Yet, its simplicity not withstanding,

the GBM model has a large number of

well-documented deficiencies that go beyond

a mere fact that it does not hit the market quite well.

So, let's go over these problems with the GBM model one by one.

First, the GBM model does not incorporate defaults or market crashes.

A bit more generally,

it produces a very small probabilities of large market moves.

When portfolio managers talk about month long periods of 20 Sigma events,

computed with the GBM model,

it appears quite clear that something is wrong with the model itself.

Third, the GBM model does not include the effects of market friction,

such as transaction costs or feedback effects from trading.

Fourth, equilibrium market models such as CAPM or the Black-Scholes model assume that

the market is in isolated system

without an exchange of capital or information with an outside world,

and this is hardly a realistic assumption of course.

Finally, for completeness, I would like to mention inconsistency of

the GBM model with various price patterns in real financial markets.

Such as short-term autocorrelations of price changes,

volatility clustering and stochastic volatility and so on.

I say here for completeness because even usually the last class of problems with

the GBM model is considered as

the main problem or main class of problems with this model,

our focus here will be in fact on

the other four deficiencies of the GBM model that I mentioned before.

Now, of course, all these deficiencies of the GBM model

are well-known and researchers proposed

many possible extensions or generalizations of this bonafide model.

If we generalize a bit,

we can classify all these models into three major classes.

The first class of models focuses on the prediction side,

and tries to improve on the set of predictors zt,

and this would be a core objective for researchers

that are hunt for Alpha signals for quantitative training.

Another and related approach would be to consider

non-linear dependencies on predictors zt.

Such approaches are often pursued with machine learning models such as for

example support vector machines or

neural networks that we talked a lot about in this course.

The objective there would be the same as in the first class of

approaches namely to improve the predictive power for better training.

The third class of extensions of the GBM model

focuses instead on modifications of the noise term.

This approach is pursued in methods based on

risk neutral valuation and used in particular in this pricing.

With methods in this third class and the volatility or noise term can

become a different function of

the state Xt and might in addition become stochastic itself.

In this later case we obtain a stochastic volatility model.

Now the common theme between all these approaches is

that they all preserve linearity of dynamics in the state variable Xt.

But in fact as we will see in the next lecture,

including non-linear effects in Xt may perhaps be

more important than such linear modifications of the GBM model.

As we will see later,

this indicates that in the sense we also need to go

beyond purely data driven machine learning and

reinforcement learning approaches or traditional financial factor

model approaches when constructing better dynamics for stock prices.

To this end we need to rely on other models and theories,

and it may sound a bit paradoxical after we spoke so much in this lectures

about data-driven and model independent approaches to talk again about models.

But a little guarded or

maybe not-so-little guarded truth of machine learning and data science is

that you cannot proceed in

a completely modern dependent way if you want to build something practical useful.

In the next video,

we will see in what sense we can do that.