# How is fundamental data taken into account when modelling stock prices with a Geometric Brownian Motion?

I have a basic understanding of the principles behind Geometric Brownian Motion and how it can be used to model stock prices, however I am confused as to how it is used in practice. In particular, how can the fundamental analysis of a company be incorporated into the GBM model?

Is it the case that information such as financial reports, social media, company news etc. is analysed to generate values for the volatility and/or drift to be then plugged into the GBM equation? Or is it the case that Geometric Brownian Motion is purely based on statistical data obtained from historical information?

It turns out that GBM with constant drift and constant volatility is not really used in real life. It is well known that volatility as well as drift may vary over time. Hence, if you want to use a model with time-varying parameters, you need to come up with a model to define $\mu_t$ and $\sigma_t$. There are classic models that use some mean-reverting process for $\sigma_t$, but then the estimation and calibration of parameters becomes really challenging and is often more related to some sort of art rather than economics. Finding a process for conditional drift is even more challenging.

The way, and whether, you use GBM differs depending on what you want to do. Indeed, vanilla option pricing for instance relies on GBM but you will actually reverse the pricing formula and back out the volatility (which is then called implied volatility) because you observe option prices. The drift used depends on how do you price the option. For instance, in the Black-Sholes framework, that is using risk-neutral probability, the drift is going to be the risk-free rate.

For investment decision purpose, like asset management, it is another story. If you are interested in pricing future stock returns (not a single stock but rather the S&P500 or any large index), there are other methods much more reliable than pure normal shocks to price. These models often relies on macroeconomic variables, and in particular consumption, that helps gives a better view of stock prices (although there are also some problems with these methods).

If you want to price a single stock, you won't use GBM neither. Indeed, as you suggested the fundamentals will be too important in that case to simply use basic GBM. Again here, you will try to use alternative methods that relies also on macro data, like some sort of variation of CAPM model.

Regarding your initial question, yes, in theory you could be able to infer some GBM calibrations with company fundamentals but I guess this is rarely used in that way.

Overall, I would say that GBM is not that used by anybody because it is the most basic model you can come up with, and have several large limitations in its "usual" form : constant mean and variance. There are strong evidence that the mean and the variance are not constant over time.

It does not necessarily have to be using historical data (you could use implied volatilities for example), but indeed fundamental analysis is not taken into account in geometric Brownian motions: you just assume returns are normally distributed with some mean and volatility and it does not change in time.

So if you want to "incorporate" fundamental analysis in a GBM, you'd need to come up with a way of translating your "findings" in terms of $\mu$ and $\sigma$.

• I think the poster is more concern on how GBM is actually being used. For example, do you plug market data into it or u come up with some parameters to check arbitrage in the market data. – SmallChess Nov 4 '15 at 3:24

Fundamentals usually do not enter the parameters of GBM. But it depends on the purpose:

• if you want to price options, then the drift is the risk-less rate and volatility is implied from other traded derivatives.
• if you want to use GBM for risk management then you usually apply statistical methods for $\sigma^2$.