Math 5632
Exam 5 - Form A 一 Tychonievich
December 14, 2022
Problem 1 /25
You are given two assets with the following statistics. It is requested that you construct a portfolio consisting of positions in these stocks that has an expected annual rate of return of 15%. Determine if such a portfolio exists, and find the variance of the return of this portfolio if it does. The correlation coefficient of the returns for these two stocks is 0.2.
|
Stock
|
Expected return
|
Variance of return
|
|
X
|
0.04
|
0.5
|
|
Y
|
0.09
|
0.2
|
Problem 2 /25
You are given two assets with the following statistics. Determine the makeup of the market (tangency) portfolio if the risk-free rate is 0.02. The correlation coefficient of the returns for these two stocks is 0.2.
|
Stock
|
Expected return
|
Variance of return
|
|
X
|
0.04
|
0.5
|
|
Y
|
0.09
|
0.2
|
Problem 3 /20
A European put has current price 23 with Greeks as in the table below. The current stock price is 90. Estimate the value of the put if the stock price increases to 100 after 6 months, using the Delta-Gamma-Theta approximation.
|
Delta
|
Gamma
|
Theta
|
Vega
|
Rho
|
Psi
|
|
-0.34
|
0.0057
|
-3.39
|
46.12
|
-107.33
|
61.43
|
Problem 4 /20
The following table contains premiums for European call options for a given stock, each of which has an expiration time of 6 months. Find an arbitrage opportunity from this data and construct a portfolio that will extract the arbitrage. Then calculate the minimum profit that your strategy will generate. The risk-free rate is r = 0.08 and the stock’s continuous dividend rate is δ = 0.02.
|
Strike
|
80
|
100
|
110
|
|
Premium
|
38
|
32
|
28
|
Problem 5 /20
Stock in CRR corp has been found to have a volatility of 30%. The risk-free rate is 6% (annual, continuously compounded), and the current stock price is 50. The stock does not pay dividends. Using the CRR model, calculate the price of a one-year European call with a strike price of 55 using a period length of 4 months.
Problem 6 /20
A stock obeys the Black-Scholes model with S(0) = 15, α = 0.05, r = 0.07,δ = 0.04, and σ = 0.3. Calculate the following expectation value:
Problem 7 /20
A portfolio consists of a number of European call contracts, all on the same asset and with the same expiration. The strike prices for these calls are 2, 5, and 8.
1. Draw the payoff diagram for this portfolio below.
2. On the next page, work out how many calls of each type you must hold to make the portfolio, given that the payoff for the portfolio at various prices is given as follows. You may indicate that a call has been written by stating that the portfolio contains a negative number of that contract. You must show complete work justifying your conclusions in this part to receive any credit.
|
Price
|
0
|
2
|
5
|
8
|
10
|
|
Payoff
|
0
|
0
|
4
|
2
|
4
|
|
Calls held
|
|
|
|
|
|