What are the advantages and limitations of predicting future stock prices using stochastic differential equations?

Question

Recently I came across the following stochastic differential equation that "predicts" the value of a given stock: \begin{equation} dS_t = \mu S_t dt + \sigma S_tdW_t \\ S_t(0) =S_0 \end{equation} where $S_t$ is the value of the stock, $\sigma$ is the volatility of the stock, $\mu$ is the drift coefficient, and $W_t$ is the Wiener process.

I do not have a finance background, but rather more computational mathematics/engineering, hence why I am asking this question. I've solved this equation numerically, setting $S_0$ to be the stock price of Google on December 28th, 2018. I choose to approximate the future stock price of Google one year from December 28th,2018, and found what I perceived to be "accurate" results. My solution was about $30 less than the actual stock price on December 28th, 2019. I was wondering what is one way to enhance the "accuracy" of this model? Moreover, what are the limitations of this SDE? Do quants use this equation or modifications of this equation, if so may someone please provide any references?

Answer 1

The SDE you are describing is called the Geometric Brownian Motion. In the end its just a model, which underlies certain assumptions, which are usually not met in the real world scenarios. There are many further extensions and variation of SDEs for modelling prices f.e. including a jump component (jump diffusion models), mean reversion (f.e. Ornstein-Uhlenbeck) etc. In the end its a matter of choice, where you should think ahead, which model might describe a certain market more accurately.

Answer 2

Take the analogy of equations modelling something in physics. Just because you write down an equation, it does not mean it has to be connected to anything in reality. It only do so to the extent you have adapted the equation and it's parameters to fit reality.

In finance things are a bit more complicated when it comes to the predicting power though. Typically one try to adapt the parameters of the processes to be consistent with data you currently can observe on the financial market (like interest rates, stock prices, exchange rates and so on). The market data of today can be seen as the analogue of the physical reality in physics.

But even if you have a very good fit to the market data of today, it does not mean you can do good future predictions with your model. It just means it is consistent with the observed data. When you simulate your process forward in time you will get different results for every try since the process is random. What you can hope for is that the probability distribution of outcomes has the right properties. In some sense it is a bit similar to predicting particle positions in Quantum mechanics. You can model the probability wave function and how it evolves in time, but you can never say exactly where the particle is located.

As other people here have pointed out, there are people trying to calculate actual predictions too, but that is not the main goal of for example the GBM model you mention.

A more typical end goal of using such and more advanced process is to be able to price financial products on the market in a way that is consistent with the observed market data. If any product available on the market is wrongly priced compared to the available market data, people will sometimes be able to use that error to get free money. This is called arbitrage.

Answer 3

The GBM model is liked by practitioners for the modelling of stock prices for the following reasons:

(i) The solution is log-normal, so the stock price distribution varies between zero and infinity: which is what we would expect from a real-world stock price.

(ii) The model has independent increments, which means the future distribution of the stock only depends on its current price (i.e. $S_0$). Future distribution does not depend on previous returns or the path taken to reach the current stock price $S_0$. This is considered a good feature by those who believe in the efficient market hypothesis.

The shortcomings of the model are for example continuity of stock prices (stocks move in small jumps (ticks) and sometimes large jumps (opening gaps, auctions)).

The most important thing to note about the model though is that it is heavily dependent on the parameters (which is somewhat obvious): the volatility and the drift. The future distribution of your stock price will depend on which $\mu$ and $\sigma$ you choose. Specifically, the $\mu$ will determine the center of your future stock price distribution and $\sigma$ will drive the width of the future stock price distribution.

The GBM model is usually only used for pricing of stock forward or options, where you only care about the expectation of your future stock price distribution. In this case, you calibrate your model to liquid forward prices so that the drift term will make the expectation of the future distribution equal to the forward prices you observe in the market (and you can already guess from this that your drift most likely won't be constant, but will have to be time dependent, to match the various forwards of different maturities). You then use this calibrated model to price forwards and options that are not traded in the market.

I have never seen the GBM model used for predicting the future stock price of a particular stock. I have seen auto-recursive (AR) time series models trying to attempt that. But I would argue that the GBM model is only good for pricing of (simple) stock derivatives, and is not a good candidate for predicting future stock prices.

If you are interested in predicting future stock prices based on past observed prices, I would try looking into AR models: there are many variations and some are better than others, but I'd argue you'll be more likely to get more accurate results with AR rather than GBM.