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Black-Scholes and Long Term Capital Management

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January 13, 2009 – Comments (1)

I ran into a reasonably coherent primer on this (Schaum’s Outline Series: “Investments”, p. 240 in chapter 21). It is an interesting equation that calculates what price an option should be, and the so-called “hedge ratio” also pops out of the calculation. It seems to be a fool-proof way of making money, and LTCM did just this using leverage. The best resource I found was on the PBS/Nova website (“Trillion Dollar Bet”). They have a very nice, interactive webpage that describes this mathematical beastie.

I have no intention of dabbling in options, but I was curious, and this is what I found. Black-Scholes-Merton uses Brownian motion as a model for the movement of stock prices. BM is a subset of stochastic/random processes. If BM sounds familiar, it describes the movement of gas molecules from high school chemistry class (for those who actually paid attention; count me out). If you plot this movement on a piece of paper, you get “lognormal” or “normal” or “Gaussian” distribution, which is the familiar bell curve. These beasties are well known and documented, and there are charts and tables that mathematically describe them, so our intrepid heroes used them as a model for stock and option prices.

Problem: it assumes that movement is reasonably small, random, and there are no external influences or biases. “Rare Events” are, well, rare: if they start to happen a lot, or the movement start to trend in a particular direction or tendency, or an external event (oh, say, Russia defaults on bonds or the world-wide financial system melts down) begins to influence in a particular way, the model no longer works, because the underlying assumptions are no longer true.  This is what happened to LTCM. Still, I find the equation interesting and might form at least a basis for investing in options.

 

1 Comments – Post Your Own

#1) On April 08, 2009 at 7:48 PM, RHaganC (< 20) wrote:

its an insurance policy that works only when it's nice outside and breaks down in a hurricane - - WHEN YOU NEED IT MOST.

sure - you will make $$$ for a bit, but a hurricane will kill you.  it only takes one day to die.

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