So, in the last video,

we spoke about challenges with modelling defaults with classical financial models.

If we summarize it,

we can simply say that the GBM model

directly contradicts the data because it does the meet defaults.

You don't even have to compare predictions of the model versus

actual stock volatility patterns as is usually done in empirical tests of the GBM model.

The model misses something very important namely, defaults.

Therefore, it cannot be a complete model as a matter of principle.

True defaults are rare events but they're

critically important for functioning of financial markets.

If you use the GBM model or a similar model for stocks,

we need to use something else such as credit spread models to incorporate credit risk.

Some sort of account for credit risk is important

for trading in at almost every time horizon,

exclusion possibly only a very short time,

whereas you're trading, intra-day trading.

So, as we see,

if we have a mathematical model with a simple formulation,

it doesn't mean yet that it's simple to use and

practice or that it's all conclusions makes sense.

So, not to contribute to essentially a long discussion of whose methods are better,

I cannot resist the temptation to tell a joke I read recently on that topic.

The joke is about a biologist,

a physicist and a mathematician,

who sit at the bar and observe two people entering a house across the street.

After a while, three people emerge from the house and leave.

Then the biologist says,

the population has replicated.

The physicist says, it's an error in the measurement.

The mathematician says, if now another person enters the house,

it will have exactly zero number of people inside.

So, okay.

Coming back to finance,

we spoke in this lesson and the previous ones about challenges of

financial modelling when using the classical concepts of competitive market equilibrium,

and related models such as the Geometric Brownian Motion,

CAPM or the Black-Scholes option pricing model.

Recall that a market in this approach is assumed to be insulated

from the outside world as it doesn't receive any new money or new information from it.

Yet, the GBM model essentially predicts that in this world,

assets will grow indefinitely large.

The very existence of the market is not

explained in the competitive market equilibrium models.

But instead, is just postulated.

I have already mentioned an alternative paradigm

to how markets can function called equilibrium,

disequilibrium, which was a term suggested by Amihood and co-workers in 2005.

In this picture, a market has a continuous access to new capital and new information.

Market-makers provide liquidity in an amount which is optimal for them,

which impacts market prices.

Investors invest or withdraw capital from the market again

in an amount that is optimal for them, and everything moves.

The only thing that is equilibrium about this,

is that it looks more or less the same pretty much all the time.

In physics, this is called a steady non-equilibrium state.

Now, what we will do next is consider

a model that can implement such a state of equilibrium,

disequilibrium in the market.

The model has to do with what we discussed in

the last week of our course on reinforcement learning,

and also to what you did in your course project for this course.

So, let's let Xt be total market capitalization of a firm.

For convenience, we can scale these by

an average market cap over the whole observation period for example.

So, we can think of Xt as

dimensionless variable whose values would be an average of the order of one.