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 FE Problem Set #19 Team LatteApril 18, 2011 The following is an extract from the Certificate in Financial Engineering (CFE) sample test. Many of these questions will require use of ExcelTM spreadsheets and application of numerical techniques. A correlation matrix is given by: The Cholesky matrix of the above correlation matrix is given by:    Given a function : Using the Gauss Legendre method (using 2 points) the value of this integral will be close to: 2.35 3.54 5.23 6.75 Given a function: Using a trapezoidal rule the value of the above integral will be close to: 105 125 188 235 The nth order Bessel function is given by: The value of the function at and order, is: 0.1657 0.3641 0.7563 0.9215 A vector of magnitude 10 along the original, positive x axis is rotated along a 45 degree counter-clockwise direction. The new coordinate system will be represented by the column vector:    A correlation matrix is given by: The eigenvalues of the above matrix are given by: 1.25 and 0.75 0.50 and 1.50 0.15 and 1.85 0 and 2 If is a one dimensional standard Weiner process, then which of the following stochastic differential equations represent a Bessel process of dimension for the variable :    The difference in interest rate differential between two countries follows a drift less arithmetic Brownian motion (ABM) given by the following stochastic differential equation: In the above equation, is the random variable (interest rate differential) following the ABM, is the volatility and is a standard Weiner process. If at time, , the interest rate differential is 1% and if the daily volatility of the interest rate differential is 0.50% then the probability that this differential will hit 3.5% at any point during the next one year is: 46.25% 65.35% 75.28% 82.15% The Laplace transform of a function, is defined as: If, is the maturity of a financial derivative contract and at any point in time, , and are constant parameters, and at the boundary condition is given by then the price of this financial derivative contract, , is a function of the underlying asset, and is governed by the following partial differential equation (PDE): Using the Laplace transform, as defined above, we can write the above PDE as Ordinary differential equation in as :    If is an Ito diffusion then the Dynkin Operator is given by:    Answers (b) (c) (c) (b) (c) (a) (d) (c) (b) (a) Any comments and queries can be sent through our web-based form. More on FE Problem >> back to top More from FE Problems More from Articles Searching for the Most Beautiful Equation in Finance Where does a Black Swan Come from? Napoleon on Wall Street: Advent of the Stochastic Volatility Models The Remarkable Power of the Monte Carlo Method Mean Reversion and the Half-Life of Interest Rates 