Issued: Week 1 of the current term
This is an INDIVIDUAL Summative Assignment.
Section 1.0
The Requirements
Question 1 (2 0 Marks)
A stock’ s terminal value S has a uniform distribution: that is, it is equally likely to
assume any value in the range (0 – 100) and will not assume any value outside of this
range. The random variable x on which this stock ’s value is based has a density
function p(x) =1 for 0 ≤ x ≤ 1 and 0 elsewhere. The stock’ s random terminal value is
f(x) =100x.
(a)
Find the distribution function P(x) for p(x)
(2 marks)
(b)
F ind the expected value of the stock ’ s terminal S value assuming it will fall within
the range (i ) 50 – 100; (ii) 0 – 50; (iii) 0 to 100.
(6 marks)
(c)
Find the variance of S in the range 0 – 100 (6 marks)
(d)
M081LON Adrian Euler
What would be the expected future cash flow (contingent on its exercise) of a call
option written on this stock if its exercise price were $50? That is, what is the
expected cash flow of the option conditional on its exercise?
(6 marks)
Question 2 (30 Marks)
Burton Gordon Malkiel (a fierce supporter of Efficient Market Hypothesi s), in his
book “ A Random Walk Down Wall Street”, claims that the daily logarithmic changes
in the closing price of stock follow a random walk —that is, these daily events are
independent of each other and move upward or downward in a random manner—and
can be approximated by a normal distribution. To test this theory, use either a printed
or electronic financial mediums (i.e. including Bloomberg) to identify/ select one
company traded on the NYSE, one company traded on the American Stock Exchange
and one comp any traded on the NASDAQ , and then carry out the following tasks:
(a)
Use Yahoo Finance or the Bloomberg terminal to o btain the daily closing stock pric e
of each of these companies of the past six consecutive weeks (so that you have 30
values per company). (5 marks)
(b)
Compute the logarithmic daily changes in the closing stock price of each of these
companies for six consecutive weeks (so that you have 30 values per c ompany) using
the formula:
˜
¯
ˆ
Á
Ë
Ê
=
-1 t
t
t
S
S
LN R
Where St is the share price in period t and St-1 is the share price in the previous period.
(5 marks)
(c)
For each of your six data sets, decide whether the data are approx imately normally
distributed by a norm al probability plot, a box and w hisker graph, and the descriptiv e
statistics summary. C ompare data characteristics to theoretical properties.
(5 marks)
M081LON Adrian Euler
(d)
Discuss in a critical manner the results of part (c). What can you say about your three
stocks with respect to daily closing prices and logarithmic daily changes in closing
prices? Which, if any, of the data sets are approximately normally distributed? Why?
If any normality deviations are observed, provide a relevant rational (i.e. base it on
information and efficient market hypothesis) to why that might be the case. Identify
the correct mome ntum links for the data set, compute them a nd provide the
appropriate interpretation.
The random – walk theory pertains to the daily logarithmic change s in the closing stock
price, not the daily closing stock price.
(15 marks)
Question 3 (50 marks)
(a)
The value of a call option is equal to its expected payoff in a risk – neutral world,
discounted at the risk – free interest rate, which can be wr itten in a generalized form as
( )
( ) [ ]
( )
( ) [ ] 0, X S MAX E e p
0, X S MAX E e c
t T
Q t T r
t
t T
Q t T r
t
– =
– =
— — –
Where [ ]
Q
E denotes expectation with respect to the risk- neutral probability measure
Q.
Using the approach in Nielsen (1992) and starting with payoff expectation formulae
above, derive the Black-Scholes option pricing formula and discuss its use.
Further on c onsider the data in the panel below related to traded options of a Stock S
(traded price at time T of £50.11 ), exercise price X = £50.11, matur ing i n one year.
The volatility is 22.00% and considers the time now to be 0. The LIBOR rate of
1.17 %. The mean and variance set has be en computed to be 4.72 and 0.08 ,
respectively.
Use the derived Black – Scholes call and put formulae derived i and the dat a given to
compute the call and put values. Interpret the results
( 20 marks)
(b )
M081LON Adrian Euler
Consider the case of the single – period model in which there is just one risky asset
with price S 1
at time 1 and one risk- free asset . Express the claim C in terms of the
risky investment, assuming that an amount , a is put into it and a b amount placed on
the risk free asset.
Show that when the intrinsic risk ℛ ( C) = 0 for all cla ims C then the underlying
probability space has effectively at most two points (so that the model is the binomial
model).
Suppose C 1
, C 2
are two claims such that ( C 1
,C 2
, S1 ) have a joint normal distributi on
and as random variables are either positively, or negatively, correlated and conditional
on S 1 . Derive an expression for of intrinsic risk of the combined claims, ℛ (C 1 + C2 ).
(10 marks)
(c )
Suppose that {X t, t ≥ 0 } is a stochastic process that may be represented as dXt = Y tdt +
Z tdWt. For (suitably nic e) functions f (x , t ) and g( x, t ) use Itô ’ s Lemma to establish the
stochastic integration – by- parts formula
( ) ( ) ( ) dt
x
g
x
f
Z f gd g fd fg d
2
t
¶
¶
¶
¶
+ + =
where f, g and the partial derivatives are evaluated at (X t
, t ). Further on for the
standard Brownian motion {Wt, t ≥ 0 }, evaluate the stochastic integral
Ú
t
0
s s
dW W .
(15 marks)
(d )
The Ornstein- Ulhenbeck process is the unique solution of the followin g equation:
Ó
Ì
Ï
=
s + – =
x X
dW dt cX dX
0
t t t
Which could also be written in a more explicit form; i f we consider
ct
t t
e X Y = and
integrate by parts, it yields ( )
t
c ct
t
ct
t t
e, X d e d X e dX dY + + = , and because
( ) 0 e, X , dt ce e d
t
c ct ct
= = , it follows that
t
ct
t
dW e dY s = and eventually
Ú
– -s + =
t
0
s
cs ct ct
t
dW e e xe X
M081LON Adrian Euler
Use the expression above to compute the mean and variance of Xt. (5 marks)
Section 2.0
The report
In the report, you should address all of the CW requirements, and you are expected to
have your own view of what is expected and how much weight to give any particular
requirement element; however you will need to plan your answ ers carefully, in order
to provide a focused answer to the questions, within the word – limit.
The report should show that you have developed your understanding of the relevant
materials in this module, therefore it is necessary that you, not only get correct
numerical answers, but also explain what is being calculated, how it is calculated, the
underlying theory, assumptions, the results at appropriate stages of the calculation,
and interpretations of the final answers.
You should justify relevant theori es used, their relevance, where possible,
demonstrate your understanding of them, by using simple equations or diagrams, or
some other illustrations and in presenting your arguments, you should also comment
on possible strengths, weaknesses and limitations .
Graphs, tables, panels, and other illustrations, should NOT be copied from books in a
“ wholesale ” fashion, instead they but should be recreated with justified data selected
on appropriate and relevant ranges.
You should justify any conclusion you reach on the basis of evidence, cross -referenced to, or quotation from, the course lecture notes, seminar material, textbooks,
other course readings, or any other reliable source you choose to use. A report
without proper referencing will not be acceptable.
You should make use of relevant articles, journals, white papers, books in your
research (which you should clearly identify) to support your work. You should also
explain the approach you have followed and should include a statement on your
attitude to risk, your own “ views”, identifying the theoretical basis for your approach .
You must make use of financial mediums/portals (i.e. Bloomberg), and other models
and tools /software and j ustify the use of the methodologies, models and techniques
used and their use fulness. Your answers should be compliant to level 7 (Master’ s)
expectations, with a considerable level of critical thinking, coherence and cohesion.
M081LON Adrian Euler
The length of the final paper (answers to all three questions with all of their subparts)
should be betwee n 2500 and 3500 words, excluding exhibits, should be referenced,
and contain a list of the sources of (i) your evidence and (ii) the theory that underpins
your analysis and commentary. Failure to do so is unprofessional and fraudulent,
and will result in a failing grade for the report and possibly the course .
The answers’ report should focus on all assignment requirements as specified in
section 1.0. And should be submitted electronically via Moodle by the deadline
indicated on Moodle (current semester).
Section 3 .0
Report Structure and Format
Assignment Report Marked on a 100% scale, but weighted at 4 0% of the overall
module mark.
Report Outline ( suggestive):
1. Stochastic Finance Mid Term Assignment
2. Question 1
a. Answer to part (a)
b. Answer to part (b)
c. Answer to part (c)
d. Answer to part (d)
3. Question 2
a. Answer to part (a)
b. Answer to part (b)
c. Answer to part (c)
d. Answer to part (d)
4. Question 3
a. Answer to part (a)
b. Answer to part (b)
c. Answer to part (c)
d. Answer to part (d)
5. Reference List
6. Appendices
Report Format:
It must be pres ented in the following format:
· All pages must be numbered
· The assignment must have a front cover stating:
M081LON Adrian Euler
o Module number
o Module name
o Title of the assignment
o Student name and number must be stated
o Submission date
o Word count
· Margins must be as follows: Top and Bottom 2.54 cm, Left and Right: 3.18 cm
(Microsoft Word default)
· Headers and footers may be outside these margins.
· Footnotes should be included within the margins.
· A reasonable number of appendices may be used for relevant supporting
information and to demonstrate your analysis; but this cannot exceed 6 Pages.
IMPORTANT: Your response to the assignment req uirements will be assessed
compared to a detailed marking scheme designed and approved by the module leader,
with the overall assignment mark, but also mark per question also effected/altered by
issues with (1) level 7 synthesis, (2) discussions and interpretations, (3) assumptions,
(4) theory, (5) coherence and cohesion, (6), in – text referencing, (7) writing skills, (8)
proper use of MS- Word and MS-Excel, (9) report layout, (10) reference list, (11)
appendix, (12) tabulated data .
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