MATH111: Simulating Financial Assets

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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.

Background

Let y_t 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 y_t cannot be negative: if there is y_t = 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:

y_t+1 =  y_t × e^xt,

where x_t is a random number that is normal distributed with

µ : the average daily increase,

σ_d : the standard deviation of the daily fluctuations.

Objective 

To be able to generate simulated data y_t, starting at t = 1, for t = 2, 3, . . . , 366, which can be used to answer financial questions e.g. around hedging and option pricing.

Personalised variable

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 σ_d = 0.01. Use the MATLAB built-in function normrnd(µ, σ_d) to generate (and store in the memory) a sequence of N = 365 random numbers x_t, 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 y_t, t = 2 . . . N + 1 using

y_t+1 = y_t × e^xt ,               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 y_t 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, . . . , v_ns , for n_s = 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?