The Derivatives-on-Line Pages

These pages are being moved! The newest version can be found here. Please update your bookmarks. If you are visiting the pages at www.qmul.ac.uk or www.qmw.ac.uk youmight be looking at an older version.

Maintained by Kyriakos Chourdakis, QM University of London. Comments, etc. are very welcome. You can reach me here.

Vist my website at thePonyTail.net! There you can find these pages in PDF, PS and other formats, relevant software, working papers, data, etc.

Site Meter next up previous contents
Next: The Black-Scholes formula I Up: Modelling: Continuous state space Previous: Itô's formula   Contents

The geometric Brownian motion

One very important special case comes when

$\displaystyle Y\left( t\right)$ $\displaystyle =$ $\displaystyle f\left[ X\left( t\right) \right]$   , where  
$\displaystyle f\left( x\right)$ $\displaystyle =$ $\displaystyle \exp \left\{ x\right\}$   ,  
$\displaystyle X\left( t\right)$ $\displaystyle =$ $\displaystyle \left( \mu -\frac{1}{2}\sigma ^{2}\right) dt+\sigma dB\left( t\right)$   .  

The diffusion $ Y\left( t\right) $ is a geometric Brownian motion, and its importance lies mainly in the fact that it is the simplest form of a diffusion that remains almost surely positive. Itô's formula would give in this case that
$\displaystyle dY\left( t\right)$ $\displaystyle =$ $\displaystyle \mu Y\left( t\right) dt+\sigma Y\left( t\right) dB\left( t\right)$   , or  
$\displaystyle \frac{dY\left( t\right) }{Y\left( t\right) }$ $\displaystyle =$ $\displaystyle \mu dt+\sigma dB\left( t\right)$   .  

In figure 7.1 one can find a sample path for a geometric Brownian motion, with $ \mu =1$ and $ \sigma =2$. One can observe that such a pattern could well match the pattern of a financial asset.

next up previous contents
Next: The Black-Scholes formula I Up: Modelling: Continuous state space Previous: Itô's formula   Contents
Kyriakos 2003-03-17