27 June 2010
15 June 2010
Option pricing
The Nobel prize-winning Black-Scholes option pricing formula is generally advanced by financial academics as the best possible way to value options. This may be true, but that doesn't mean it is any good.
It is further worth noting that the formula (often used for dynamic hedging) models security prices as follows:
dS = mSdt + sSdW
Where S is security price, mS is the average rate of change of the security (drift rate), t is time, sS is standard deviation of the security, and W is a geometric Brownian function.
The problem here is that stock prices generally can be modeled using geometric Brownian motion. However, there is (more or less) a hard stop to trading at market close. Despite this fact, orders can be placed after market close, to be executed immediately upon the opening of the market. For this reason, we will rarely find extreme jumps between the underlying security price at closing and opening. The possibility of what is effectively discrete motion should make us realize that using the geometric Brownian without at least including some fudge factor for the possibility of discontinuous motion (though approximating the Brownian with a discrete probability distribution such as the binomial would be preferable) will occasionally but decisively give us bad pricing information.
Against those who would suggest that the error in the formula will be relatively minor, we advance the anecdotal but highly compelling evidence that the year after Mr. Scholes won his Nobel prize, the hedge fund he was helping to run (Long Term Capital Management) imploded in a multibillion dollar disaster that left financial markets reeling.
It is further worth noting that the formula (often used for dynamic hedging) models security prices as follows:
dS = mSdt + sSdW
Where S is security price, mS is the average rate of change of the security (drift rate), t is time, sS is standard deviation of the security, and W is a geometric Brownian function.
The problem here is that stock prices generally can be modeled using geometric Brownian motion. However, there is (more or less) a hard stop to trading at market close. Despite this fact, orders can be placed after market close, to be executed immediately upon the opening of the market. For this reason, we will rarely find extreme jumps between the underlying security price at closing and opening. The possibility of what is effectively discrete motion should make us realize that using the geometric Brownian without at least including some fudge factor for the possibility of discontinuous motion (though approximating the Brownian with a discrete probability distribution such as the binomial would be preferable) will occasionally but decisively give us bad pricing information.
Against those who would suggest that the error in the formula will be relatively minor, we advance the anecdotal but highly compelling evidence that the year after Mr. Scholes won his Nobel prize, the hedge fund he was helping to run (Long Term Capital Management) imploded in a multibillion dollar disaster that left financial markets reeling.
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