Download ADMISSIONS EXERCISE MSc in Mathematical and Computational

MScMCF 2015
ADMISSIONS EXERCISE
MSc in Mathematical and Computational Finance
For entry 2015
• The questions are based on Linear Algebra, Probability, Partial Differential
Equations, and Algorithms. If you are still studying for a degree and are
yet to take or complete courses in these areas, please indicate so here.
Please specify the titles and dates of courses which you are due to take/are
still taking.
• You should attempt all questions and show all working.
• Stating the answers without showing how they were obtained will not attract credit.
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Statement of authenticity
Please sign and return the following statement together with the solutions. Your application
will not be considered without it.
I certify that the work I am submitting here is entirely my own and unaided
work.
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Signed
Date
2015
2
Partial differential equations
1. Consider the Black-Scholes problem
∂V
∂2V
∂V
+rS
+ 12 σ 2 S 2
− r V = 0,
∂t
∂S 2
∂S
S > 0, t < T,
(1)
V (S, T ) = f (S),
where σ > 0, r ∈ R and T > 0 are constants and f : R+ → R is a given function.
(i) Given that V (S, t), the solution of (1), is infinitely differentiable in S > 0 and
t < T , show that
∂V
V1 (S, t) = S
(S, t)
∂S
also satisfies the partial differential equation in (1).
(ii) Assume that V (S, t) and all its t and S-partial derivatives are differentiable with
respect to the parameter r, for all S > 0 and t < T , and any value of r. Deduce
that
∂V
(S, t)
ρ(S, t) =
∂r
satisfies the problem
∂ρ 1 2 2 ∂ 2 ρ
∂ρ
∂V
+ 2σ S
+rS
−rρ = V −S
,
∂t
∂S 2
∂S
∂S
S > 0, t < T,
(2)
ρ(S, T ) = 0.
(iii) Given that (2) uniquely determines
ρ(s, t) and assuming that both V and
S ∂V /∂S are order o 1/(T − t) as t → T − , show that for S > 0 and t ≤ T
∂V
ρ(S, t) = (T − t) S
(S, t) − V (S, t) .
∂S
(iv) You may assume that if K > 0 is a constant and
log(S/K) + (r − 12 σ 2 )(T − t)
p
d=
,
σ 2 (T − t)
1
Φ(x) = √
2π
Z
x
e−p
2 /2
dp,
−∞
then
V (S, t) = e−r(T −t) Φ(d)
is a solution of the Black-Scholes equation in (1), for S > 0, t < T , σ > 0. For
this particular V , and assuming S > 0 and t < T , find
f (S) = lim V (S, t)
t→T −
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and ρ(S, t) =
∂V
(S, t).
∂r
Turn Over
Linear algebra
2. Consider the following simple model for an economy. There are only four assets, A1 ,
A2 , A3 and A4 . Initially, at time t = 0, their values are
A1 = $5,
A2 = $5,
A3 = $10,
A4 = $4.
Later, at time t = 1, there are three possible states of the world, ω1 , ω2 and ω3
(viewed from time t = 0) and these can occur with probabilities p1 = 1/4, p2 = 1/2
and p3 = 1/4, respectively. In state ω1 the values of the assets at t = 1 are
A1 = $6,
A2 = $3,
A3 = $12,
A4 = $9
A3 = $12,
A4 = $3
A3 = $9,
A4 = $3.
while in state ω2 the values at t = 1 are
A1 = $9,
A2 = $6,
and in state ω3 they are
A1 = $12,
A2 = $6,
In Parts (i), (ii) and (iii) below, assume that it is possible to own a fraction of an
asset but that it is not possible to sell an asset you do not own.
(i) By constructing a specific example, show that it is possible to set up a portfolio
of assets at t = 0 which is guaranteed to have a value of $30 at time t = 1,
regardless of which state of the world occurs at time t = 1.
(ii) Find the guaranteed profit, or loss, for this portfolio.
(iii) Show that there are, in fact, an infinite number of such portfolios and find the
maximum and minimum profits which can be made on such a portfolio.
(iv) If you are allowed to sell assets you don’t own, what happens to the maximum
profit and the maximum loss you can make on such a portfolio?
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Probability
3. Let {Zi }i∈Z be independent identically distributed random variables with zero mean
and unit standard deviation. For (a0 , a1 , ..., ar ) a sequence of r real numbers and
j ∈ Z, let
r
X
Yj =
ai Zj−i .
i=0
(i) For r = 1, with (a0 , a1 ) = (1, 1), give an expression for the probability density f
of Yj , given that Z has density g.
(ii) For r = 1, suppose a0 = 1 and |a1 | < 1. Show that Yj can also be written
X
Yj = Zj −
ρ(j−i) Yi
i<j
for a constant ρ that you should determine in terms of a1 .
(iii) For general r > 0, calculate the covariance of Yj and Yk , for j, k ∈ Z.
(iv) For r = 2, state and prove a general sufficient condition on (a0 , a1 , a2 ) such that
Yj and Yj−1 are independent regardless of the probability distribution of Z.
(v) For r = 2, give an example for which Yi and Yi−1 are uncorrelated but are not
independent (and show these properties hold).
(Hint: To show they are not independent, find appropriate values of (a0 , a1 , a2 )
with a1 6= a0 , suppose Z only takes the values ±1 with equal probability, and
consider the possible values of Y2 when Y1 is at its maximal or minimal value.)
4.
(i) In a model for a production process, the number of widgets successfully produced (Y ) before the kth failure follows a negative binomial distribution with
parameters (k, p), that is,
y+k−1
P (Y = y) =
(1 − p)k py ,
y
where k ∈ N and p ∈ (0, 1). The machine in question shuts down as soon as the
number of failures reaches some fixed (unknown) level r ∈ N. In order to estimate
this number, an observation Y = 10 was obtained (when the machine shut down),
and p is known to equal 0.25. Calculate numerically the maximum likelihood
estimator of r given Y , that is, the value of r which maximizes P (Y = y) for
k = r.
(ii) For your estimated value of r, what is the probability that the machine will shut
down before successfully producing three widgets?
(iii) For a fixed value of r, calculate the expected value of αY , for α > 0.
(iv) Using Markov’s inequality or otherwise, show that for any q ∈ [p, 1], there exists
a constant C such that P (Y > x) ≤ Cq x .
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Turn Over
Algorithms
5. This is the pseudo-code of the algorithm maxValInd which takes a finite list L of real
numbers, and returns the pair (VAL, IND), where VAL is the value of the maximum
entry in L, and IND is an index of VAL in L. Note that IND need not be unique.
function maxValInd(L)
let IND = 1
let VAL = L[IND]
for i = 1 to length(L)
if L[i] > VAL then
let IND = i
let VAL = L[IND]
end if
next i
return (VAL, IND)
end function
In this algorithm,
– the function length(L) takes a finite list L=[a1 , . . . , ak ] of arbitrary length, and
returns the number of elements k in L,
– the expression L[i] returns the ith element in the finite list L=[a1 , . . . , ak ].
(i) Construct the algorithm sumToN that takes a non-empty and finite list L of real
numbers that are sorted in increasing order and a real number N, and
– if there exists two indices i and j, such that 1 ≤i<j≤length(L), and
L[i]+L[j]=N, then sumToN(L,N) returns the pair (L[i],L[j]); in the case
when multiple pairs of indices exist with the required property, the function
returns one of the valid pairs,
– if the equality L[i]+L[j]=N is not satisfied for any indices i and j, such
that 1 ≤i<j≤length(L), the function returns the empty pair (,).
Ensure that sumToN completes the search in at most c*length(L) steps, for some
positive constant c that does not depend on L.
(ii) Show that your algorithm returns the correct value. Moreover, briefly explain
why your implementation of sumToN completes in at most c*length(L) steps.
(iii) Construct the algorithm eliminateRowCol that takes an n by m matrix A of real
numbers and sets the entries to zero in each of its columns and rows exactly
when the original A has at least one zero entry in them.
Ensure that the algorithm uses an additional storage of at most n+m variables.
You may use the following notation:
–
–
–
–
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rows(A) returns the number of rows in the matrix A,
columns(A) returns the number of columns in the matrix A,
A[i,j] returns the value of the jth entry in the ith row of A,
A[i,j]=a modifies the value of the jth entry in the ith row of A to a.
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(iv) Present a brief complexity analysis of your implementation of eliminateRowCol; that
is, find a function f , such that eliminateRowCol(A) completes in f (n,m) steps whenever A is an n by m matrix.
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