Let's talk about models with more than one brand in motion. You don't have to look at the slide yet the details. Let me just tell you general story about this. So, the if you do calibration of the model or something if you know from statistics the methods of principle component. Then studying options data, you can find out that there is more than one significant principle component. Which means really that you need in your model more than one stochastic factor driving the model. So that's where there is a need to introduce modern motions. Alternatively, what modern and sophisticated models do, they also jumps to the price, okay? They also jumps to the price of the stock. And we will talk a little bit about that in the next set of slides. Or you combine both, okay? So the most sophisticated models would have both. Several brand emotions. Maybe stochastic volatility is to casting this stochastic bath. Plus jumps maybe jumps in the stock price. Maybe also jumps in the volatility. So you can combine all of these things. And that gets you at a very geneneral than black shelves, right? So combining extra brand emotions, extra stochastic economic factors with jumps. All right, but for for the time being we are not talking about jumps. We're just going to add an additional brand in motion. Let me skip to the slightly examples first. Then it's going to be easier to understand the general theory, Here are two well known examples. The first one is historically an early stochastic volatility model called Heston's model. In which the stock is something like black shows. Except instead of sigma here, I'm writing square root of V of t. Where V stands for volatility. And I'm writing everything directly under the pricing probability Q. So I replaced you by r. But this V volatility is going to be assumed to be also another stochastic process. This type inheritance model with some other brown emotions ZQ. And you may want to have some correlation between your stock and the volatility of the stock. So let's assume that WQ. And ZQ have correlation role, okay? That's the assumption in the Heston's model. Now why exactly like this, right? So here there's a little bit of economic reasons and a little bit of mathematical reasons. Here you have positive constant A, positive constant B. And positive constant gamma. This type of drift is called mean reverting drift. We are going to see it also in the interest rate models. And this is because you don't really want your volatility to have exponential growth. It's not the case that like stocks or stock market on average, the volatility goes up. That's just not the case. It oscillates around certain values. So in fact here, this B has the interpretation of the long term mean. It can prove mathematically that the expected value of V is going to converge to be in the limit. So why is that? Well look, if these below B then this is going to be positive. And the DT term will push and the process up to B if it's below B, okay? And on the other hand, if V is larger than B and this is going to be negative. And this is going to push down V towards B, okay? So in both cases whether it's up or down. This is going to be very positive or negative and is going to push it back to B. So that's intuitively as time goes by the expected value converges to B. Fine, so that's kind of the economic reason. You put a linear function here for mathematical trackability. But the linear function, which is mean reverting, it pushes you back to this B. Which has the interpretation of the long term mean of the volatility. Now, why square root here? Well, You want something here so that it doesn't go below zero, right? If you have only constant then brand emotions could push it below zero. You don't want to have volatility negative. Why square root? That's for mathematical reasons. It makes particularly Heston's model. At least if you have independent brand emotions here. You can get explicit formulas for the standard vanilla options. And even if you have correlation and it's more the model is more tractable, okay? So square root here is strictly for mathematical reasons. There is no strong economic reason why to have square root here. All right, so this is a typical stochastic volatility. In fact the most famous probably stochastic volatility example model. And so how do you price options here? Well, again, two ways. You can either try to compute expected value under these dynamics, under the risk need the probability. Or you can try to solve the partial differential equation corresponding to the price. How is that going to look like? So now the price is going to be a function. Price of an option of a European, Yeah, part independent option is going to be a function of time as C(t,s,v), okay? As soon as you introduce another process you have to plug it in as a variable in your functions. Which are expected values related to these stochastic processes, okay? So you have to put a v here. Okay, that's important. I always mentioned this in my classes and many students forget. No matter how many times as I said. If you introduce additional stochastic factors in your model. You have that variable as an extra variable in the option price. And now it's going to be similar as with options into stocks. And to underlying you can just pretend that V is another underlying. That's another stock in which you cannot invest. In fact, these days there are volatility swaps that are derivatives on volatility. So you can actually trading in volatility. But when Heston invented his model or suggested his model, there was no possibility to actually trade in volatility. So this was really an incomplete market model. V was not another asset. V was just another factor. But you can think. So let's stay in that context. Let's assume we cannot trading in volatility. So this is an incomplete model. Nevertheless, you can think of V as an as an asset which is there. But risky as which is there, but you cannot trade in it. Okay, so then the formulas are going to be, the partial differential equation is going to be the same as with too risky assets when we have options into risky assets. Meaning zero has to be equal to partial direct respected T and then one half second derivatives times their volatility squared. So I have a squared, then I have square root of the square, which gives me we and then here I have gamma square square to the square that gives me also V which is factored out here. Okay, so this is a secondary motive terms with the squares of these DW terms, then this is the usual black shows part from the drift of the stock here and from discounting, this is the same as in black shows. And then I have to have first derivative think Lita's rule. First, the liberty respect to V times the drift of V it's this part AB minus V. And then because I have correlation, I multiply the volatility is, it's going to be S square three times gamma square to be square. Tahrir square to end square root of V, so we will give me AV. And I have S comma and I have to multiply by correlation raw and that multiplies the second mixed derivative respect to S and V, okay. Pretty much the same if you look back at the case of 22 assets to underlines. So that's a partial differential equation and you may try to solve it numerically enhancements model, you can do better than just brute force, numeric six. And this is what it looks like there is another more recent model called Sabra model. C S A B R model which is kind of a combination of the previous constant elasticity variance C E V model namely, you put you put the stock price in here in the denominator one over as to the beat of T, but you multiply by something which is not constant. But in this case the martingale multiply by something which is in fact he had geometric brand emotion of multi D C Q. And these are again correlated brown emotions. It's just another popular model, okay. Kind of swept under the rug a problem here theoretically conceptually a problem, which is that this is in fact if you don't trade and volatility, this is an incomplete market model which means there are many possible prices. There are many cues. Okay, so under different queues. Under all queues, you will have our here. That's not an issue. However, under different cues you might have different parameters AB and gamma here or alpha down here. So you could have you could have different parameters actually. Gamma is not going to be different, but you could have different A and B. So theoretically it's a problem. How do we know which Q to use? There are many cues which will give you different no our trash prices possibly if your claim cannot be replicated and there are claims here which cannot be replicated, typically they will not be replicable. What do you do? Well, you assume that the market knows what's doing, and the market has a queue under rigid prices, everything. And then you try to estimate from the available data what that Q is, okay. In this model, for example, it has this model it would mean find A and B. That in fact corresponds to the market data that you observed. Okay. More precisely. So in some sense, it would be implied A and B. And implied gamma the same way we're talking about implied volatility. Yeah. Let me just talk about that. This isn't practical issue which is really the basics of what is done in practice here. I just want to mention it so that you know about it, but we are not going to go into details. So, what roughly speaking, you would do something like this, you would suppose you are working as a quantum investment bank. You would, so you have your model, let's say pass this model and you would observe frequently traded options, liquid lee traded options on whatever underlying you are interested in. And you would see those prices, today's prices of options with many maturity, as many strike prices are on the same underlying on the same stock. Let's say, okay. And then you have a model and you observed, you can compute theoretical crisis forgiven forgiven parameters AB and gamma, let's say in the previous example. That's check again. So it was, yeah, it was just AB and gamma. Well there was also raw actually and then you would compare theoretical prices which depends which is a function in this case AB roll and gamma, you compare it. So you subtract markets price, the observed price. So, I'll just call it market price, the traded price or whatever options you are looking at. Okay, let me get this right market price and then maybe you are doing something like mean square there. So you look at the square there between your theoretical model and your market price and then maybe you want to. So this is the simplest method. You can be more sophisticated about this, but maybe you simply add this sum over all frequently liquid lee traded options traded, some our traded options. What? That's not legible so I'm going to right again. So some over traded options that you observe the prices and then you may want to choose AB roll and gamma to minimize this, minimize AB rogue. Okay, so this is a journalist concept of implied volatility here. You're implying A and B, raw and gamma. You're all you're all your parameters. It's a four parameter model rather than just one parameter model which was sigma and black scholes. So you're going to get a better fit. The seller is going to be smaller typically than if you have just one parameter. Okay, the price you pay, the model is more complicated and if you have too many parameters there is problems of over-fitting. But in principle this is what you do and then what? And then you choose your a, b row and gamma that you get optimally by minimizing something like this. You choose those to price custom made derivatives that you're in Western bank is selling to private customers let's say, okay? So that's typical. You have a model, you test it or you calibrated you fit it. So these are the expression of calibration or fitting of the model. You fit it in a way similar to the market data, option data that you see today. And then use those parameters of the model to price other things which are traded [COUGH] over the counter, which you trade directly with your customers. Fine, so this is what is done. There is going to be a problem theoretically because in principle you do this every day. And if your model is completely correct, how should a, b and row and gamma change from one day to another? They should not change. They should stay the same if your model assumes that they are constant, right? If your model assumes they're constant then they should stay constant. However, no model is going to be completely correct. So your parameters will always change from one day to another. So it's not really theoretically consistent but you work with it. So ideally what you want is to have as few parameters as possible and parameters which don't change too much from one day to another, okay? So you want parameters which are stable to collaboration from one day to another and you don't want too many of those parameters. You also want presumably mathematically tractable model because this is actually computational intensive even with supercomputers or whatever. If you have to do this for a lot of options, a lot of stocks, a lot of underlines. It helps a lot if you can compute this more or less fast, either explicitly or with a nice numerical method. Because you have two computers every time for every a, b row gamma, or you're minimizing you're a b or your parameters. And so you have two computers for a lot of parameters if you're maximizing over those and that can be time consuming numerically than computation is time consuming. Okay, so this is one of the main uses of the theory that I'm doing here is how people use these small, sophisticated, complicated models like this. They are trying to fit the observed options data by the models, by maximizing our parameters in the model. And once they're happy with that fit to the data, they believe their model is more or less pricing the same way the market does, and then they can trade based on that, okay? That's the main idea.
