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Stochastic Models Probability Walt Pohl Universität Zürich Department of Business Administration February 28, 2013 What is Probability? Mathematically, it’s just a list of axioms: 1 Probabilities between 0 and 1. 2 Probabilities of all possibilities add up to 1. 3 If two things, A and B can’t both happen at the same time, then probability of A or B equals probability of A + probability of B. But when do these axioms apply? Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 2/1 The Three Justifications of Probability 1 2 3 Subjective probability Objective probability Market probability Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 3/1 Objective Probability Objective probability – a probability statement is an objective fact about the world. If you flip a fair coin a large number of times, it will come up heads 50% of the time. But what about events that only happen once? Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 4/1 Subjective Probability Subjective probability – a probability statement is a subjective belief how likely something is. Usually formatted as betting odds. You believe that a coin is fair if you’re willing to bet 0.5 to win 1 that it comes up heads. In practice, probability is frequently subjective by necessity. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 5/1 Market Probability Market probability – the betting odds assign in the marketplace. In 2010, SkyBet gave Switzerland a 55% chance of beating Honduras in the World Cup. Did SkyBet believe that Switerland has a 55% chance of winning? Probably not. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 6/1 Probability Distributions We usually work with real-valued probability distributions: for each range [a, b], we can calculate P(a ≤ x ≤ b). Sufficient to know the cumulative distribution function (cdf): F (a) = P(x ≤ a) Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 7/1 Discrete Probability Distributions Can only take on a finite number of values: a1 , . . . , an . Can be given by probability mass function, f (a) = P(x = a). CDF is F (a) = X f (ai ). ai <a Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 8/1 Continuous Probability Distributions Can take a continuous range of values. Can be given by probability density function, f (a). CDF is Z a f (x)dx. −∞ Probability of any single real number happening is zero. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 9/1 Other Possibilities Discrete and continuous probability distributions are not the only possibilities. Let X be normally distributed, and define a new random variable as min(X , 0). Then this is a discrete random variable 50% of the time (exactly equal to zero), and a continuous random variable 50% of the time. There are more exotic possibilities, related to fractals. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 10 / 1 Working with Probability Distributions We typically want to know certain things. Moments: Mean Variance Quantiles: Median Extremes: 5%, 95% We get these values either analytically or by Monte Carlo simulation. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 11 / 1 Aside on Quantiles For continuous distributions, quantiles and the CDF are closely related. The q-th quantile is a such that P(x ≤ a) = q. If F is the CDF, then F −1 (q) is the q-th quantile. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 12 / 1 Analytically For discrete distributions, you can compute everything by brute force. For continuous distributions, you can do it only certain cases. Most important case: normal distribution. √ 1 2πσ 2 e −(x−µ)2 2σ 2 We can show the mean is µ and the variance is σ 2 . Let Φ be the CDF. Then Φ−1 (q) is the q-th quantile. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 13 / 1 Monte Carlo Simulation We simulate draws from the distribution on a computer. We then work with these statistically – to compute the mean, standard deviation, or quantiles, we use the sample quantities computed from the random draws. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 14 / 1 Why Does Monte Carlo Work? Monte Carlo simulation generates a simulated CDF, which approximate the true CDF. Glivenko-Cavelli Theorem: As the number of draws goes to infinity, the simulated CDF converges to the true CDF. Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 15 / 1