# 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?

-

## migrated from math.stackexchange.comApr 4 '13 at 9:20

This question came from our site for people studying math at any level and professionals in related fields.

At the moment, $t \mapsto S_0 e^{-100t}$ is a good model :-(. – copper.hat Apr 3 '13 at 17:43
I don't know too much, but Lévy processes are used for this. – Bunder Apr 3 '13 at 18:46
Well, GBM is not really widely accepted anymore for most of sophisticated investors. Could you explain why you would need that for? – SRKX Apr 4 '13 at 12:15
@SRKX Well, GBM is not really widely accepted anymore for most of sophisticated investors". Can you shed some light on what is and how are sophisticated investors classified in your definition – ash Apr 8 '13 at 15:52
I would say that most firm who indeed use models to try to make profit out of derivatives mispricing are trying to do use by assuming that the GBM model is inconsistent. They are betting on the fact that they have a better model. – SRKX Jun 6 '13 at 14:37

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.

-

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.

-

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.

-