Now, let's talk a bit about other problems with the GBM model that node called auto correlations and volatility clustering. Namely, let's talk about something very important in finance which is corporate defaults and market crashes. I put these two categories of events on the same footing here because they are similar in nature. When we look at the single security, default can be described as a sudden drop of the stock price to zero, and if instead we consider a model for all stock simultaneously, then we can consider events of drops off multiple stocks as markets crash events. Corporate defaults are similar to transitions to a so-called absorbing state. If a system gets into an absorbing state, it cannot escape it and this is why such state is called an absorbing state. Now, as we just said, the 0 price level x equal 0, who serve as natural absorbing default boundary for stock price dynamics. However, the problem is that in a GBM model, the boundary x equal 0 is unattainable. That is it cannot be reached. So, strictly speaking, defaults are incompatible with the GBM model and because they do not exist in the GBM dynamics. Now, the reason for the absence of default is normally not viewed as a drawback of the GBM model is that truly absorbing default that Xs or 0 is usually replaced by some other level X larger than 0 as default boundary. The default itself is described as a level crossing event for diffusion, described by the GBM process but actually in a different variable. So this is indeed the only meaningful way to describe defaults when the drift term in diffusion is linear as in the GBM model. But let's know that such events of level crossing is not exactly the same as a transition to a truly absorbing state where the stock price would stay forever. Instead, when we model defaults as level crossing events as we normally do in financial models for credit risk, we proceed in the somewhat artificial way. We just stop looking at the stock once it crosses some level on its way down. At this moment from the point of view of the model, we just declare the stock to be in default and just stop looking at it from that point on. It does not matter with this purely mathematical approach that immediately after the moment of first crossing, the stock may well come back and continue raising and falling and so on. So even though such view of corporate defaults as level crossing events underlies many models such as the Merton firm value a model which we'll discuss later, this construction is therefore somewhat artificial. It's very different from say the way raging transitions are usually modeled. In this approach, a corporate default is modeled as a truly absorbing state. If a firm goes into the state, it will stay there forever. An alternative to using in-stock prices themselves to describe default is to use absorbed market credit spreads as predictors of defaults. This is good because there are liquid markets for credit default swaps that can be used to find markets implied views of probabilities of default for a number of stocks, a huge number of stocks. So one possible approach would be to introduce additional state variables such as the guest expressed as drivers of defaults and calibrate them to the CDS data. But this approach brings another problem as we now have to ensure consistency between the stock price dynamics and spread dynamics. The same multi-model demonstrates that spreads are not independent of stock prices. Therefore if we reduce credit spreads are separate state variables, we generally end up with quite a complex model of joint dynamics of prices and spreads, where we also have to ensure some sort of consistency between the stock price and spread dynamics. So, the first conclusion even though it's not normally viewed as a deficiency of the GBM model is that the fact that it doesn't contain defaults is certainly a drawback of this model. Now, another problematic point with the GBM model is an unbounded asset growth in this model. Let's take another look at the stock price equation of the GBM model in the form that we put before and this is shown in Equation 5 on this slide. Now, again, the important point here is that the drift in this equation is linear in Xt. If we now take expectations in both sides of Equation 8, we can obtain an equation for the mean value that we call x bar and it's shown in Equation 6 and its solution describes an exponential growth of the mean as a price. So to reiterate, this behavior is obtained as a consequence of linearity of the drift, and equivalently, we can say that it's a consequence of the fact that the GBM model is scale invariant. It does not change if we scale, the asset price Xt to Alpha times Xt, where alpha is a positive parameter. So we can conclude that on average, you should get infinitely rich in the long run if only the GBM model go through. But how realistic is this assumption? In classical financial models based on the competitive market equilibrium concept, a market is assumed to be a closed system without any exchange of capital or information with an outside world. But how can you get infinitely rich in such scenario? We can also ask the same question differently. How can the market grow infinitely large without any income of money or capital? This appears somewhat puzzling implication of conventional financial equilibrium models. But we might conjecture that such behavior is absorbed as simply an artifact of not incorporating some effects of market saturation or alternatively competition of market participants for the same limited value in stock prices that can be exploited for investments. Therefore, we can assume that there are some effects of saturation in the market that may show up when market grows very large and if such effects only show up very late when the market is already very high, they will be almost unnoticeable for levels that are much lower or rather for normal levels prevailing in markets in normal regimes.