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Comp_Fin_HW5 - Jaime Frade Computational Finance Dr Kopriva...

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Jaime Frade Computational Finance: Dr. Kopriva Homework #5 0. Executive Summary In this assignment I developed a program which evaluates definite integrals given the intregrand f , the closed interval [ a, b ], and a relative tolerance for the error. I will use an adaptive quadrature method that uses Simpson’s rule with an estimate for the errors involved. I will then use the equation to solve applications in the Black-Scholes equation. To ensure accuracy, I tested several functions with known and calculatble results. (Advice from collegue, similiar to HW3). Each test case function was specifically chosen so that I could calculate the exact value and the error estimate in theory. The numerical results from the code were compared to the exact values and the error was given with a certian tolerance. Each test case was proven correct and behaved according to the error estimates. I obtained the following for each question in this homework: (b) . P (1) = 0 . 5 + 0 . 341342151 = 0 . 841342151 (c) . Value of Call: 98 . 5671380 (d) . Implied volatility: 7 . 6089457% 1
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1. Statement of Problem In this homework assignment there were 4 assigned problems. (a). Write an adaptive quadrature function that uses Simpson’s rule as its quadrature rule. Test your routine on some integrals for which I know the exact values and that demonstrate how the method behaves. (b). Find the value P at x = 1 in the function below, accurate to at least six significant digits. Explain the acute of accuracy in my result. P ( x ) = 1 2 π x -∞ e - x 2 2 dt (1) (c). Given the data below, compute the value of the call, C , to at least three significant digits using the equations below and (1). State the number of figures. Given the following: Time of expiry ( T ) 54 365 Current time ( t ) 0 Interest rate ( r ) 6 . 75% = 0 . 0675 Volatility ( σ ) 13 . 5% = 0 . 135 Exercise Price ( E ) 3425 . 0 Asset Price ( S ) 3441 . 0 Using the information above and (1), compute the value for the call , C , using the following equations: C ( S, t ) = SP ( d 1 ) - Ee - r ( T - t ) P ( d 2 ) , (2) where d 1 and d 2 are determined by d 1 = log S E + r + σ 2 2 ( T - t ) σ T - t (3) d 2 = log S E + r - σ 2 2 ( T - t ) σ T - t (4) 2
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(d). Given the data below, compute the implied volatilty , σ to at least four significant digits using the equations below and (1). Given the following: Value of call ( C ) 94 . 0 Time of expiry ( T ) 117 365 Current time ( t ) 0 Interest rate ( r ) 6 . 75% = 0 . 0675 Exercise Price ( E ) 3475 . 0 Asset Price ( S ) 3461 . 0 In this problem, I will determine the implied volatility by using the methods from previous homework. Using the Newton-Rhapson method, I will calculate with a certain error. 3
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2. Description of the Mathematics There are several concepts of mathematics used in this homework assignment: (a). Statistics. In a given Gaussian standard normal probability function, (1), to make calculations in part(b) less complicated and less prone to error. I will use the properties of integrals to split up into sum of two separate integrals where one is known. Let x = 1.
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