Python随机过程代写|BENG0091 Coursework 1



– You need to provide all MATLAB, Python or equivalent code that you have developed as part
of your submission to Turnitin. This is compulsory. Include clarifications/comments in your
code whenever you feel appropriate.

– You need to submit one version of the code that is executable. Unless the code is executable
locally reaching the same results as those in your report, it will not receive full marks.

o One option is to have the code in the submitted document in a state where we can
copy it off your submission and execute. A few tips to assist you in the process:

Please note that line numbers left in the code often creates an issue with
executability. Python codes embedded in LaTeX can also create problems with
executability. Please ensure that prior to submission, you can copy the code back
from the document you plan to submit and execute it, just to double check.

o If you do not want to worry about the Turnitin version being executable or not, you
can additionally choose to use Datalore as suggested. A detailed video on how to
use it is available in Moodle. Please note that submitting via Datalore is optional.

– Your submission (excluding the space taken up by your code) should be no more than 15
pages and contain no more than 15 Figures. Clarity is expected in the Text, in your Figures,
and in your codes. A single figure/image cannot comprise of 10 illegible plots, please use
your reasoning when preparing your report.

– Please make sure that you address the answer for each section or question at its respective
slot. e.g. a correct answer to section (a) provided as response to section (b) will not be
considered for marking.

– You need to develop your own code. You are not allowed to use pre-existing toolboxes to
conduct stochastic simulations, for example. However, the use of standard Python packages
such as pandas or NumPy are acceptable. Regarding random number generators (r.n.g.), you
are only allowed to use a/the uniform r.n.g. available in the programming language you
chose (MATLAB, Python etc.). Uniqueness of your scripts will be assessed and will contribute
to your mark.

– To achieve full marks in each question, your methodology needs to be correctly
implemented and your code needs to be original (i.e. your own work).

– You will be allowed to submit your work multiple times until the deadline. The Turnitin
submission will be made available weeks before the deadline. Please note that it is your
responsibility to ensure that the submission is made on time. Late submissions, SORAs and
ECs will be handled by the Admin Team, not your tutors.

Problem Statement

Trading in the stock market is literally risky business. Given that risk is unavoidable, the key question
that arises is how a risk-averse investor can construct a portfolio that yields the desired returns at
the lowest possible risk. Efficient frontier theory, pioneered by Nobel Laureate Harry Markowitz,
provides an answer to this question, and allows us to identify investment portfolios that strike the
best balance between risk and return. While this theory makes certain assumptions that are
probably oversimplifications (notably, assuming that past performance is an indicator of future
trends and that asset returns are normally distributed), the theory is extremely important in
understanding the effect of diversification in investing, and has been extended in attempts to better
capture the real market behaviour (e.g. post-modern portfolio theory, Black–Litterman model, etc.).

In this coursework, we will explore the main concepts of modern portfolio theory and we will try to
come up with efficient portfolios of stocks of three fictitious companies:

Nano-Automata Corps (abbreviated as NAC)

Quantum Cryptographers Inc. (abbreviated as QCI)

Inter-Galactic Telecommunications (abbreviated as IGT)

Data analysis of the stock prices of these companies in the past 10 years has shown that the daily
closing prices of NAC adhere to the following stochastic process:

where X n , n =1,2,… are i.i.d. that follow the Laplace distribution, X n ∼Laplace (μ,b) , n denotes the
day, and nmax the maximum number of days we want to predict the stock price for. The Laplace
distribution is continuous with two parameters, b and µ, and has the following probability density:

From the data analysis we know that X n ∼Laplace (0.00051 0.0162 , )
The prices of the stocks of QCI and IGT can be expressed as follows:

Things to do

1. We will first do some preliminary work that will enable us to simulate stochastic realisations of
the prices of the stocks of interest.

(a) Derive an explicit mathematical formula for the cumulative distribution function of the
Laplace distribution. [2]

(b) Describe a simple approach for the generation of random deviates from the Laplace
distribution. [4]

(c) Write a program that generates 100000 samples from the Laplace distribution with µ = 0.5
and b = 0.3 (use the approach you developed in the previous question; do not use built-in
functions for this). To verify that your method works correctly, show a histogram of these
samples and a line-graph of the probability density function (both in the same plot). [6]

2. We are now able to perform the stochastic simulations of the prices of the three stocks of
interest. We will assume that each year contains 260 days of trading (roughly 52 weeks of 5
working days each), and we will simulate stock prices for a 5-year window, i.e. nmax =1300 days.

(a) Write a program that simulates the stock price sequencies S S S n n n NAC QCI GT , , , 0,1,2,…, I n = nmax .

To do this, first generate three random streams, one for X n , n n =1,2,…, max , one for Yn and
similarly, one for Zn , using the method you developed in Question 1. Thus, you are allowed
to use only a uniform random number generator available for the programming language of
your choice (MATLAB, Python etc.), initialised with a seed value of your choice.

Subsequently, use these sequences in the application of the equations that give the stock
price per day, e.g. for QCI, you will evaluate the expression S X Y S nQCI = + + ⋅ 1 0.5 1.7 ( n n n )⋅ QCI −1
recursively. Present graphs of the stock prices per day in the same plot. [12]

(b) Repeat the procedures of Question 2(a) with two more different random seeds and
produce plots of the stock prices ped day. You will present two separate plots, one for each
random seed. Each of these plots will contain three line-graphs, one for each stock (NAC,
QCI, and IGT). Comment on the observed behaviour in the three “scenarios” that you
simulated and plotted in Questions 2(a) and 2(b). For instance, how does the magnitude of
the fluctuations (volatility) compare with the overall “average” behaviour of a stock (drift)
in the three different scenarios? Which stock(s) would you pick for your portfolio if you
were an investor? [10]

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