Question 1 Part A: Using the file Stock Prices, CAPM and ROE.xlsx posted on Blackboard
1) Generate the variance covariance matrix and correlation matrix for all stocks (ignore the portfolio column)
2) Make sure that the variance covariance matrix is fully symmetric
3) Build a variance minimizing portfolio choice model with expected returns and summing up constraints.
4) Assuming non-negativity, generate the EV-Frontier using a strict equality on the income constraint for returns starting at 7%, 10%, 13% up to 40%. a. Record the portfolio standard deviation for each return b. Graph the EV frontier
5) Relax non-negativity and regenerate the model as in (4).
6) Compare and interpret the two graphs in the context of say a mutual fund versus a hedge fund.
7) Assuming that short selling is allowed, add a riskless asset to the mix with a risk free return of 4%. Generate solutions from 4% to 40% and record portfolio standard deviations for each.
8) Plot this curve along with the results in (5). (You can invert the x and y axis if you like).
9) Summarize the stock portfolio solutions for an expected return of 25% for each of the three models (4,5,7). Compare the solutions along with their respective standard deviations.
10) What does the comparison in (8) show? How does this relate to the Capital Asset Pricing Model?
11) Using the correlation matrix develop a model to simulate portfolio returns for solutions at 25% without risk free borrowing and without short-selling (i.e. questions 4). What is the probability that portfolio returns will fall below 20% or rise above 30%?. Question 2. Option Pricing You have been asked by a client to price out a number of possible options on a stock over a six-month period (125 days). The current stock price is $25, and its expected risk neutral drift rate is 5%/year. It is also known that the annual volatility is 35%.
12) The first option is a straight forward vanilla European put option that will pay your client if the stock price falls below $20 as at the last trading day at the end of 6 months.
A) What is the expected payoff to this option?______________.
B) What price would you charge the client for this protection before any transaction fees?__________
13) Your client then asked you to price an option that would payoff only if the average price over the 6 months was less than $20 (e.g. an Asian option).
A) What is the expected payoff to this option?______________. B) What price would you charge the client for this protection before any transaction fees?__________
14) The next option considered was even more exotic. Your client asked you to price an option which would trigger a European put option, with a strike price of $25, if and only if the lowest price observed in the last 30 days of trading was below $20.
A) What is the expected payoff to this option?______________.
B) What price would you charge the client for this protection before any transaction fees?__________
C) What is the probability that the option will be exercised? ___________________