# 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 yt be the opening value at day 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.

Objective

To be able to generate simulated data yt, starting at = 1, for = 23, . . . , 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 IDb = 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 = 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= 2 . . . N + 1 using

yt+1 = yt × ext ,               t = 1 . . . N .

(c)   Plot the value of your stock showing the number of days = 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 be the value of your initial investment after 1 year, i.e., y366 with y1 = £100. Using a for loop, generate with your code a sequence v1, v2, . . . , vn, 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 (£) which is known to be measured by

(c) Complete the table

 ns 10 20 100 200 1000 10,000 (v) s

What would you say that your investment has gained on average after one year?

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