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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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