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ISE 563 Homework 4
发布时间：2024-05-09
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ISE 563 Homework 4
Due Date: May 4th, 2024
Exercise 4.1.
Estimate the GARCH(1,1) parameters for the daily stock data in the attached excel file.
What are the GARCH parameters and the forecasted volatilities 30, 365, and 730 days into the future?
What are the parameters if the errors are t-distributed with f degrees of freedom?
Exercise 4.2
Given the following portfolio
• One unit of stock S with initial value $100
• $100 in a money market (MM) risk-free account (Risk-free)
• One 3-year Put option on S with strike of $80.
Other assumptions
• Risk free rate term structure of 3% flat
o Risk-free rate follows a Vasicek process with a=0.10, b=0.03, and sigma = 0.01. Term structure flat at the risk-free level
• Stock Real World (RW) growth rate of 8%
• Stock RW volatility of 17%
• Implied volatility of 20%, flat term structure and no skew
• 30% correlation between stock return and change in risk- free rates
Simulate 100 (or more) RW scenarios using 12 monthly steps and calculate the following.
a) What is the mean, median, std dev, VaR (90%), and Expected Shortfall (90%) of this portfolio in 1 year?
b) What is the mean, median, std dev, VaR, and Expected Shortfall of the same portfolio if you rebalance the Stock and MM to the 50%/50% weights each month?
Exercise 4.3
Given monthly US Treasury rates in the Excel file.
• Calculate the PCAs using monthly changes for the following specifications
o Simple differences: [x(t) – x(t-1)]
o Log differences: ln[x(t) / x(t-1)]
o Displaced log differences: ln[(x(t)+2%) / (x(t-1)+2%)]
• What percent of the variation is accounted for by the first 3 Principal Components?
• What would be a 2-standard deviation confidence interval for the first Principal Component over a 1-month horizon? A 12-month horizon?
• Redo using annual changes and compare your 12-month confidence intervals?
Exercise 4.4
Use a 3-step trinomial tree to value the following real option to either expand protection or abandon a project. The underlying commodity is C with the initial value of 20 and a risk-free rate of 4%. The risk neutral process for C is:
C/dC = [θ(t) − 0. 1 ln(s)]dt + 0.2dz
The future prices for years 1, 2, and 3 are $21, $22, and $23. Cashflows are based on C are received in years 1, 2, and 3. The profit for each unit is given by:
Profit = 1.2*C - 25
. What is the value of the profits with no optionality?
. What is the value of the project if you can abandon at times 1 and 2 for $5?
. What is the value of the profits if you can double production at time 2 for $10?
