# Modeling the Stock Market [closed]

Hi I was wondering what is the model that best describes the price movement of the stock market?

1. A Brownian motion Process with drift?
2. An Ornstein Uhlenbeck_process? (where the long term mean is described by an ax+b function for the drift component)
3. A model that uses stochastic volatility?
4. Jump Diffusion Models?
5. Or something else?

(I placed these 4 points to act as multiple choice bullets. I don't know how to make this question more specific, that is what I want to know, which model best fits the stock market.

Which model more realistically fits the stock market, which one will show volatility clustering, price appreciation, market shocks(jump diffusion), show leptokurtic distributions, mean reversion etc.

I see many models used but I haven't yet seen which one is standard for realistic simulations. I placed the 4 points up there as a guide to which class of mathematical models I'm referring to.

I realize that there can be many good ways to answer this question. I believe though that only a few models will approximate the way the stock market moves and behaves. So IDK.)

Thanks

## closed as too broad by lehalle, Bob Jansen♦May 30 '16 at 13:58

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Modeling the stock market ... for the purpose of option valuation ? (or for something else?). – noob2 May 26 '16 at 19:42
• The first person that I worked with in finance modeled the stock market this way: "The stock market is a process designed to take the maximum amount of money from the maximum amount of people in the minimum amount of time". – JoshK May 27 '16 at 13:14
• Agreed with @noob2, you really need to define some context to make this answerable. – Bob Jansen May 30 '16 at 13:58

## 1 Answer

Given the well-known stylised facts of equity markets, I would go for a generic stochastic volatility model where

• log-asset prices, hence geometric returns, are driven by a standard Brownian motion (although this would explain the lack of returns' auto-correlation, it would also boil down to assuming their independence, which is a stronger assumption).
• volatility of returns is stochastic (would explain heavy tails), mean-reverting (would explain volatility clustering), negatively correlated to the spot evolution (would explain skew). Some may even consider modeling (log-)volatility using fractional Brownian motions instead of standard Brownian motions (better emprical behaviour).
• volatility and price should be subject to random jumps both in timing and in size (blame it on Murphy's law). Common practice is to use compound Poisson processes to model that feature.

Now, if you are working under the real-world measure and looking at using such a model for forecasting purpose, the complex parts is how exactly would you calibrate such a model to historical data?

If you are on the other hand working under the risk-neutral measure and that you (manage to decently) calibrate your model to observed vanilla option prices, the question is then, how to can you guarantee that your model will also embed a plausible forward smile dynamics so that it produces decent prices for more exotic structures?

• I'm assuming some kind of mean reversion to a general mean(that's that is slowly going up), and whenever something happens in the news it gets shocked either way and then reverts to mean. Causing cluster volatility. The problem with straight stochastic volatility is that in my opinion it doesn't model volatility clusters well, where as jump diffusion with mean reversion does. idk though – hammond May 26 '16 at 21:23
• You should indeed always develop/use a model based on your needs. The field of application is so large that it's hard to give a generic answer to your question. What is it exactly that you are trying to achieve with your model? Also nothing forbids stochastic volatility + jumps. – Quantuple May 26 '16 at 21:56