 Hi everyone, it's MJ and in this video, I'm going to attempt to explain what ito's lemma is Essentially, why do we need ito's lemma because with ito processes? How do we integrate when we have this Brownian motion function over here remember ito's process is when you're drifting your dispersion of functions and like with say the vina process where they were constants and remember the vina process is just a more generalized version of your Brownian motion or think of this as white noise So make sure you've seen these three videos and you're comfortable with the theory because now This is where things get difficult. So ito's lemma we've got ito's process and What we're gonna do is Like I said, we're not gonna explain how he did it. We're just gonna see how it's useful We're gonna use it as a lemma because mathematically this is insane But essentially k ito came up with this idea saying well hold on if I create a function g of x and t it's going to follow this following Form and like I say how he figured that out. We don't know what we do know There's there is the proof, but it very much is above my pay grade I don't understand it what we're gonna do though and especially what they they ask for us in the actual exams is just to Understand it or how to use it. So this is the tricky step if you can figure out how he came up with that step Fantastic well done your IQs past 500, but for the rest of us mere mortals Let's just examine this function because what this function is of itself. It's an ito's lemma where we have Drift is going to be equal to this term over here and our variance or our dispersion is this term over here Remember we're taking the squared because this variance not standard deviation. It's very important Don't get that confused. I sometimes do anyway, let's now take this ito's lemma and look at it with our stock price model which we were building in the previous video and if we take okay, this is our our model for the stock price and we use ito's lemma and we put You know this function of g and the set of every time we have x and t we're changing it You know with our mu and stock price and segment of stock price. We get this following idea over here Now this is the next big step was to say okay Well, what should this function of g be and we say if this function of g is equal to the natural log Now, why do we use the natural log? Well, we use it because it has some nice mathematical properties It makes things a lot simpler and you're gonna see that if we didn't have the natural log I mean that we have to deal with this monstrous math So the natural log is gonna make things simpler when we look at black skulls later We're gonna see that they start using different functions for g and they use that for the pricing of their derivatives But in this model we assume or we let our g function equal the natural log and What's lovely about that is look how everything starts to simplify. So these big ugly terms in this Drift part over here become nice and simple When we're using the lint of s So much so that we get to this very nice and actually manageable thing over here Where what we're seeing is okay, hold on by using the lint of s I'll mu and our sigma have now become constants. They're no longer linked to the stock price Which means we've now reduced it back to a vina process and Since it's a vina process We can use that idea that we spoke about in the vina video that the change in the value of lint s over time Interval t is going to be equal to the normal distribution with given mean and given variance Which means that the lint of s of t This means what is the stock price going to be in the future? It's natural log is given by this idea here and essentially what how this is different from the one above it is This is the change in value. So it was lint s t minus lint s 0 I've just moved lint s 0 into the distribution and now we can use this to model future stock Prices and that is a very high level view of ita's lemon Like I said, I'm not going into the mathematics because I don't know how he did that You don't need to know that for the actual exams that very much is Above our pay grade, but this is how ita's lemma is used so that we can now Integrate and have a model for the future stock price. Anyway, thanks so much for watching and in the next video We're gonna look at how this is applied to the black skulls formula