So in the last video, we discussed the problem of unbound growth in a GBA model and outlined how it can be solved using the model, even though the latter model creates its own problems that we are yet to resolve. And now, I would like to talk about another fundamental problem with the GBM model, namely, the problem will and solve defaults. Let's discuss first what happens if a company defaults. Wwhen people talk about defaults in finance, default can be of different sort. In a credit world, default is technically considered as non-payment under an obligation to pay. For example, a missed coupon payment by a bond issuer is considered default. But if such non-payment happened in fact because of some technical glitches and was paid a few days after the coupon date, then this is called the technical default, and it does not lead to the company that issued the bond being closed. But in other cases, consequences maybe more drastic. For example, if a company goes bankrupt, this is also called default in the industry jargon. And this can be mathematically described as a drop of the stock price of that company to zero, and staying there forever. And another class of default called restructuring can also be thought of as resetting a firm to zero so that the old stock drops to zero. The old firm dies, and the new firm emerges instead. From now, let's just agree to use the word default for events when the stock price drops to zero, and stays there forever, and leave aside more technical terms such as technical default that I mentioned. So if the actual process of default can be expressed mathematically as simple as an event of a stock price dropping to zero, things should be simple there, right? We actually have a model for the stock price, and it's called the geometric Brownian motion model that we discussed earlier. So let's just watch for events when the stock price touches zero in the GBM model. This will be our defaults in this model. And because the model is so simple, transition probabilities in this model are simple to compute, so the whole default thing should be easy, right? And actually, it's not right at all. Such events simply cannot happen in the GBM model, because in this model, the zero level is inaccessible. In other words, the GBM model is incompatible with default, they cannot occur in this model. And this practice, of course, known to everyone who studied financial engineering, but implications of this factor are rarely discussed, so let's talk a bit about it. If defaults cannot occur with the GBM model, we should say that the model is wrong in a worst case, or incomplete in a best case, which would essentially be more or less the same but just differently phrased. Corporate defaults are rare events with probabilities of the order of 1% or so, but awareness of these events is very essential to the markets. If these events are ignored by the markets, then credit spreads will be equal zero, which is solely not the case. So credit markets and their observable, such as press or credit default swaps, or CDS, which we discussed earlier in this facilitation, indicate that these events are taken into account by the market, but now by the GBM model. And then the next thing we could try with GBM model is to draw a default boundary for the stock at some non-zero level and say that the default probability would be the probability to cross this boundary. And this sounds almost the same as the Merton model that we discussed in the previous lectures. But not exactly the same, because here, we talk about a level crossing event for the stock itself. So if you try this, you would quickly find that such model would be disastrous in the sense that you would not be able to calibrated it to both stock prices and credit spreads. The stock, in this model, would either too quickly run up away from the default or too quickly cross the default boundary. And this is of course the reason you've never heard about such a model of default. What is used instead, there's a smart way to describe defaults while still used in the GBM model. And this is done in the Merton model, which we discussed when we talked about bank failures. To remind you of what the Merton model does, instead of relying on the observable process of a stock price, it uses the GBM model to describe a non-observable process for something called firm value. When the GBM process for the firm value crosses the default boundary at debt level of the firm, the firm defaults. And in this case, equity value drops to zero simply by the way of enforcing a specific bonded condition. And know the fact that it's done in this way, in a Merton model. In other words, the stock price goes to zero in this model by decree and not as a result of some internal dynamics, there are no other internal dynamics in the Merton Model other than the event itself. So the Merton model appears to a capital structure of a firm, it uses a capital structure of a firm to build a model of corporate default and describes it in terms of a process for the film value. And this leads to a very mathematically attractable model, but at the same time, it leads to a lot of questions when you try to use this model in practice. First, the firm value process is unobservable, so you cannot directly measure its of volatility. Second, the actual default boundary is hard to know exactly, and in the Merton model, it's given by of a single bond issued by the firm. But in reality, that has a more complex structure, so you will never know exactly what number to use in numerical formula. But even if you know what number to use and the summation of this number from data will invariably bring some noise again. So the conclusion is that in the Merton model, we actually never know the default threshold with certainty. There are also some extensions of the Merton model that explicitly acknowledge that and construct models that fluctuates in default threshold. Now, if the threshold position is actually uncertain, then the default event itself should be uncertain, too. And this means that, strictly speaking, the Merton model and similar models that when treated properly are not able, these models are not able to predict with certainty if the fault occurred or not. Now, we can also work to this very important observation that in the GBA model, the dynamics is completely reversible. If you'll wait only a little bit after their firm value just crosses the boundary, and remember, the boundary itself is uncertain, as we just said, then the firm can actually bounce back to a shortly after crossing the threshold. Now, imagine that you look at the firm only at discrete times as is usually happens in practice. In this case, when you know that the firm is closed to the boundary, you will not be sure if it actually already crossed the boundary or not just because of this uncertainty. So when you take all these real life complexities of actual modeling of default into account, you will find that the model is in fact unable to say was confident at each particular moment in time if the default already happened or not. And in this sense, it starts to resemble the famous Schrodinger cat in quantum mechanics who is either alive or dead, but we can't know for sure unless we open the door of its cage. But in this case, it will be dead with certainty because it will be killed by the radioactivity activated the moment we open the door. Well, before closing this video, I want to mention that to model, credit people usually use other models with credit spreads as additional state variables, which we already mentioned before. This certainly has more flexibility than the Merton model in describing joint dynamics of store prices and spreads. But the main challenges with this approaches would be higher model complexity. As now, you have more parameters to estimate, as well as consistency between the dynamics of spreads and stock prices, especially in a region of high spreads and low stock prices, which is most important for modeling defaults.