HW5_LIMO - 642:623 Computational Finance Report of Homework...

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642:623 Computational Finance: Report of Homework 5 Due on Feb 14, 2012 1:00 pm Tues 6:30PM Mo Li
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Mo Li 642:623 Computational Finance : Report of Homework 5 Explanation The task of this assignment is to pricing the Vanilla, Geometric and Arithmetic Asian call option with difference schemes variance reduction. Then testing program is built based on the code’s structure of Joshi. Meanwhile, we compare the efficiency and accuracy of the following variance reduction methods: Park-Miller random generator with antithetic variates, Park-Miller random number generator with stratified samples of specified strata and QMC method using Sobol sequences. In this case, we employ closed-form formula of pricing of vanilla options and geometric Asian options for result checking. Closed-form option price Algorithm Vanilla Option The closed-form formulae of vanilla option with underlying stock price process under Black Scholes Model is derived clearly. Formula for European call option c ( t, x ) = xe - N ( d + ( τ, x )) - e - KN ( d - ( τ, x )) Formula for European put option p ( t, x ) = e - KN ( - d - ( τ, x )) - xe - N ( - d + ( τ, x )) where d ± ( tau, x ) = 1 σ τ log x K + r - a ± 1 2 σ 2 τ Geometric Asian Option Closed-form option pricing formula of geometric Asian Call option with the underlying assets being geometric Brownian motion is clearly derived. Briefly, we use formula c G = Se ( b - r ) τ N ( d 1 ) - Ke - N ( d 2 ) for direct pricing. Where d 1 = ln( S/K ) + ( b + 0 . 5 σ 2 A ) τ σ A τ and d 2 = d 1 - σ A τ The adjusted volatility and dividend yield are given as σ A = σ 3 b = 1 2 r - d - σ 2 6 But for the closed-form pricing of arithmetic asian option, a exact solution does not exist. Implementation We create class GeometricAsianCF in which we implemented the closed-form pricing formula of Vanilla option and Geometric and Arithmetic Asian options. The prototype of GeometricAsianCF is defined as following Page 1 of 14
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Mo Li 642:623 Computational Finance : Report of Homework 5 class GeometricAsianCF { public : 5 GeometricAsianCF( unsigned long NumberOfTimes_): NumberOfTimes(NumberOfTimes_){} double closedformVanillaPrice( double Expiry, double Spot, double Vol, double r, 10 double d, double Strike) const ; double closedformGeometricAsianPrice( double Expiry, double Spot, double Vol, 15 double r, double d, double Strike) const ; virtual ˜GeometricAsianCF(){} 20 private : unsigned long NumberOfTimes; }; With the member variable NumberOfTimes to be the number of the discretely chose time of monitored price spots. We have the functions of vanilla and geometric Asian option implemented by closedformVanillaPrice and closedformGeometricAsianPrice respectively with most of the parameters self-explainable. Spe- cially, the parameter d is the dividend yield and r to be the risk free interest rate. Park-Miller Uniforms with Antithetics 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 the generated uniformed random number to be u i +1 = x i +1 /m The Antithetic Scheme is discussed in Assignment 2. We recall it briefly here. We generate pairs (
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