Chapter V - Numerical & 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
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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 ???
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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
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KINDS OF OPTIMIZATION PROBLEMS Unconstrained optimization Constrained optimization
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Chapter V - Numerical & Simulation Techniques in...

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