Chapter III - Numerical & Simulation Techniques...

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Unformatted text preview: Numerical & Simulation Techniques in Finance FRE 6251 Chapter III Numerical Integration Edward D. Weinberger, Ph.D, F.R.M. Adjunct Assoc. Professor Polytechnic University edw@panix.com OUTLINE Chapter III Numerical Integration Statement of Problem Many approaches, only some described here Rectangular Integration Romberg Integration Gaussian Quadrature THE PROBLEM ( 29 , dx x f b a Find a reasonable approximation for maximizing accuracy for a given amount of work (measured in function evaluations) EXAMPLE (from credit risk) -- -- - - - - 5 5 2 2 N N 2 N N 2 2 2 df d vf B c uf A e df d vf B c uf A e f f N( x ) is standard normal distribution, A , B , c , d , u , and v are all constants. MANY APPROACHES Adding up values of integrand at cleverly chosen points in range of integration (This lecture). Rectangular Integration Classical Formulas (eg. Trapezoidal, Simpson) Romberg Gaussian Quadrature MORE APPROACHES Solving Initial Value Problem to find F ( b ) . Good if F has sharp bends. Analytic integration of some approximation of f , eg. Chebyshev polynomials, spline fits Fast Fourier Transform Monte Carlo ( 29 ( 29 , = = a F x f dx dF THE CURSE OF DIMENSIONALITY FOR INTEGRALS Each independent variable of function increases computation effort by factor of M (except for Monte Carlo) M Steps M Steps M 2 steps 2 ind. variable 3rd ind. variables M steps FIRST ALTERNATIVE: RECTANGULAR INTEGRATION ( 29 ( 29 - + =- - = N a b i a x x f N a b dx x f i b a N i i , 1 a b EXAMPLE...
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Chapter III - Numerical & Simulation Techniques...

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