Mathematical and Computational Finance

The questions are based on Linear Algebra, Calculus, Probability, PartialDierential Equations, and Algorithms. If you are still studying for a de-gree and are yet to take or complete courses in these areas, please indicateso here. Please specify the titles and dates of courses which you are dueto 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.1Statement of authenticityPlease sign and return the following statement together with the solutions. Your applicationwill not be considered without it.I certify that the work I am submitting here is entirely my own and unaidedwork.Print NameSignedDate2016 2Linear algebra and calculus1. (a) Let T : R3 ! R3 be a linear transformation. Prove the equivalence of thefollowing statements.(i) R3 = ker(T) im(T);(ii) ker(T) = ker(T2);(iii) im(T) = im(T2):(We write R3 = ker(T) im(T) if, for all v 2 R3, there exists x 2 ker(T) andy 2 im(T) such that v = x + y, and ker(T) \ im(T) = f0g:)(b) Magical squares are 3 by 3 matrices with the following properties: the sum of allnumbers in each row, and in each column, and in each diagonal is equal. Thisnumber is called the magical number. For example0@4 3 89 5 12 7 61Ais a magical square, and the magical number is 15.(i) Prove that the set of magical squares forms a vector space with the usualmatrix addition and scalar-matrix product.(ii) Find a basis of the vector space of magical squares and determine its dimen-sion.2. (a) Evaluate the following integrals, indicating your working:(i)Zx5exsdx; (ii)Zdxx3 ?? x;(iii)Z0sin123(x)dx; (iv)Z 1xdt1 + et:(b) (i) Find the second derivatives of f(x) =R x2=2??1 ex??t2=2dt andg(x) =R x2=2??1 e??(x2+1)t2dt.(ii) Derive the solution of the ordinary dierential equationd2ydx2 = f(x); x > 0; y(0) = 0;dydx(0) = 0;in the formy(x) =Z x0h(t)dt;for a function h which you should determine in a simple form.2016 3 Turn OverPartial Dierential Equations3. Consider the heat equation@u@t=@2u@x2 :(a) Show that if u(x; t) = t() where = x=pt and is a constant, then ()satises the ordinary dierential equation??120 = 00;(where 0 d=d):(b) Show that Z 1??1u(x; t)dx =Z 1??1t()dxis independent of t only if = ??12 . Further, show that if = ??12 thenC ??12= 0where C is an arbitrary constant.(c) From this last ordinary dierential equation, and assuming C = 0, deduce thatu(x; t) =Apte??x2=4tis a solution of the heat equation (here A is an arbitrary constant).(d) Show that as t tends to zero from above,limt!0+1pte??x2=4t = 0 for x 6= 0and that for all t > 0 Z 1??11pte??x2=4tdx = Bwhere B is a (nite) constant. Given thatR 1??1 e??x2dx =p; nd B.(e) What physical and/or probabilistic interpretation might one give to this solutionu(x; t)?2016 4Probability4. A group of n players is divided into two teams (a red and a blue team) according tothe following procedure.First a number X is chosen randomly form the set f1; 2; : : : ; n ?? 1g, with all valuesequally likely. Then X of the n players are chosen to form the red team, with allpossible sets of size X equally likely. The remaining n ?? X players form the blueteam.(a) Find the mean of the size of the red team.Consider one particular player, called Player A.(b) Find the probability that the team containing Player A has size k for k =1; 2; : : : ; n ?? 1.(c) Find the mean of the size of the team containing Player A.After the teams have been chosen, each team selects a captain, who is equallylikely to be any member of that team.(d) Find the conditional distribution of the size of the team containing Player A,given that Player A is the captain.2016 5 Turn Over5. For k; > 0, the Gamma (k; ) distribution is dened by the following density functionfk;(x) =(1k??(k)xk??1e??x= if x 0;0 if x 0. Show that (X+Y ) Gamma(k1+k2; ). [You may assumewithout proof that Z 10(1 ?? x)k??1xm??1dx =??(k)??(m)??(k + m)for k;m > 0:](b) Derive the distribution ofPni=1 Xi where Xi Gamma(ki; ) for i = 1; : : : ; n areindependent random variables for some k1; : : : ; kn; > 0.(c) For > 0 and n = 1; 2; : : : ; let the random variable Zn be dened asZn =Un ?? E[Un] pVar[Un];where Un Gamma(n; ). Using part (a), part (b) and the Central Limit The-orem (or otherwise) show that Zn converges in distribution as n ! 1. Specifythe limiting distribution.[You may assume without proof that the Gamma(k; ) distribution has nite rst andsecond moments for k; > 0:]2016 6Algorithms6. This is the pseudo-code of a recursive algorithm (a routine calling itself) calculatingthe n!, where n is assumed to be a positive integer:function VAL=factorial(N)if N=1 thenVAL=1elseVAL=N*factorial(N-1)end ifreturn VALend functionThe pseudo-code of a non-recursive implementation could look like thisfunction VAL=factorial(N)VAL=1for I=1 to NVAL=VAL*Iend forreturn VALend function(a) Write a pseudo-code for a recursive function fibonacci(N) that takes a positiveinteger N as input argument and returns the Nth Fibonacci number f(N), wheref(1) = f(2) = 1 and f(N) = f(N ?? 1) + f(N ?? 2) for N > 2.(b) Write a pseudo-code for a non-recursive version of fibonacci(N).(c) Assuming that the computational cost of adding two integers is one unit, estimatethe total computational cost of the function in part (a) and the function in part(b) in terms of N.2016 7 End of Last Page