Let's go back to where I talked a little bit about the theory of this, so I told you what in practice is done in theory. So in practice doesn't really matter if your model is incomplete or complete because there is only one Q that the market uses. Okay, that's the view, there is only one Q that the market uses, so you just estimate that Q. Even though there are many Qs many recent neutral probabilities, there is only one which the market uses. And if you have enough data, you estimate that Q the way I just described. There is a problem if you are in a new market which doesn't have the data right? If you suddenly have their regulation in like it was the case for energy markets, suddenly you have new derivatives on energy, electricity. And you are new in this and there is, no, it's a new market, there is no history, then you cannot do that. There is no liquidity traded derivatives, then it's a hard problem I don't have much to say about that. It's a hard problem to choose which model, which parameters to use if there are no liquidity traded options or derivatives. All right, going back to kind of a general case with two brand emotions. So suppose I put here two brand emotions in us, but independent brand emotions. So W2 to have correlation with this one I put one and we I could have done other way around. Doesn't really matter, I could have put two grand emotions and being only one in us. But okay, I chose to put two grand emotions W1 and W2 which are independence in us and one brand in motion and we in the volatility. And then the signals in some way depend on volatility like inheritance model for square root of the right dysfunctions were square roots. But in general you can have something like this and this is under the actual so called physical measure, probability. I want to show here that there are many possible martingale and I was just tell you how to construct them. Okay, so for any process kappa which doesn't look into the future, so adapted process, you can construct a risk neutral probability Q kappa. So there is really infinitely many continuously, infinitely many of those for any process Kappa, what do you do? I'm just first going to look at the 2nd equation here, you change W2 by adding kappa dt, okay, what does this mean? This means really the integrated form as usual It means that W2 Q kappa Tis W2 of T equal sign in between plus integral 0 to T kappa of StS that's what it means. Okay the sum of theory still holds I can change by integral is not by multiplying by T but I changed by adding an integral like this. Okay that's what I mean here and then similarly for the W1, this part is the same as in black shows mu minus or sigma except they may depend on time. In which case I also mean something like this with the integral, I mean the integrated forward and then I also subtract sigma to kappa. All I want to do is to make my discounted stock price martingale which means I want to replace mu by R here. Now if you plug this dW1 here sigma one, sigma one will cancel mu and mu will cancel, you will get your R here. But you will also get minus sigma 2 kappa, however that will cancel when you replace the W2 with this one. Because here you will have sigma two times Kappa, so sigma 2 kappa minus sigma2 kappa will cancel and you will have R dt here. So for any change give some type of change of probability like this, you do get rdt plus some martingales plus dW terms. So for any kappa you have a martingale probability, have a risk to the probability. Okay this is theoretically what's going on in a model where you have to run emotions and only one risky asset and you only have to make one risky asset. The martingale, you are not making the other one of martingale discounted. Then you have many ways to do that, okay and as I told you already in practice you then just. You make assumptions in particular, you make assumptions typically like inheritance model that this alpha and gamma and mu, sigma one or kappa if you want, you assume that their constant okay? That's the easiest case, you assume that those parameters are constant and then you estimate them from the market data, market options data okay? So if you do this, change what happens to V, you get alpha minus kappa gamma in the drift, otherwise stays the same and change W2, W 2Q. All right, and then you can write down a partial differential equations, similar ideas before, Well, you can write a partial differential equation if you have constant parameters or at least the true mystic parameters or parameters which are deterministic functions of your obvious stochastic factors here. If you introduce additional randomness, you may need additional terms in your partial differential equation okay? But if you have constant, so you assume typically that you have constant parameters and then you can write down your partial differential equation same as before. This is standards vary greatly respect the time okay now my total volatility sigma one squared plus sigma two squared because I had to brand emotions in us. And then 2nd derivative respectable opportunity gamma squared then I have the drift terms. This is the black scholes usual term and then I have the drift of V and d Q and then I have the mixed term. Give me the mixed second derivative times the, in this case sigma two gamma because sigma to gamma was the correlation. If you look back, you have gamma and sigma two multiplying the W2's and then I get nothing from these two guys W2 and W1 because they have zero correlation and these guys have one correlation one. So I just multiply gamma, siogma two S, that's where this time comes from, actually there is a type of here as is missing, right? That should also be and should be also an S here [COUGH] Okay, and then this paragraph at the bottom tells you what I already told you on that blank slide that you calibrate this. And then you choose your alpha, kappa, gamma, so that you match as well as possible, the observed market prices of liquidly traded options. Okay, now there is another difficulty here, which is, it's not your function, your function. See here, as I said before, it's going to depend on t,s and V and V is not as directly observable as S such as the stock price. You see it's time is also, what time is what is, well, pretty well then, do you have to estimate? So,it's a little bit tricky to compute these prices also because they depend on something which is not quite directly observable. This will appeal to here, you have to although these days is much more better defined what people think is volatility because they trade options. Okay but that's that's how you do it since the classical to you write the model you estimate, calibrates the parameters. You compute the price either as a solution to the differential equation or by expected values. If it's a solution to the differential equation, it depends on time, on the underlying and on the value, current value of volatility too, all right that's it.
