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MATH111: Simulating Financial Assets
Instructions: Answer sheets are available on Canvas. Once you have com- pleted your project, you should convert the file to a PDF and submit on Canvas. This project contributes 25% to your overall mark.
Let yt be the opening value at day t of a financial asset traded at the Stock Exchange in London. If y1 is its opening value today, then y366 would be its closing value in a year’s time. The value of the asset fluctuates from day to day. The values yt cannot be negative: if there is yt = 0 for any day t, the company would be bankrupt - a situation that we do not study here. A widely used statistical model is the following:
yt+1 = yt × ext,
where xt is a random number that is normal distributed with
µ : the average daily increase,
σd : the standard deviation of the daily fluctuations.
To be able to generate simulated data yt, starting at t = 1, for t = 2, 3, . . . , 366, which can be used to answer financial questions e.g. around hedging and option pricing.
This project contains variable which depend on your student ID. Before start- ing you will need to define the following in MATLAB: a=your student ID, b = max(mod(a,10), 1), c = max(mod(a,7), 4).
Exercise 1: [40 Marks]
(a) Choose for the daily average increase µ = b × 10−c and for the daily volatility σd = 0.01. Use the MATLAB built-in function normrnd(µ, σd) to generate (and store in the memory) a sequence of N = 365 random numbers xt, which are normally distributed using the seed a.
(b) Starting from an opening stock value on day one of y1 = £100, calculate the value of the stock yt, t = 2 . . . N + 1 using
yt+1 = yt × ext , t = 1 . . . N .
(c) Plot the value of your stock showing the number of days t = 1 . . . 366 on the x-axis and the value yt on the y-axis. Add the plots and a printout of your code to your answer sheet.
Exercise 2: [60 Marks]
(a) Let v be the value of your initial investment after 1 year, i.e., v = y366 with y1 = £100. Using a for loop, generate with your code a sequence v1, v2, . . . , vns , for ns = 10 simulated outcomes after one year. You should re-set your seed a from Exercise 1.
(b) Calculate an estimate for the average value of your investment after one year (v) (£), i.e.,
and an estimate of its error s (£) which is known to be measured by
(c) Complete the table
What would you say that your investment has gained on average after one year?
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