# Chapter V - Numerical &amp; Simulation Techniques in...

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Numerical & Simulation Techniques in Finance FRE 6251 Chapter V - Optimization Edward D. Weinberger, Ph.D, F.R.M. Adjunct Assoc. Professor Polytechnic University edw@panix.com

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OUTLINE Chapter V – Optimization Why optimization is important in finance Example: Maximum likelihood estimation of GARCH(1,1) model Kinds of optimization problems Some unconstrained optimization methods Simplex method (NOT linear programming) “lin min” Fletcher Powell Conjugate gradient
THE PROBLEM OF OPTIMIZATION Given continuous function, F, of possibly vector valued x, find x that minimizes F. Note that F may have, per Numerical Recipes Fig. 10.0.1, local optima discontinuities a global optimum outside of domain of interest F evaluations dominate computational effort, so want to minimize need for these

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EXAMPLE: Maximum Likelihood Estimation WE THINK THAT A PARTICULAR PRICE SERIES IS GENERATED BY THE GARCH(1,1) PROCESS R k+ 1 = σ k B k , B k ~ N (0, 1) σ 2 k+ 1 = α σ 2 k + β σ 2 0 + γ R 2 k HOW TO CHOOSE α , β , γ ???
ANSWER: Maximize joint probability of observations … { } ( 29 ( 29 1 2 0 2 1 2 2 1 1 2 2 / 2 1 2 1 2 1 2 1 , 2 ,..., , ,..., , ,..., , ,..., , Pr 1 2 2 - - = - - + + = = = - k k k k N k k r N N N N r dr e dr dr r d r r r f dr dr r d R R R k k γ β σ ασ σ πσ Choose α , β , γ to maximize the joint probability of observing what actually has been observed; namely, the product of the independent probabilities of observing R k , R k+ 1 , R k +2

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OR THEIR LOGARITHM… ( 29 ( 29 ( 29 1 2 0 2 1 2 2 1 2 1 1 2 2 2 1 2 ln 2 1 2 / ,..., , ln - - - = - + + = - - = k k k k N k k k N r r r r r f γ βσ ασ σ πσ So optimum obtained by finding α, β, γ = 1 – α - β
KINDS OF OPTIMIZATION PROBLEMS Unconstrained optimization Constrained optimization

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## Chapter V - Numerical &amp; Simulation Techniques in...

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