Stock Price Models Beyond Geometric Brownian Motion
Summary
The document surveys alternatives to geometric Brownian motion for describing stock prices, especially when the goal is pricing derivatives. It identifies local volatility as a deterministic extension of the Black–Scholes framework and names Heston and SABR as stochastic volatility models. It also notes that combining local and stochastic volatility is possible, though described as harder to implement and less commonly used in practice. Lévy processes are mentioned as another model family.
A separate response emphasizes jumps: discontinuous price moves are features that diffusion-only models may fail to capture, and they can matter when valuing options and other contingent claims. The discussion is an overview rather than a comparison backed by data, calibration results, or implementation details. One further answer proposes a non-drift model based on buy and sell orders and chart displacement, but the document does not provide evidence establishing its acceptance or reliability. Model choice therefore depends on the intended application, such as representing volatility patterns or pricing derivatives, and the material does not prescribe a single best model.
Key ideas
- Local volatility extends the Black–Scholes framework with a deterministic volatility structure.
- Heston and SABR are cited as stochastic volatility models used in derivatives contexts.
- Lévy processes and models with jumps offer ways to represent discontinuous price changes.
- Jumps can matter when valuing options and other contingent claims.
- The document offers a model overview without comparative evidence or a recommended universal choice.
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Full text
# Stochastic modeling of stock price process
# Stochastic modeling of stock price process
Apart from the model of Geometric Brownian motion is there any other "widely accepted" stochastic model to characterize the dynamics of a stock price process?
## Answer by FKaria (score 2)
https://quant.stackexchange.com/a/7689
There are many, which are mostly generalizations of the Black-Scholes model (Geometric Brownian Motion).
For Equity stocks, the most widely used (IMHO) is the deterministic generalization of Black-Scholes model, the Local Volatility model. Followed by stochastic volatility models such as Heston or SABR, also there is a generalization of the Local volatility model with stochastic volatility but not used as much because is harder to implement.
Of course, there are a lot of different models such as Levy processes, but the ones I indicated here are the ones that you would see implemented in practice for derivatives pricing.
## Answer by sets (score 2)
https://quant.stackexchange.com/a/8174
In addition to local volatility and stochastic volatility models, discontinuous jumps are also an important component of stock price movements that are cannot be properly explained by diffusion models.
This article: "Which model for equity derivatives?", gives an overview of the rationale behind discontinuous jumps and their importance for modeling the evolution of equity prices, in particular when assessing the fair value of options and contingent claims.
## Answer by myoptionexpr (score 1)
https://quant.stackexchange.com/a/8186
Though not widely accepted, an alternative, non-drift option pricing model is detailed here.
Essentially, we derive a relation between buy and sell orders and visual displacement on a chart.
$$\left (\frac{2b_o-t_o}{t_o-b_o}\right )=\frac{\Delta_p}{\Delta_b}$$
The buy and sell orders themselves can be used in a normal distribution to find the probability of a stock rising above a strike price.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.