Monday, August 9, 2010

Black-Scholes and Monte Carlo

Aside from the lattice method discussed in yesterday's entry, the two most important methods of valuing the expense of stock options as compensation are: the Black-Scholes-Merton (BSM) formula, and a Monte Carlo simulation.

BSM is here.

This amounts to valuing an option to buy a share of stock by making certain simplifying assumptions. The devisers of the formula were explicit about these assumptions. One of the more important of them is that extreme price changes are very rare, because the movements in the value of the underlying stock follow lognormal distribution. Many scholars have quarrelled with this, but BSM does still offer a useful approximation of the value of such options.

Still: the "Monte Carlo" method has by far the cooler name, paying homage as it does to the city whose image I've uploaded with this entry.

Through the miracle known as a digital computer, modelers can run thousands of simulation paths covering possible price movements in the underlying asset -- the stock - and can assign a value to the option for each of these paths. The present value of the average of all those values is the Monte Carlo value of the option.

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