Chapter III

# Chapter III - Numerical& Simulation Techniques in...

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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 [email protected] 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 π π 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 in...

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