BENG0091 Coursework 1

matlab代写 | report作业 | Stochastic Calculus | math – 这是一个关于ios的题目, 主要考察了关于ios的内容,是一个比较经典的题目, 涉及了matlab/report/Python/ios等代写方面, 这是值得参考的lab代写的题目

To be submitted on Moodle by 24 th Mar 202 3.

Please read the following guidelines before starting the work.

Guidelines

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 subject to significant and unavoidable risk. Hence, 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 portfol ios 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, BlackLitterman 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:

Elysium Investment Forecasting (abbreviated as EIF)

Centurion Energy Solutions (abbreviated as CES)

Quantum Nanomaterials Technologies (abbreviated as QNT)

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

( ) 1 max

EIF
0
EIF 1 EIF 1,2,...,

7.

n^ n = n  nn

=

=+

S

SXS

where X n , n =1,2,… are i.i.d. that follow the Laplace distribution, X n Laplace(, 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 , and has the following probability density:

( )

1 
exp
2

x
px
bb

=

X^

From the data analysis we know that X n Laplace(0.00031 0.0097, )

The prices of the stocks of CES and QNT can be expressed as follows:

( )

CES
0
CES CES
1 1,2,.

31. .

3

n 1 0.3 n 0.8 n = n  n.

=

= + +

S

SXYS

( )

QNT
0
QNT QNT
1 1,2,.

25. .

9

n 1 0.2 n 0.9 n = n  n.

=

= + +

S

SYZS

where Y n Laplace(0. 00048 ,0. (^0108) ) i.i.d. and Z n Laplace(0.00012,0.0077) i.i.d. for n = 1,2,3,… Note also that the sequencies of X n , Y n and Z n are independent with respect to each other, e.g. X 1 is independent of all Y n and Z n , n =1,2,…, etc.

Things to do

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 1000 00 samples from the Laplace distribution with  = 1. 5
and b = 0. 8 (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 ]

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. n max= 1300 days.

(a) Write a program that simulates the stock price sequencies SSS n EIF, n CES, n QNT, n =0,1,2,..., n max.
To do this, first generate three random streams, one for X n , nn =1,2,...,max, one for Y n and
similarly, one for Z n , 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 EIF, you will evaluate the expression SXYS n CES= + 1 0.3( n +0.8 n ) n CES 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 (EIF,
CES, and QNT). 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 ]

We will continue with the calculation of certain quantities that will allow us to apply the efficient frontier approach to the data we generated for the prices of the three stocks over time.

(a) Pick one of the three scenarios you simulated (whichever you want), and for each of the
stock price sequencies, calculate the daily returns as follows:

ax
1

1 1,2,..., m

ABC
ABC n
n ABC
n

nn

=

S

DR=

S

where ABC will be each one of the stocks in discussion (EIF, CES, QNT). Present, in three
separate plots, one for each stock, histograms of the daily returns you calculated. Discuss
the appearance of these histograms with reference to the plots of the stock price
movements you plotted earlier. [10]

(b) Calculate and  report the following quantities to 3 significant figures:

Annualised mean returns:

( )

max

max^1

11

1

where:

ABC ABC

ABC ABC
n

n
n n
=

+

=

AR=DR

DRDR

Report one number for each stock, ABC = EIF, CES, QNT.

Annualised (co)variances of the daily returns. The following formula is used to calculate variances when ABC = DEF, but also covariances when ABC DEF (where ABC and DEF can be any of the stock symbols: EIF, CES, QNT):

( )( )

max

max^1

260

1

whe :
1

re

ABC DEF ABC DEF

ABC DEF ABC ABC DEF DEF
nn
n

n
n

=

ACV=DCV

DCVDRDRDRDR

Thus, report 6 numbers in total: 3 variances, one for each stock, and 3 covariances, one
for each pair: EIF-CES, EIF-QNT, CES-QNT. [9]

We now have the data necessary to apply efficient frontier theory. To this end, we will consider an ensemble (collection) of portfolios, all of which invest in the three stocks under consideration. However, each portfolio will have different weighting factors per stock, e.g. in one portfolio the allocation may be 40% EIF, 20% CES and 4 0% QNT; for another portfolio the allocation may be 30% EIF, 35% CES and 35% QNT. Note that the weighting factors will have to sum up to 1 (i.e. 100%).

(a) For the scenario you picked in Question 3(a), create an ensemble of 2000 portfolios, each
with different allocation. Thus, for each portfolio you will generate 3 weights that sum up to
1 and represent the relative numbers of shares bought for each of the 3 assets (EIF, CES,
QNT). Then calculate the expected (annualised) return and variance of each portfolio
according to the following formulas:

EIF, CES, QNT

ABC
ABC
ABC

Return w

=AR

EIF, CES, QNT EIF, CES, QNT

ABC DEF
ABC DEF
ABC DEF

Variance w w

=ACV

where wABC is the weight of the stock of ABC. Present a scatter plot of the return in the y-
axis with respect to the standard deviation, i.e. Variance  in the x-axis. Find the portfolio
with the minimum variance, highlight it on your plot (e.g. with a marker with different
shape or colour) and report its allocation (i.e. the weights). [ 15 ]

(b) Discuss the results you have obtained. Compare the weights of the minimum-variance
portfolio, with the weights of riskier portfolios that may have better or worse returns
than the minimum-variance one. How does the above plot guide you in choosing an
allocation for your portfolio? [ 12 ]

Finally, present your mat lab Python (or other) codes as Appendixes to your report or preferably upload them to Datalore. If presenting your codes in an Appendix make sure these are copy- pasted as text, not as figures, and that they dont contain the codes line numbers (see the guidelines in the first page for more detailed instructions). [ 2 0]