Limits of Generalized Brownian Diffusion Models for Asset Prices
Summary
The document reviews limitations of modeling asset prices with a positive, continuous diffusion whose drift and volatility depend on the current price. It notes that this broad model family can represent many no-jump processes driven by Brownian motion, but may omit features observed in markets.
The listed concerns include continuous paths that cannot capture jumps, strict positivity that excludes default, thinner tails than observed large moves, symmetric relative-return distributions despite downside asymmetry, and the absence of volatility regimes or clustering. A further response questions whether drift and volatility should depend only on price and Brownian-history information, since external events may affect them; it points to stochastic-volatility models as one way to address changing volatility. These are qualitative critiques, not empirical tests or a comparison of specific model fits, and the stated limitations do not imply that diffusion models are useless for every application.
Key ideas
- Continuous Brownian diffusion paths cannot represent jumps in asset prices.
- A strictly positive price process cannot reach zero to model default.
- Normal relative returns may understate extreme moves and fail to capture downside asymmetry.
- The basic setup does not represent clustered volatility or distinct market regimes.
- Drift and volatility tied only to price history may omit effects from external events.
Tags
Full text
# Shortcomings of generalized Brownian motion for asset price modelling # 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 by user1157 (score 5, accepted) https://quant.stackexchange.com/a/10113 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. ## Answer by Phun (score 1) https://quant.stackexchange.com/a/9599 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.
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.