高斯分布, 音樂, 遐想 在金融市場生存遊戲, 整天思想數和情(emotion). 拋磚引玉: 世間許多事 , 3 sigma 之內, 規矩與方圓 (“不盡合理”) 當家; 3 sigma 之外, 情當家, 野性的很 ,常常是驚天地泣鬼神, 包括事件發生, 過程, 結局, 影響…. 正態分布 ，也稱為高斯分布. A Gaussian distribution significantly underestimates the probability of a large price or rate movement. A Gaussian distribution may underestimate the probabilty of a 3 sigma price movement by a factor of 10. In other words, the chance of a 3 sigma movement is potentially 10 times greater than that predicted by a Gaussian probability curve. distribution of value movements in the financial markets follow Levy or Cauchy distributions Supposedly, The Cauchy distribution is an example of a distribution which has no mean, variance or higher moments defined. Its mode and median are well defined and are both equal to x0. zt Uncertainty The perils of prediction May 31st 2007 From The Economist print edition “IT'S tough to make predictions, especially about the future,” said that great baseball-playing philosopher, Yogi Berra. And yet we continue to try, churning out forecasts on everything from the price of oil to the next civil war. Nassim Nicholas Taleb, a professor of the sciences of uncertainty (who gave us “known unknowns”), has no time for the “charlatans” who think they can map the future. Forget the important things: we can't even get it right when estimating the cost of a building—witness the massively over-budget Sydney Opera House or the new Wembley Stadium. The problem is that almost all forecasters work within the parameters of the Gaussian bell curve, which ignores large deviations and thus fails to take account of “Black Swans”. Mr Taleb defines a Black Swan as an event that is unexpected, has an extreme impact and is made to seem predictable by explanations concocted afterwards. It can be both positive and negative. Examples include the September 11th 2001 attacks and the rise of the internet. Smaller shocks, such as novels and pop acts whose popularity explodes thanks to word of mouth, can also be Black Swans. Humans are bad at factoring in the possibility of randomness and uncertainty. We forget about unpredictability when it is our turn to predict, and overestimate our own knowledge. When researchers asked a group of students to choose a range for the number of lovers Catherine the Great had had, wide enough to ensure that they had a 98% chance of being right, a staggering 45% of them got it wrong. Why didn't they guarantee being correct by picking a range of none to ten thousand? After all, there were no prizes for keeping the range tight. The answer is that humans have an uncontrollable urge to be precise, for better or (all too often) worse. That is a fine quality in a watch-repair man or a brain surgeon, but counter-productive when dealing with uncertainty. Mr Taleb cut his philosophical teeth in the basement of his family home in Lebanon during the long civil war there (another Black Swan), devouring books as mortars flew overhead. By the time he began work as a financial-market “quant” in the 1980s, he had already become convinced that the academic mainstream was looking at probability the wrong way. He remains a maverick, promoting the work of obscure thinkers and attacking Nobel laureates. All he is trying to do, he says, is make the world see how much there is that can't be seen. Why, he asks, do we take absence of proof to be proof of absence? Why do we base the study of chance on the world of games? Casinos, after all, have rules that preclude the truly shocking. And why do we attach such importance to statistics when they tell us so little about what is to come? A single set of data can lead you down two very different paths. More maddeningly still, when faced with a Black Swan we often grossly underestimate or overestimate its significance. Take technology. The founder of IBM predicted that the world would need no more than a handful of computers, and nobody saw that the laser would be used to mend retinas. Nor do we learn the right lessons from such eruptions. Mr Taleb argues convincingly that the spectacular collapse in 1998 of Long-Term Capital Management was caused by the inability of the hedge fund's managers to see a world that lay outside their flawed models. And yet those models are still widely used today. This is ridiculous but not surprising. Business is stuffed full of bluffers, he argues, and successful companies and financial institutions owe as much to chance as to skill. That is a little unfair. Many blockbuster products have their roots in bright ideas, rigorous research and canny marketing, rather than luck. And corporate “scenario planners” are better than they used to be at thinking about Black Swan-type events. Still, this is a small quibble about a deeply intelligent, provocative book. Deftly weaving meditation with hard-edged analysis, Mr Taleb succeeds in bringing sceptical empiricism to the masses. Do not expect clear answers. He suspects that crises will be fewer in number but more severe in future. And he suggests concentrating on the consequences of Black Swans, which can be known, rather than on the probability that they will occur, which can't (think of earthquakes). But he never makes professional predictions because it is better to be “broadly right rather than precisely wrong”.
