(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 2

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)

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)