Stochastic Finance

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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