So, before we go back to the topics of modeling market dynamics, using our RL inspired framework, I would like to go back as I said almost to square one in finance and talk about one of the most famous financial models, namely the Geometric Brownian Motion model or GBM model. This model was proposed by Paul Samuelson in 1965 as a way to improve on a specific deficiency of the previous model of Louis Bachelier from 1900. Bachelier proposed a Brownian motion with a drift as a model for stock prices. But prices in such model can become negative which is a problem with the model. So Samuelson suggested an improvement in form of the GBM model that is shown in Equation 1 here. We can view it as a model for the stock value Xt with a linear drift function f of x given by Xt multiplied by a sum of a square root rf, Alpha signals zt or with weights w and a noise term with intensity Sigma times Xt. Now, the difference with the Arithmetic Brownian Motion or ABM model of Bachelier, is that in ABM model, both the drift and the volatility terms are constant. Now, unlike the ABM model of Bachelier, the GBM model only admits positive values for the stock price, Xt. A simple way to see it is to know that the GBM model is equivalent to the ABM model for the log price instead of the price. As the algorithmic function is only defined for positive values of this arguments, this implies that the price in GBM model can only be positive. So the price in the GBM model cannot be negative which is good news. Now, I would like to show a discrete time version of the GBM model that we will need going forward. This is really simple. We just a place infinite decimal increments by finite increments and regroup terms. These gives us Equation 3 on this slide. So it simply says that the growth in a time step Delta t is a linear with rate are equal to deterministic part, rf, plus wz plus the noise term that is proportional to Gaussian white noise side. We can also equivalently express this equation as an equation for a new value X of t plus delta t as shown in Equation 4 here. Now, either in a continues, taking either in the continuous-time formulation or the discrete-time version, the GBM model is one of the most popular models in finance. It's used in many financial models/ most famous ones include the Capital Asset Pricing Model or CAPM of William Sharpe and the Black-Scholes model. The GBM model is simple and highly tractable, which are probably the main reasons it still remains the main power horse of modern finance to these days after it was proposed in 1965 by Samuelson. 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.