You are Asked to Estimate the Parameters in the Black-Scholes model: Maths Assignment, NZ

1. In this question, you are asked to estimate the parameters in the Black-Scholes model by using the NASDAQ100 index data ranging from 1 January 2013 to 31 December 2018.

(a) Download the NASDAQ100 index data from the Yahoo Finance website and import the data (S marla into SAS. Hint: The NASDAQ100 index data have 1510 daily observations in this period. Please check that you get the correct Index data

(b) Using the formula r(ti)= In (r5N, calculate the daily log-returns of the index data.

(c) Visualize the index data and the daily return data in SAS by using Proc Gplot.

(d) Using your results in (b), calculate the mean and the variance of the log return data.

(e) Find the empirical 5% and 95% quartiles of the log return data by using SAS.

(f) According to the Black-Scholes model, the daily log return is normally distributed. Using your (5 marks) results in (d) as the parameters for the Black-Scholes model, find the theoretical 5% and 95% quartiles of the log return data in the Black-Scholes model. Compare the theoretical results with the results in (e).

2. (a) Suppose that security is presently selling at $150, the nominal interest rate is 3% per annum (S marks) compounding continuously, and the volatility of the security return Is 25% per annum. Use SAS to find the no-arbitrage cost of a European call option that expires in three months and has a strike price $152.

(b) Suppose that the current price of a stock is $100. and a European call option with a strike price (7 maths) S98 expires in 6 months. Suppose that the risk-free Interest rate is 3% per annum compounding continuously. If the market price of this option is S4.5. Apply the New-Raphson method in 3 steps to find an estimate of the implied volatility of the stock return.

(c) Suppose that the current price of a stock is $50 and the risk-free interest rate is 4% per annum compounding continuously. You observe that the market prices of a set of European call options on this stock with strike prices Kt = 40, K2 = 45. K3 = 48. Ka = 50, K5 = 52, 4 = 55, K7 = 60. all of which expire in 6 months, are = 11, c2 = 6.8. = 4.5. = 3.3. 4 = 2.5. 4 = 1.8. c2 = 1. respectively. Use SAS to plot the implied volatility smile.

3. Assume that the current interest rate is 4% per annum compounding continuously. Consider the Vatitek model with the parameters: a = 100. p = 0.05 and a = 0.02.

(a) Write a SAS function to generate one sample path for the interest rates within I year from now. with a step.size a.r = 1 week. Plot the sample path, and add the horizontal line of long-term mean interest rate as a reference line.

(b) Use the Monte Carlo simulation in MS to estimate the mean and variance of the interest rate at the end of year 1.

(c) Compare your simulation results in (b) against the theoretical results obtained from the Va&ek model. That can you conclude?

4. Assume that the current Interest rate is 5% per annum compounding continuously. Consider the Cox-Ingersoll-Ross (in short, an) model with a = 100. p = 0.05 and a = 0.025.

(a) Write an MS function to generate one sample path for the interest rates within I year from now. with a step-size At = 1 week. Plot the sample path, and add the horizontal line of long-term mean interest rate as a reference line.

b. Use the Monte Carlo simulation in SAS to estimate the mean and variance of the interest rate at (7 marks) the end of the year I.

(c) Compare your simulation results in 1b) against the theoretical results obtained from the aft model. What can you conclude?

5.  The parameters of a GA80111.11 model are estimated as co = 0.000003. a = 0.12 and = 0.83.

(a) What is the long-run average volatility?

(b) What is the equation describing the way that the variance rate reverts to its long-run average?

(c) If the current volatility is 2091 per annum. calculate the expected daily volatility in 20 days.

(d) Suppose the strike value of a variance swap maturing in 20 days Is 0.02, and the principal Is 15 marks) 1.000.000. Using your answer in (c). calculate the expected payoff of the long position of the variance swap?