Let's start with where we left it in the last week namely

with a simple model that I presented in the last week.

I showed you a model of an open market with an exchange of

money with an outside world and market frictions.

Let me remind you how this model is obtained.

The model is formulated in

a discrete-time setting that is quite similar to the setting of

convex portfolio optimization of both and

co-workers that we discussed in our course of reinforcement learning.

The model can be formulated for either a single stock or a portfolio of stocks,

but here for simplicity,

we will discuss a single stock version.

The model is defined by three simple equations shown in this slide.

The first equation is the growth equation.

It says that at each timestep,

a firm with a total market cap X_t,

the price evolution is made in two steps.

First there is an ejection or withdrawal of money by investors

in the market in the amount u_t in the beginning of the period.

In addition, the investors are paid the dividends on

the stock and the amount c times X_t times delta t,

where c is the dividend rate.

So, the net flow of cash from the market to investors

would be the difference phi minus c. After that,

the new value X_t plus u_t minus c times

X_t delta t grows at rate r_t that is shown here in the second equation.

The third term here is the market impact term.

The reason we have the same quantity u_t in

the first and second equations is quite simple.

In the first equation,

u_t enter says money injection at the beginning of the interval.

But injecting new money in the market in

the amount u_t means trade in exactly that amount of the stock.

Due to market frictions,

this produces the third term in the second equation,

which is proportional to u_t with a market friction coefficient mu.

Lastly, the third equation here specifies the form of the new money supply function.

We use the simple quadratic polynomial for u_t

as a function of X_t as displayed in this equation.

Please note that this polynomial has

a linear and quadratic terms but not a free acts independent term.

We set such term to zero to make sure that

outside investors do not invest in a stock with zero value,

because for a publicly-traded stock,

zero price means default.

Now, as we already did in the last week,

we can plug this expression for u_t and substitute it into the first equation,

where we also use the second equation.

If we neglect the quadratic term,

Johnson Delta t in the resulting expression and take the continuous-time limit there,

we obtain the model given in equation two with parameters defined in equation three.

We have three parameters theta,

kappa and g here that are determined in terms of original parameters phi,

lambda and mu of the original system of three equations.

I called this simple model the Quantum

Equilibrium-Disequilibrium model or QED for short to emphasize two things.

First, this model suggests a simple implementation of

an equilibrium-disequilibrium paradigm for

markets that were suggested by Amihud and co-authors.

Second, effects of noise are very important for the dynamics of this model

as a classical stochastic system is equivalent to a quantum system,

and moreover, some stochastic phenomenon such as decay,

the case of metastable states can be

conveniently described using methods of Quantum Physics,

hedging the world Quantum here might be justified as we will see later.

Now, equation two describes the model with

the non-linear drift given by a cubic polynomial.

So, in this model,

we have non-linear drift and three extra parameters relative to the GBM model.

If you now take the limit when there is no money exchange with the outside world,

that is phi and lambda are both zero,

then also kappa and g will be zero, and in this case,

we will formally recover the GBM model from the QED model.

We can also get the same result if we keep

our phi non-zero but instead take mu and lambda to zero.

We can also consider another limit when we keep kappa non-zero but take lambda to zero.

In this case, we obtain equation five from this slide.

This model is called the Geometric Mean Reversion model or GMR,

and it was studied by Dixit and Pindyck who

applied it to real options and commodity assumptions.

It was also studied by Ewald and Yang who found

conditions for the existence of a steady state in this model.

If we in addition take the limit of zero volatility in the resulting model,

we will obtain the Verhulst model that I presented in the last week.

So, we can call this limit the Verhulst limit.

We will continue with the analysis of this model in our next video.