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