ECN603 Asset Pricing/Workshop 5
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ECN603 Asset Pricing/Workshop 5
1.
A stock whose price is $30 has an expected return of 9% and a volatility of 20%. In Excel, simulate the stock price path over 5 years using monthly time steps and random samples from a normal distribution. Chart the simulated stock price path. By hitting F9 observe how the path changes as the random sample change.
2.
Suppose that a stock price has an expected return of 16% per annum and a volatility of 30% per annum. When the stock price at the end of a certain day is $50, calculate the following:
(a) The expected stock price at the end of the next day.
(b) The standard deviation of the stock price at the end of the next day.
(c) The 95% confidence limits for the stock price at the end of the next day.
3.
What is the price of a European call option on a non-dividend-paying stock when the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30% per annum, and the time to maturity is three months?
4.
What is the price of a European put option on a non-dividend-paying stock when the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the volatility is 35% per annum, and the time to maturity is six months?
5.
A stock price is currently $50. Assume that the expected return from the stock is 18% per annum and its volatility is 30% per annum. What is the probability distribution for the stock price in two years (assume the behavior of the stock is as specified in the lectures)? Calculate the mean and standard deviation of the distribution. Determine the 95% confidence interval.
6.
Calculate the value of an eight-month European put option on a currency with a strike price of 0.50. The current exchange rate is 0.52, the volatility of the exchange rate is 12%, the domestic risk-free interest rate is 4% per annum, and the foreign risk-free interest rate is 8% per annum.
7.
Suppose that a portfolio is worth $60 million and the S&P 500 is at 1200. If the value of the portfolio mirrors the value of the index, what options should be purchased to provide protection against the value of the portfolio falling below $54 million in one year’s time?
8.
Consider again the situation in Problem 7 . Suppose that the portfolio has a beta of 2.0, the risk-free interest rate is 5% per annum, and the dividend yield on both the portfolio and the index is 3% per annum. What options should be purchased to provide protection against the value of the portfolio falling below $54 million in one year’s time?
9.
A financial institution has the following portfolio of over-the-counter options on sterling:
|
Type |
Position |
Delta of Option |
Gamma of Option |
Vega of Option |
|
Call |
−1,000 |
0.5 |
2.2 |
1.8 |
|
Call |
−500 |
0.8 |
0.6 |
0.2 |
|
Put |
−2,000 |
-0.40 |
1.3 |
0.7 |
|
Call |
−500 |
0.70 |
1.8 |
1.4 |
A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8.
(a) What position in the traded option and in sterling would make the portfolio both gamma neutral and delta neutral?
(b) What position in the traded option and in sterling would make the portfolio both vega neutral and delta neutral?
10.
Consider again the situation in Problem 9. Suppose that a second traded option with a delta of 0.1, a gamma of 0.5, and a vega of 0.6 is available. How could the portfolio be made delta, gamma, and vega neutral?
11.
A put option on the S&P 500 has an exercise price of 500 and a time to maturity of one year. The risk free rate is 7% and the dividend yield on the index is 3%. The volatility of the index is 20% per annum and the current level of the index is 500. A financial institution has a short position in the option.
a) Calculate the delta, gamma, and vega of the position. b) How can the position be made delta neutral? c) Suppose that one week later the index has increased to 515. How can delta neutrality be preserved?
12.
A bank’s position in options on the dollar–euro exchange rate has a delta of 30,000 and a gamma of 一80, 000 . Explain how these numbers can be interpreted. The exchange rate (dollars per euro) is 0.90. What position would you take to make the position delta neutral?
2023-05-23