Stochastic Finance

Stochastic Finance

Stochastic Finance
M081LON
[NEW ] Mid-Term Assignment
Issued: Week 1 of the current term
This is an INDIVIDUAL Summative Assignment.
Section 1.0
The Requirements
Question 1 (20 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)
Find 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)
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 Hypothesis), 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 company traded on the NASDAQ, and then carry out the following tasks:
(a)
Use Yahoo Finance or the Bloomberg terminal to obtain the daily closing stock price
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 company) using
the formula:

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 approximately normally
distributed by a normal probability plot, a box and whisker graph, and the descriptive
statistics summary. Compare data characteristics to theoretical properties.
(5 marks)