# 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.

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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.
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I would that the geralized brownian process assumes independent increments while imperically there is statistically significative autocorrelation in assets returns. – Drmanifold Feb 3 '14 at 17:11
Well somewhat. While it is true that clustering involves autocorrelation, the latter does not necessarly implies the former. – Drmanifold Feb 3 '14 at 17:16
Good point. You could add this point as an answer to the question... – user1157 Feb 3 '14 at 17:24

I dont know what you want to hear, but i have several points for you:

• The main driver of uncertainty is a Wiener process, which goes back to the discrete binomial model for stock prices. In reality the main stochastic source could be something completly different.
• $\alpha$ and Vola $\sigma$ are depending directly on your stockprice. Why should they? the could easily depend on your Wiener-Process like in the CIR or Vasicec model.
• The Drift $\alpha$ and Vola $\sigma$ a depending both of $S_t$ and therefore no exogenous impacts can be modeled. Consider: $\alpha$ and $\sigma$ are both adapted on your Wiener-Filtration coming form $W$. This filtration at time $t$ can be interpreted as all the information you saw in the stockprices up to that point. In reality the Vola or Drift may change due to things that happen not depending on the stock market. Think of small random disturbends genereted by news in politics or catastrophes. For this are Stochstic Volatilty models needed.
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I don't understand your second point - do you mean that they should be stochastic (like stochastic vol)? As Anna points out below the fact that in an SDE no jumps are modelled is one of the most important drawbacks in my mind. – Richard Jan 30 '14 at 9:47