MFE5130 – Financial Derivatives First Term, 2021-22 Final Examination
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MFE5130 – Financial Derivatives
First Term, 2021-22
Final Examination
Exam Duration: 2 hours
Instruction
1. Total Marks: 100 points.
2. Answer ALL questions.
3. You must show all the steps in order to get full mark for each question.
4. When using the attached standard normal distribution table, do not interpolate.
Use the nearest z-value in the table to find the probability. Example: Suppose that you are to find Pr(Z < 0.759), where Z denotes a standard normal random variable. Because the z-value in the table nearest to 0.759 is 0.76, your answer is Pr(Z < 0.76) = 0.7764.
Use the nearest probability value in the table to find the z-value. Example: Suppose that you are to find z such that Pr(Z < z) = 0.7. Because the probability value in the table nearest to 0.7 is 0.6985, your answer is 0.52.
1. (20 points) The 1-year futures price of Stock A is currently $1,000. The volatility of the futures price is 30%. The continuously compounded risk-free interest rate is 7%. A
1-year American call option on the futures contract has a strike price of $1,000. Use a 3-period forward tree to determine the current price of the American option.
2. (20 points) The current price of a stock is 50. The stock will pay a single dividend of 0.75 in 1 month. The continuously compounded risk-free interest rate is 6%. The following table shows the premiums of 6-month European call options on the stock of various strike prices:
|
|
|
|
50 |
5.19 |
|
60 |
1.96 |
A trader is considering two investment strategies. The first is to long a 50-60 6-month bear spread consisting of selling a 50-strike 6-month call and purchasing a 60-strike 6- month call. The second is to long a 50-60 6-month collar consisting of purchasing a 50- strike 6-month put and selling a 60-strike 6-month call.
Determine the range of stock prices in 6 months for which the profit for the long bear spread is greater that the profit for the long collar in 6 months.
3. (25 points) Let X(t) and Y(t) be two stochastic processes.
Suppose that X(0) = 1.5 and Y(0) = 2.3. Under the probability measure P, the dynamics ofX(t) and Y(t) are given by
dX (t )= 0.06dt +0.18dZ1 (t )+0.32dZ2 (t)
dY (t )= 0.12dt +0.27dZ2 (t)
where Z1(t) and Z2(t) are two independent standard Brownian motions under P.
「 ( 1 X(2)+1 Y(2) 1 X (2) )]
4. (15 points) Assume the Black-Scholes framework. For a European put option on a stock, you are given:
(i) The expiration date is T.
(ii) The strike price is $40.
(iii) The volatility of the European put option is 124%.
(iv) The volatility of the underlying stock is 20%.
A cash-or-nothing put option and an asset-or-nothing put option are on the same underlying stock and have the same expiration date as the European put option. Their details are given in the following table
|
|
Payoff |
Price |
|
Cash-or-nothing put option |
pays $1 if ST < $40 and 0 otherwise |
$0.43 |
|
Asset-or-nothing put option |
pays ST if ST < $40 and 0 otherwise |
$P |
where ST is the stock price at the expiration date T.
Determine P.
5. (20 points) Currently, the price of a stock is $47. The stock does not pay dividends. A
European call option on the stock has a price of $8.85. A European put option on the stock has the same strike price and expiration date as the European call option, and the price of the European put option is $5.66.
Some number of months ago, the price of the stock was $50. The price of the European call option was $12. 13, and its delta was 0.726. The price of the European put option was $4.53. At this time, a market-maker sold 100 of the call options and immediately delta-hedged the position. The market-maker has not ever rebalanced the position. Suppose that the continuously compounded risk-free interest rate is a constant and remains unchanged over time. Calculate the current market-maker’s profit.
2022-12-13