stats and financial maths

Answer ALL questions.
1. (a) Briefly describe the difference between Forward and Futures
contracts and explain the role of the Clearing House in the latter.
(5 marks)
(b) The price of a forward contract on a share is given by
0 0
rT F S e =
where
0
S
is the current share price, r is the risk-free rate of
interest and T is the length of the contract. Suppose that a share is
currently trading at £12, r = 6% and a 3-month forward contract is
available on the share. Show how a risk-free profit can be made if
either,
(i)
0 F = £11
;
(ii)
0 F = £14
and state the amount of profit in each case. You may take long or
short positions on the share, the bank or the contract.
(6 marks)
(c) Three friends Amy, Susan and Catherine are members of an
investment club. They each have a total of £10,000 to invest in
three companies A, B and C. On 1st August 2019, the following
prices were available (prices in £).
Company
A B C
Share price 40 20 30
Call option price 0.80 1.20 4.20
Put option price 7.61 4.61 1.51
Strike price, K 48 24 28
Question 1 continues on the next page.
6G5Z3004_2021_B4 Page 4/8
Question 1 continued.
The friends made the following individual investments (no dealing
charges of dividends were paid).
Amy bought 30 A shares, 400 C puts and saved the rest of the
money in the bank.
Susan bought 50 A shares, 400 B calls, 100 C puts and put the rest
of the money in the bank.
Catherine bought 100 A puts, 50 B shares and spent the rest on C
shares.
The options expired after six months and the bank interest rate
was 5% compounded continuously (assume that fractional
numbers of shares may be purchased). If, at this time, A shares
were worth £42, B shares £12 and C shares £22, find the value of
each of the friends portfolios and, hence, the returns they made
on their original investment of £10,000. (9 marks)
6G5Z3004_2021_B4 Page 5/8
2. A steel cable has a breaking strain, T tonnes, whose distribution follows
a Rayleigh distribution with density function,
2
exp for , 0
( , ) 2
0 otherwise
t t
t
f t

  
      − 
=    

A random sample of n such cables is subject to test resulting in
independent breaking strains of
1 2 , , ,
n
t t t
tonnes.
(a) Show that the log-likelihood function for

can be written as,
2
1
1
( ) ln( ) ln( )
2
n
n
i
i
i
i
t
l t n  

=
=
= − −
 
and, hence, that the maximum likelihood estimator of

is given
by,
2
ˆ 1
2
n
i
i
t
n

= =

(7 marks)
(b) Given that
2 E T  = 2  
, show that the maximum likelihood
estimator of

is unbiased. (2 marks)
(c) Show that the distribution function of T is given by,
2
1 exp 0
( ) 2
0 otherwise
t
t
F t 
    − −   
=    

(4 marks)
Question 2 continues on the next page.
6G5Z3004_2021_B4 Page 6/8
Question 2 continued.
(d) A random sample of 10 cables had breaking strains (tonnes),
2.35 2.47 2.36 2.46 2.29 2.30 3.51 4.55 5.43 7.61
(i) Find an estimate for

based on this data, assuming the
breaking strengths follow a Rayleigh distribution.
(1 mark)
(ii) The safe working load (SWL) of a cable is defined to be one
sixth of the median breaking strain. Find the maximum
likelihood estimate of the median breaking strain, and hence,
estimate the SWL of the cable from this data.
(4 marks)
(iii) Estimate also the proportion of cables that would fail at the
safe working load.