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A. Basic Calculus [1

5

%]

A1. The function f (x) is de…ned as

f (x) = exp

�

x

3

� x

�

:

Show that by writing f (x) as f;

df

dx

=

�

3x

2

�

1

�

f:

Use Leibnitz’formula to di¤erentiate this equation n times. Hence show that,

at x = 0;

f

(n+1)

0 = �f

(n)

0 ; if n = 1

f

(n+1)

0 = �f

(n)

0 + 3n (n� 1) f

(n�2)

0 ; if n > 1;

where f (n)0 denotes the n

th derivative of f; evaluated at x = 0:

A2. The integral In is de…ned, for positive integers n; as

In =

Z 1

0

�

1 + x2

�

�n

dx:

Using a reduction formula deduce that

In = 2n (In � In+1) :

Hence or otherwise

show that

I

4

=

Z 1

0

�

1 + x2

��4

dx =

5�

32

A3. If f is a di¤erentiable function of u and v and the variables (u; v) are related to

x and y by the formulae

u = xy; v = y � x;

show that

@f

@x

= y

@f

@u

� @f

@v

:

Determine the corresponding formula for

@f

@y

: Verify these formulae by direct

substitution in the special case when

f = u+ v2:

2

B. Linear Algebra [15%]

B1. Show that the linear system

2x+ y + z = �

6

�

2x+ y + (� + 1) z = 4

�x + 3y + 2z = 2�

has a unique solution except when � = 0 and � = 6:

If � = 0 show that there is only one value of � for which a solution exists, and

…nd the general solution in this case. Discuss the situation when � = 6: Hint:

In the augmented matrix swap the …rst two columns and the …rst two

rows before row reduction. Consider each case of � separately.

B2. Given that detA means ‘determinant of the matrix A’, solve the equation

det

0BB@

x a a a

a x a a

a a x a

a a a x

1CCA = 0

B3. For the matrix A given by 0@ 1 0 10 1 0

1 0 1

1A

…nd a matrix P such that D = P�1AP is diagonal and calculate the form of D:

3

C. Probability and Stochastic Calculus [30%]

C1. The Moment Generating Function (MGF) M� (X) for the random variable X

is de…ned by

M� (X) =

E

�

e�x

�

=

Z

R

e�xp (x) dx

=

1X

n=0

�n

n!

E [xn]

where p (x) is a general probability density function.

Consider the probability density function p (x)

p (x) =

�

� exp (��x) x � 0

0 x < 0

where � (> 0) is a constant.

(a) Show that for this probability density function

E

�

e�x

�

=

�

1� �

�

��1

Hint: You may assume � > � in obtaining this result.

(b) By expanding

�

1� �

�

��1

as a Binomial series and equating with

1P

n=0

�n

n!

E [xn] ;

show that

E [xn] =

n!

�n

; n = 0; 1; 2; ::::

(c) Calculate the skew and kurtosis.

4

C2. Consider the di¤usion process for the state variable Ut which evolves according

to the process

dUt = ��Utdt+ �dXt; U (0) = � (1)

Both � and � are constants. dXt is an increment in Brownian motion.

(a) Show that a solution of (1) can be obtained by using an Integrating Factor

and Stochastic Integration to give

Ut = �e

��t + �

�

Xt �

Z t

0

exp (�� (t� s))Xsds

�

:

(b)Write (not derive) the forward Fokker-Planck equation for the steady state

transition probability density function p (U 0; t0) for this process, where a primed

variable refers to a future state/time.

By solving the Fokker-Planck equation which you have written above, obtain

the steady state probability distribution p1 (U 0), which is given by

p1 =

r

�

�2�

exp

�

� �

�2

U 02

�

:

By looking at p1; write down the mean and standard deviation for this distri-

bution.

5

D. Di¤erential Equations [40%]

D1. Consider the following Cauchy-Euler type equation

1

2

�2S2

d2V

dS2

+ (r �D)SdV

dS

� rV = 0;

for the unknown function V (S) and where r > 0 , D � 0 and � > 0 are all

(known) constants. It is to be solved together with the following conditions

V (0) = 0; V (S�) = S� � E;

dV

dS

����

S=

S�

= 1:

S� and E are constants.

Solve this Euler equation and apply the …rst two conditions to show

V (S) = (S� � E)

�

S

S�

�m+

;

where

m+ =

1

�2

�

�

�

r �D � 1

2

�2

�

+

q�

r �D � 1

2

�2

�2

+ 2r�2

�

> 0:

Explain why m+ > 0:

Now using the third condition

dV

dS

(S�) = 1; calculate S� (this means dV=dS

evaluated at S = S�):

6

D2. Consider the following Black-Scholes problem consisting of a PDE and …nal

condition at time T;

@V

@t

+

1

2

�2S2

@2V

@S2

+ (r � rf )S

@V

@S

� rV = �C (S; t) ;

V (S; T ) = S

for the unknown function V (S; t) : �; r; rf and T are all constants.

Suppose that C (S; t) has the form C (S; t) = f (t)S: By writing V (S; t) =

� (t)S; show that

� (t) = e�rf (T�t) +

Z T

t

exp (�rf (� � t)) f (�) d� :

Hint: Substitute V = � (t)S into the PDE and solve the resulting ODE.

Then use the …nal condition to arrive at the result.

7

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