Mathematical Finance Coursework
• This Coursework is marked out of 100 and counts for 20% of your final mark. Work is to be submitted by midnight of the 1st May 2025 and will be marked in the following couple of weeks.
• Your coursework should consist of a PDF file with your report and all your results.
• The maximum number of pages of the PDF is 5 in 12pt font size. Adding content not directly relevant to the question will not give you extra marks. There is no restriction on the word count.
• You will need the file Data.csv which can be found also on Blackboard. This file contains daily prices, not including weekends and holidays when the market is closed; as an average over the past five years you can assume that each year has 252 days.
• Make sure to explain how you did all your calculations and reference appropriately any information source you use including citations.
• You can use a programming language or spreadsheet but not AI systems. It is not necessary to include any programming code in your answer file.
• All the graphs should have appropriate titles, captions, and labeled horizontal and vertical axes.
• Express quantities with 4 decimal places precision in terms of percentages eg 3.9278%.
• Marks for late work will be reduced by 10% for each working day, or part of a working day, after the deadline. No marks will be obtained for submissions that are later than 5 working days. If you have difficulties in handing in the work by this deadline you should contact me well before the deadline.
• You may discuss the problems with others, but you must write up the solutions independently.
• I have set up a Discussion Board for you to post questions (see further information on Blackboard). I will regularly read it and reply. If you have any questions please do not email me directly - first post them on the Discussion Board so that others can benefit from the answer - unless you have a personal issue in which case you should email me of course.
1 Nasdaq
1.1 Explain what is the Nasdaq Composite (from now on we will refer to it as “Nasdaq" only), its stock composition (ie. what type of companies it includes) and the weighting of the stocks. From the file Data.csv find the Nasdaq daily historical data and plot it as a function of time. [5 marks]
1.2 Assume that the Nasdaq data follows a Geometric Brownian Motion. Using the for-mulas given in class estimate the daily values of the drift coefficient (µ) in units of [day]−1 and of the diffusion coefficient (σ) giving its correct units. [10 marks]
1.3 In fact the formula given in class for σ is not quite right. Can you correct it? Justify your answer. [10 marks]
1.4 Using the estimates for µ and σ, calculate what is the probability that the Nasdaq will increase by 50% after one year. [10 marks]
2 Optimal Portfolio Construction
In this exercise, you will learn how to build optimal portfolios by using real market data from the file Data.csv. You will be asked to build a portfolio composed of combinations of four specific stocks.
1. Plot on the same graph the daily historical data for the four stocks AMD, CSCO, NFLX and SBUX corresponding to the last five years. [5 marks]
2. Calculate the mean and standard deviation of the daily returns of the four stocks. [10 marks]
3. Calculate the β coefficients for the four stocks and, using the CAPM, predict their daily returns over the same period. Assume that the risk-free rate is 3% per year and that the Nasdaq is a good proxy for the Market Portfolio. Compare the CAPM predictions with the returns and risks that you calculated in the previous question. Comment on the comparison both with respect to the risks (σ vs β) and the returns (mean return vs CAPM return). [20 marks]
4. Calculate the correlation matrix of the four stocks. [10 marks]
5. Obtain the portfolio which has the minimum risk. What are its average risk and daily return? To do this you can either program a piece of code or you can do it analytically. [20 marks]