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Unformatted text preview: Quantitative Approach in Finance &QuantsÂ¡: Financial quantitative analysis and/or its methods MPT (modern portfolio theory) quants approach STAT (statistics/econometrics) quants approach Why do we need to use a quantitative approach to study Â¢nance? Normative: How should people behave (theoretical) Positive: How people behave (empirical) Strength and weaknesses of quantitative approach Y. F. Chow (CUHK) Financial Economics 2009Â£10 First Term 1 / 60 Notes on the Theory of Choice Choice under Certainty Elements of the Decision Problem Binary Relation Operators Preference Orderings Choice Criterion Numerical Representation Application: An Illustration of Capital Theory under Certainty Choice under Uncertainty A Model for Uncertainty Contingent Claims Stochastic Economy Y. F. Chow (CUHK) Financial Economics 2009&10 First Term 2 / 60 Elements of the Decision Problem Whereas the objective and question asked seem to be almost trivial, an abstract rigorous treatment is extremely simpli&ed by the introduction of some mathematical elements. De&nition (Choice Problem) A choice problem is a triple ( X , R , O ) where the three elements are given as follows: Choice set X : Set of possible alternatives, which can be &nite, countable or uncountable. It can be an abstract set or a topological space. Relation operator R : Device to sort alternatives. Choice criterion O : Rule to discriminate between ordered elements of choice set. Y. F. Chow (CUHK) Financial Economics 2009Â¡10 First Term 3 / 60 A relation operator does de&ne an ordering on the choice set. If for X , Y 2 X we have that X R Y then the relation operator establishes a property that holds true between the choice elements X and Y . Such a property therefore does establish a binary relation between two elements. Typical examples of the relation operator R will be Â¡weakly preferredÂ¢or Â¡indi/erentÂ¢. Other important examples in the theory of decision making will be Â¡more riskyÂ¢, etc. Those among the readers trained in mathematical analysis will clearly recognize the relation with equivalence relations established on sets of numbers. The most universal choice criterion in neoclassical economics and &nance is optimality. We quite often will assume that an individual will choose from all elements of the choice set the optimal one, where optimality will be de&ned by a relation operator that acts on the choice set. Y. F. Chow (CUHK) Financial Economics 2009Â£10 First Term 4 Â¤ 60 Example (Project Evaluation / Capital Budgeting) This problem is almost trivial but let us describe the diÂ¡erent elements: X : Possible projects available characterized by their cash &ows. R : Net present value rule. For X , Y 2 X , X R Y , NPV ( X ) & NPV ( Y ) ....
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 Fall '09
 Prof.Chow
 Finance, Economics, Optimization, Utility, Y. F. Chow

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