HW3_LIMO - 642:623 Computational Finance Report of Homework...

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642:623 Computational Finance: Report of Homework 3 Due on Feb 7, 2012 1:00 pm Tues 6:30PM Mo Li
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Mo Li 642:623 Computational Finance : Report of Homework 3 Explanation The task of this assignment is to implement the L’Ecuyer uniform random number generator and the Fishman acceptance-rejection algorithm for generate normally distributed numbers in this assignment. Based on these work, we compare the performance of the combinations of each uniform random number generators with each normal random number generating algorithm. Closed-form option price Algorithm Closed-form option pricing formula of Black Scholes model is the same as we have discussed before. Briefly, we use formula for European Call c ( t, x ) = xe - N ( d + ( τ, x )) - e - KN ( d - ( τ, x )) for direct pricing. Where d ± ( tau, x ) = 1 σ τ [ log x K + ( r - a ± 1 2 σ 2 ) τ ] Implementation Implemented by C++ function of double BSVanillaCallClosedform( double Expiry, double Spot, double Vol, double r, 5 double Strike); With most of the parameters self-explainable. Especially, the parameter r is the risk-free rate. MC option price with Park-Miller unform generator Algorithm Park-Miller is a method of generating uniform random numbers, a linear congruential generator. With the following formula x i +1 = ( ax i + c ) mod m where x 0 = seed a = multiplier c = shift m = modulus then we have u i +1 = x i +1 /m to be the i + 1th generated random number. Here, in the example of Park-Miller generator, we have a = 168707 m = 2 31 - 1 c = 0 which has a period of 2 31 - 1. Page 1 of 7
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Mo Li 642:623 Computational Finance : Report of Homework 3 Implementation We use the code implemented by Joshi for generating random numbers. Joshi’s code constructed two main classes for this algorithm. Class RandomBase in head file Random2.h is the base class of all generator class with the method of GetGaussians(MJArray& variates) of generating normal random numbers by inverse transform. Class RandomParkMiller in head file ParkMiller.h is is derivative class of RandomBase , which implements the method of generating uniform random numbers by Park-Miller Algorithm by function GetOneRandomInteger() . And inheriting the normal random number generating function of RandomBase . MC option price with L’Ecuyer uniform generator Algorithm L’Ecuyer uniform generator employing the idea of combined generator to increase period length but preserve same simplicity. Construct two elementary congruential random number generators as following x 1 ,i = ( a 11 x 1 ,i - 1 + a 12 x 1 ,i - 2 + a 13 x 1 ,i - 3 ) mod m 1 with a 11 = 0, a 12 = 63308, a 13 = - 183326 and m 1 = 2 31 - 1. And x 2 ,i = ( a 21 x 2 ,i - 1 + a 22 x 2 ,i - 2 + a 23 x 2 ,i - 3 ) mod m 2 with a 21 = 86098, a 12 = 0, a 13 = - 539608 and m 1 = 2145483479. And we get the uniform random number in (0 , 1) by two more steps of x i = x 1 ,i - x 2 ,i mod m 1
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