Saturday, November 13, 2010

The financial crisis and economic meltdown has taught us one big lesson: managing risk failed at every stage. homebuyers failed to assess the risks that come with the benefits of owning a home through subprime mortgages - receiving little or misleading information about the risks involved. banks, hedge funds, etc invested in bonds and collateralized debt obligations, backed by subprime mortgages as well. credit agencies and insurance firms failed to appreciate the assumptions and limited data being used to value debt-backed securities.

risk is at the centre of all this. can we have a risk-less asset, by the way? more about this in the next post...

for now, a nice aside:

Louis Bachelier, an obscure French mathematician, wrote a thesis on speculation and defended it with the great Henri Poincare standing as thesis advisor. Bachelier, Paul Samuelson claims, beat Einstein in analyzing what is called Brownian motion (the physicist did so in 1905). even the efficient markets doctrines were anticipated by the Frenchman! Bachelier’s thesis was found to involve price changes subject to absolute Gaussian distribution. by contrast, limited liability common stocks can raise and fall but by definition, their prices cannot be negative. so, Samuelson suggested replacing the distribution by geometric Brownian motion, based on log-Gaussian distributions.

so, what is Brownian motion?

in fact, Bachelier worked on the Paris bourse and seems to derive the inspiration for his thesis from there. “Among the commodities sold on the bourse were various government bonds called rentes. But one could also buy options on these rentes—a form of “derivative.” You bought the right to purchase such a bond at a later time at the price for which it sold at the time you bought the option. Until then you did not own the bond but just the option to buy it at a later time. If the price had gone up you would be in the money and if it went down you lost whatever it had cost you to buy the option. Unless you were a clairvoyant you would have to guess at the future price of the bond.

What Bachelier wanted to do was to replace clairvoyance by mathematics. To do this he needed to make some assumption of how stock prices evolve. He decided that at any given time it was as likely for a stock to go up as down. You might at first think that this means that a stock price would never get anywhere. But after a first up, say, the stock has a fifty-fifty chance of going up as down and thus moving further away from its starting point. In short the price of the stock, Bachelier decided, takes a “random walk.”

In his thesis Bachelier presents mathematics of this, along with examples. One of the things he shows is that the price evolves away from its initial price as the square root of the time elapsed (days, say)—not an obvious result. It is like the random walk of a drunk whose distance from his starting point increases with the square root of the time elapsed except here the object taking the random walk is the stock price.

What he did not know was that he had solved an outstanding physics problem. Early in the nineteenth century a Scottish botanist named Robert Brown had observed that microscopic particles suspended in liquids had a jiggling movement which we now call Brownian motion. It was Einstein in one of his great 1905 papers who presented the theory of this movement as a random walk induced by the bombardment of the suspended particles by the molecules of the liquid. The equations Einstein arrived at are identical to the ones in Bachelier’s thesis, which he had never heard of.”

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