Every once in a while the Geometric Brownian motion (GBM) breaks down due to the problem with Euler scheme in implementing a stochastic differential equation. Spurious paths are those Monte Carlo paths which cross zero!

We have talked about this problem in a separate article in some detail and discussed why a GBM breaks down when the stochastic differential equation of a GBM is implemented in a finite difference scheme. For certain values of absurdly high volatilities the GBM gives negative values of the asset prices.

A GBM should never cross zero and in fact should not even come close to zero. But for some values of random numbers it will. Why?

Take the stochastic differential equation (SDE) of a GBM:

In the above, is the Weiner Process and is the stochastic asset price. The Euler scheme for the above SDE that transforms it into a difference equation is:

Here, is a random normal number, i.e. . This scheme will converge to the mathematically correct description of GBM only in the limiting case of becoming infinitesimally small, i.e. . But if the time slices are finite, i.e. if are small, but finite such as one month, one week, one day, one hour, etc. then depending on the parameters one can always draw a random normal number, , such that

There is a small but a finite probability of the above happening. In fact, if the time slice is finite, then it is only a matter of time when the above will happen, i.e. the random normal number, , drawn will be less then the expression on the right hand side. And if that happens then the value of will become negative!

Such a path where the asset price becomes negative is called a spurious path. One can easily test the above using an Excel spreadsheet and running a very large number of iterations with sufficiently big time slice, i.e. and reasonably large values of drift and volatility.

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