ADMISSIONS EXERCISES. MSc in Mathematical Finance. For entry in 2020

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1 MScMF 2019 ADMISSIONS EXERCISES MSc in Mathematical Finance For entry in 2020 The questions are based on Probability, Statistics, Analysis, Partial Differential Equations and Linear Algebra. If you are still studying for a degree and are yet to take or complete courses in one or more of these areas, please indicate so here and please specify the titles and dates of courses relevant to these areas which you are due to take or are still taking. You should attempt all questions and show all working. Stating an answer without showing how it was obtained will result in no credit for that answer. 1

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3 Probability 1. (a) Let X be a random variable that takes only non-negative values and has a finite (non-negative) expected value. Show that P(X a) E(X) a for a > 0, where E[ ] is the expectation operator. (b) Let Y be a random variable with moment generating function M(t) = E ( e ty ). Show that P(Y b) e tb M(t) for b > 0, t > 0. (c) Consider a standard normal random variable Z N(0, 1). It has the probability density function φ(z) = 1 e z2 /2 for z R. 2π Obtain the moment generating function of Z. Hence obtain an upper bound on P (Z a) as a function of t. By optimising over t, or otherwise, show that P(Z a) e a2 /2 for a > Let X N(0, 1). Derive the distribution of the random variable X Let X 1, X 2, X 3,... be an infinite sequence of independent, identically distributed random variables with finite mean µ and finite variance σ > 0. Show that the partial sums S n = X 1 + X X n have the property that where the symbol 1 lim n n S n d µ, d means the convergence is in distribution Turn Over

4 Statistics 4. A collection of independent random variables X 1,..., X n are modelled with a common distribution defined by 0 if x < 0, P(X k x) = (x/β) α if 0 x β, 1 if x > β, for fixed parameters α > 0, β > 0. (a) Write down the probability density function of X k. (b) Find the maximum likelihood estimators (MLEs) of α > 0 and β > 0 based on the observed values X 1,.,., X n. (c) The length, in millimetres, of cuckoo s eggs found in hedge sparrow nests can be modelled with this distribution. For the data 22.0, 23.9, 20.9, 23.8, 25.0, 24.0, 21.7, 23.8, 22.8, 23.1, 23.1, 23.5, 23.0, 23.0, evaluate the MLEs of α > 0 and β > 0. (d) Using the estimated values for α and β and assuming that cuckoo eggs volumes, in millilitres, satisfy the relationship V = 3 π L3, where L is the egg length in millimetres, a give an estimate for (i) the average volume of a cuckoo s egg, (ii) the maximum possible volume of a cuckoo s egg. a This formula takes into account the ellipticity of the egg

5 Analysis 5. (a) State Rolle s Theorem for continuous functions on bounded intervals of R. (b) Let f : R R be such that f has derivatives of all orders and f(x + 1) = f(x) for all x R. Prove that for each n = 1, 2,... there exists some y n such that f (n) (y n ) = 0. (Here f (n) denotes the n th derivative of f.) (c) Let g(x, y) = (e x + 1) y (e x2 e 2x 1 ) y + (e x2 1). For any fixed x R, show that the equation g(x, y) = 0 admits a solution y(x) 0, and lim x 0 y(x) = (a) Show that the log function is concave, i.e., that if 0 < x < y and t [0, 1] then log ( tx + (1 t)y ) t log(x) + (1 t) log(y). (b) let p, q (1, ) and 1 p + 1 q = 1. Prove that for any x = (x 1,..., x n ) R n and y = (y 1,..., y n ) R n we have n ( n x k y k x k p) 1 ( n p y k q) 1 q. k=1 k=1 k=1 This is one version of Hölder s inequality and can be stated as x y 1 x p y q, where x y = (x 1 y 1,..., x n y n ) and for r [1, ), x r = ( n k=1 x k r ) 1 r. [Hint: consider the first part of the question when answering the second part.] Turn Over

6 Partial Differential Equations 7. Assume V (S, t) is a sufficiently differentiable function so that it satisfies the Black- Scholes digital call problem V t σ2 S 2 2 V S 2 + r S V r V = 0, S > 0, t < T, S { 0 if S < K, V (S, T ) = 1 {S K} = S > 0, 1 if S K, (1) where r, σ > 0, T and K > 0 are given constants. (a) Use the chain rule with the following change of variables τ = σ 2 (T t), x = log(s/k), v(x, τ) = V (S, t), k = r/σ 2, to reduce (1) to v τ = 2 v x 2 + (k 1 2 ) v k v, x R, τ > 0 x v(x, 0) = 1 {x 0}, x R. (2) (b) Now let v(x, τ) = e α x+β τ u(x, τ), for some as yet unknown constants α and β. Show that using an appropriate choice of α and β, problem (2) can be reduced to the initial value problem for the heat equation u τ = 2 u, x R, τ > 0, x2 u(x, 0) = e (k 1/2)x 1 {x 0}, x R. (3) (c) For τ > 0 the solution to the initial value problem u/ τ = (1/2)( 2 u/ x 2 ) with u(x, 0) = f(x), a is u(x, τ) = 1 2πτ show that u(x, τ) = e (k 1 2 )x e 1 2 (k 1 2 )2τ Φ(d 2 ), where Φ(z) = 1 2π z (d) In terms of the original variables, show that f(y)e (y x)2 /2τ dy e q2 /2 dq and d 2 = x + (k 1 2 )τ τ. V (S, t) = e r (T t) Φ(d 2 ), d 2 = log(s/k) + (r 1 2 σ2 )(T t). (4) σ 2 (T t) a There are restrictions on what f(x) can be here, but we are not going to worry about that now

7 Algebra 8. (a) Consider the system of linear equations A x = b where A is an m n real matrix, and the column vectors x and b are elements in R n and R m respectively. Show that A x lies in the column space of A for any x. Deduce, or prove otherwise, that a solution x exists, for given A and b, if and only if the augmented matrix (A b) has the same rank as A. (b) Let t R and define a matrix A t by A t = 0 1 t 1 t 1 t 1 0 Determine the rank of A t for any t R. Let b R 3. For which t R does A t x = b have a unique solution? (c) Determine all vectors b R 3 such that the system of linear equations A 0 x = b has no solution.. (d) Determine 3 3 invertible matrices P, Q, such that P A 0 Q = End of Last Page