modeling - Shortcomings of generalized Brownian motion for asset price modelling

I'm simply interested on hearing some views on which shortcomings arise by using the (multidimensional) SDE $$dS(t)=S(t)\alpha(t,S(t))dt+S(t)\sigma(t,S(t))dW(t)$$

as a model for asset prices.

I know this is indeed quite general question, but I've often encountered this in my studies and most likely you guys have a lot more insight into this than I can figure out myself.

Answer

I would like to add a few more points to @Phun's already very good answer:

The question is interesting because generalized Brownian motion already covers a lot of cases:

This example includes all possible models of an asset price process that is always positive, has no jumps, and is driven by a single Brownian motion for each asset.

(Shreve, Stochastic Calculus for Finance II, p. 148)

Shortcomings:

• Brownian Motion is continuous, i.e. no jumps in the stock price paths.


• It cannot become zero, whereas companies can default.


• The likelihood of large price movements is smaller than observed in real markets. See for example Mandelbrot's criticism.


• The distribution of relative movements following the normal distribution is symmetric, in practice a common pattern is: many small movements up, and fewer but larger movements down.


• In practice large movements tend to be clustered together, followed by long periods of little movements, i.e. no regimes. See for example "The clustering of stock price movements", by Malkiel et al., 2009.