a constant cash (i.e., risk-free) rate of 1%, and that the stock price follows a gBm with a constant volatility of 10%. (a) Draw the portfolio's payoff at maturity as a function of the spot price at maturity. (b) What is the cost of the porfolio? (e) Compute the "delta of the portfolio. How could Mimi hedge against the risk of this portfolio at this moment? (d) It turns out that only European calls instead of puts are available in the option market at the moment. To avoid being sacked on the spot, please help Mimi form another portfolio consisting of only European calls but yielding identical payoff as the above portfolio at maturity. (Hint: Use the put-call parity.) 4. (Ex: 7.1 of Tsay (2010, 3rd ed)) Consider the daily returns of GE stock from January 2, 1998, to December 31, 2008. The data can be obtained from CRSP or the file d-ge9808.txt. Convert the simple returns into log returns. Suppose that you hold a long position on the stock valued at $1 million. Use the tail probability 0.01. Compute the value at risk of your position for 1-day horizon and 15-day horizon using the following methods: (a) The Risk Metrics method. (b) A Gaussian ARMA-GARCH model. (c) An ARMA-GARCH model with a Student-t distribution. You should also estimate the degrees of freedom. (d) The traditional extreme value theory with subperiod length n= 21