Stochastic Finance

M081LON Adrian Euler

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)

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 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)