Solution to SDE being Evolution of Price Process

Question

I am trying to explain the concept of a solution to SDE being the model for the evolution of a price process. How would you do this to someone who doesn't have a financial engineering background?

Consider the following SDE: with the condition $X_0=1$,

$$dX_t=\sigma(X_t)dW_t+b(X_t)dt$$

My question is about the layperson intuition behind the usual language:

The dynamic evolution of a price process is modeled as a solution of a stochastic differential equation of the above form.

Some of the questions I have to answer in a very easy language is:

1. What is SDE?

2. What is a solution to SDE?

3. What is the evolution of a price process?

4. Why are we modeling the evolution as a solution to SDE?

5. What makes such modeling a legitimate one compared to a chimpanzee throwing a dart to predict the evolutionary path of a stock?

p.s. The explanation is not to involve BM, Girsanov type tranformation, equivalent martingale measure, etc FYI. Just for a layperson.

Answer 1

To make it easier, assume that the process $X_t$ corresponds to the price of some asset.

1. The SDE you have given is a probabilistic model for price changes (i.e. $\mathrm{d}X_t$). The model postulates there is some deterministic (non-random) part $b(X_t)\mathrm{d}t$ which simply corresponds to known changes over time (perhaps a trend) and there is a stochastic (random) part $\sigma(X_t)\mathrm{d}W_t$ which allows for some unpredictable, rough changes creating the daily noise of prices jaggedly going quickly up and down.

2. The solution to an SDE is a stochastic process which behaves precisely as the model postulates it, i.e. the changes of this stochastic process can be decomposed into a deterministic and stochastic component. It satisfies the properties which the model imposes.

3. The evolution of a price process corresponds to a time series or realisation (path) of the price process. It merely shows how the prices have played out over time.

4. We could simply say "because it works fine". The theory of stochastic processes naturally fits the context of financial markets. It allows us to build ''fancy'' models (e.g. via SDEs) and gives us the tools to find solutions to these models and do further analysis with the model (e.g. compute the probability of a future price $X_T$ being greater than a threshold $K$). This, in turn, helps us to understand (i.e. price and hedge) financial products (e.g. derivatives ... in which billions of dollars are traded). In particular, we can build SDEs to reproduce the behaviour of assets which we observe in the market (e.g. Markov property, mean reversion, fat tails, asymmetry, etc.) This is why stochastic calculus is so popular, it just works fine for the applications we have in mind.

5. I suppose it's not hard to argue that modern models for stock prices, interest rates etc. are slightly more sophisticated than a simple random walk which may emerge from a chimpanzee throwing a dart. Incorporating stochastic volatility, jumps, etc., we can match many stylised facts and build (hopefully) reasonable models to mimic real-world dynamics of financial markets. The models are (by definition) never perfect but are certainly better than a chimpanzee. The enourmous success of mathematical finance over the last 50 years should prove that quite a few people believe it is rather worthwhile to employ such models and methods.

Answer 2

You might describe the SDE as a "random process simulator". Given the SDE and some simple computer code, you can produce hundreds of simulated trajectories for the future price. The solution of the SDE is the set (or "ensemble" to use the technical term) of all these possible trajectories. All these outcomes are equally likely, and we don't know which one in specific will occur.

Thus the SDE does not "predict what will happen" like the monkey does. It only provides statistical information about the trajectories. For example we can say that on average X% of the trajectories will pass above (or below) a specific level N months from now. This type of information is very useful if you are going to take "bets" on future outcomes, it allows you to set the odds for the bet.

This type of modeling is legitimate because we can compare the frequency of observed events to those predicted by the model and thus, in principle, "falsify" (i.e. reject, in the sense of Karl Popper) the model if the results are not in agreement with reality. Furthermore because the profit and loss of the "betting business" relies on these models we have an economic incentive to do so.