北京FRM一级强化_&aring

北京FRM一级强化_å

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1 FRM一级强化班讲义 Quantitative Analysis 讲师:程黄维 FRM 开发:FRM教研组 日期:2011.03.20 地点: ■北京 □上海 □深圳 上海金程国际金融专修学院 上海金程国际金融专修学院
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100% Contribution Breeds Professionalism 2-74 74 THE CONTENT Fundamentals of Probability Fundamentals of Statistics Monte Carlo Methods Estimating Volatilities ¾ Mean, variance, ¾ Skewness, kurtosis ¾ Normal distribution ¾ T distribution ¾ Estimation ¾ Hypothesis testing ¾ Liner Regression ¾ Geometric Brownian Motion ¾ Inverse transform method ¾ EWMA ¾ GARCH
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100% Contribution Breeds Professionalism 3-74 74 Fundamentals of Probability
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100% Contribution Breeds Professionalism 4-74 74 Random Variable (rv) ¾ What is random variable (rv) ¾ For financial markets, where stock prices , exchange rates , yields , and commodity prices can be viewed as random variables.
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100% Contribution Breeds Professionalism 5-74 74 Univariate Distribution Functions ¾ A random variable X is characterized by a distribution function: ¾ Cumulative Distribution Function F (x) = P(X x) ¾ Frequency Function or Probability Density Function (p.d.f.) : f(x) () x Fx f udu −∞ =
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100% Contribution Breeds Professionalism 6-74 74 Univariate Distribution Functions
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100% Contribution Breeds Professionalism 7-74 74 Moments - mean ¾ A random variable is characterized by its distribution function. Instead of having to report the whole function, it is convenient to summarize it by a few parameters, or moments . ¾ The expected value for x, or mean, is given by the integral: ¾ which measures the central tendency, or center of gravity of the population () ( ) uE X x f xdx +∞ −∞ ==
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100% Contribution Breeds Professionalism 8-74 74 Quantile ¾ The distribution can also be described by its quantile, which is the cutoff point x with an associated probability c: ¾ Generally, define this quantile as Q(X, c) VAR( c) = E(X) Q(X, c) () x Fx f udu c −∞ = =
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100% Contribution Breeds Professionalism 9-74 74 VAR as a Quantile
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100% Contribution Breeds Professionalism 10 10 -74 74 Variance ¾ Another useful moment is the squared dispersion around the mean, or variance: ¾ Standard Deviation: Volatility, risk SD( X) = σ 22 E[(X-E(X)) ] =
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