Fractals, Mandelbrot, Bachelier and Volatility
A Rogue mathematicians search for answers. From Tripple Helix.
How do you measure the rough and jagged coastline of the United Kingdom? Or the sharp, seemingly arbitrary rise and fall of a stock-price? To the layperson, the answer to the first question might seem a straightforward matter of getting on a boat and making a trip. 1 The answer to the second question might be observing the markets for long periods of time and trying to discern patterns within the graphs (much like technical analysts do today). However, mathematicians aren’t known for their love of fieldwork. This is the story of a rogue mathematician’s search for an answer to questions like these, questions which have to do with how we measure ‘roughness’ in the world around us: from the sharp edges of a stock price graph to the uneven surface of a cauliflower. 1 It tells the story of how different kinds of ‘roughness’ can be described by different kinds of statistical distributions, and how we may have been using the wrong distribution to price our bonds and derivatives all along.
Mandelbrot’s Story: From World War II France to ‘60s America
Mandelbrot was born in Warsaw in 1924 and privately tutored by an uncle who despised rote learning. His first exposure to algebra and his self-discovery as a mathematician followed his family’s relocation to France in 1936. After being relegated to the countryside at the onset of war in 1944, he was hidden by French resistance members in a Lyon school. For every question the professor asked, Mandelbrot would describe a geometrical approach to yield a fast, simple solution. He passed in this way through a series of elite French universities as well as Caltech. He returned to Paris for his PhD, then proceeded to the Institute for Advanced Study. It was during these heady years that Mandelbrot developed the fractal, to which we now turn. 1
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