Well, we are here in the last set of lectures last week of the course, which is about derivatives in the fixed income markets. We have talked about bonds and fixed income and the interest rates and yields in chapter two. But that was all in from the deterministic point of view, you just compute your yields today. You don't model them as moving randomly in time, then you repeat the procedure tomorrow and so on. Now we want to have models which model the interest rates and yields as stochastic processes. Let's see how to do that. Well, let's first look at the last formula. Last equation here in this slide, everything is going to be based on that. This is the price of a bond today, which is a small t with maturity big T. And this is our idealized bond, zero coupon bonds, no coupons, pure discount, zero coupon Bonds, which pays one dollar at maturity. We assume that the price and maturity is the value of maturity that it pays is $1. In practice it will be mostly $100. But we assume here $1 and well this is just a generalization of the present value formula. If the interest rate is continuously compounded but constant, then $1 will have today just the value e to the -r(T-t). All right, you can multiply one by this. But here this counting has to be done like this with this integral here into the minus will just rewrite it here, t to T r(u)du. Because the interest rate is moving in time, you have to integrate and that's how you do discounting, right? So you discount your $1. But because r maybe random, you also take the expectation and you take the expectation under the pricing probability. I don't write Q here. There is this first paragraph that tells you, I am now going to not use the notation Q. There will be only the pricing probability, there will be no actual probability. So I'm not going to put Q or star or anything here. Everything is modeled directly under the pricing probability, right? So previously I had, I would put Q here, that's what I mean now too. But I'm not writing it the expectations that always going to be under the pricing probability, okay? So I'm not going to write it and this is just up here. This is just the discrete version of the same thing. You discount $1 because the interest rate can change from one period to another. It's (1+0) times (1+r1) times (1+r2) and so on. You discount for each period with one plus corresponding r, okay? So we are going to look at this and we are going to model once you look at it like this. Well, there are two things you could do. You could model the bond prices directly or you can model the short rate and then compute bond prices from here. Historically, what happened is when people started being more sophisticated by modeling fixed income markets, they were modeling the short rate directly, okay? Why not bonds? Well, the reason is if you model bonds directly, like in the black Scholes model is we model the stock directly. We didn't model the rate of the stock or the volatility of the stock. Well, later we also model volatility in the stochastic volatility models, but in principle that was directly through modeling the stock. Here, we are not going to do that, we are not going to model the bond prices were going to model the short rates. What is the reason for that? The reason is there is many bond prices of the same time, there is many maturities and you will have to model each maturity. You have a model for each maturity, but then you have to worry about whether you have arbitration your model, whether you can make arbitrage by trading in bonds with different maturities. Now, in this approach, if you model bond prices like this under the pricing probability, automatically the bond prices will be such that there is no arbitration the market. Why? Because if you do it like this, the discounted bond prices are going to be martingale. And because of that, by our meta theorem from one of the fundamental theories that was surprising we did before, there will be no address, right? We even prove that if there exists martin probability under which discounted prices are martingale, there is no arbitration. And under this approach, this kind of bond prices will be matching less. And under the pricing probability and there will be no arbitration. That's why this is convenient to model the interest rate and then compute bond prices from this and there will be no arbitration in your model. Now, why if you're curious mathematically, why do we know that this kind of bond prices are matching less? Well, let's look at it. You have e to the minus integral 0 to t, r(u)du times the bond price that's discounted bonds, okay? That's equal to expectation. I'm going to put this discounting factor inside because it's known at time t. So you can go outside or inside and if I put it inside, I'm going to have e to the minus integral 0 to t, times e to the minus integral small t. To casualty. That's just e to the minus integral from 0 to capitol T. All right, you add these two integral 0 to small t, small to capital T. You get from 0 to capital T out of u(d) times nothing, that's 1, okay? Now this thing is a martingale. Why is this a martingale? Well, any process defined, we may have talked about this before, but if I define empty to be expectation T of some fixed and a variable X. Which is known as a future time casualty, that's martingale. And this is a fixed, now this is not a fixed random variable doesn't depend on T, it's fixed. And therefore these expectations form martingale. Why is this martingale? Well this is if you take expectation time at a time. Sorry like this. If you take expectation at time s of Mt. That's Es(Et)X from the definition of Mt. But then only the expectation with less information survives the law interested expectations. So this is going to be (Es)X but that's Ms and that's your martingale property expectation of the future is today's, okay? So any time you define Mt like this, it's a martingale, and that's exactly what we have here for the discounted bond price. It's expected value, conditional expected value of a fixed rate and variable and therefore it's a martingale and therefore there is no arbitration in this model. That's why people thought of historically first of modeling bond prices like this. In fact modeling short rates and then bond prices from here. Now, you're not going really to use these models to price bonds. The way you use this in practice is, okay, I'm going to have my model from the interest rate but I'm not sure which parameters I should use in the model, but I can compute theoretical bond prices from here. I can observe the actual bond prices in the market and then I can find the parameters so that my model is calibrated well that it fits the data well, okay? It's calibrated well to the bond price data, which is really the yields, right? So every day you look at the bond prices or the yield curve and you try to find parameters of your model. So that this right hand side that you compute in your model is close to the left hand side which are observed bond prices today or observed yields equivalently. Okay, this is the same what we talked about before about calibrating models with statistical activity or otherwise. The same thing you do here except you don't have to calibrate two options data. You calibrate directly to the bond price data. Now, what is going to happen is your model parameters to calibrate it today may not be the same as the model parameters calibrated tomorrow. Which is theoretically inconsistent because those parameters are supposed to be constant. But again, that's the usual problem. And you just hope to find a model in which these parameters would not change too much from one day to another. That's the idea, okay? To repeat, the idea is model the short rate, compute theoretical expectations and choose parameters of the model. So that theoretical expectations are close to the observed bond prices, okay? Calibrate your model. Okay, so that's the main idea and I am first going to look at the screen time, couple of the three time examples, most of it in practice is continuous time models, but I'll just spend a little bit of time on the screen time. So for example, one of our goals in this set of slides or these several sets of slides for the last week is eventually to be able to price a call option on a bond. Let's say that the bond has maturity call at T. And the call option on the bond has maturity Tao, which is less than capital T, right? So effectively what we have to compute if it's in continuous time expected value under the pricing probability of discounted payoff discounted up to maturity, which is Tao. What is the payoff? The payoff is going to be the bond price minus the strike price, positive part of that. And the bond price at maturity can be written as expectation at maturity Tao of the discounted $1 discounted from Tao to maturity of the bond which is capital T minus the strike price, okay? So that's how it looks like in general. And this is what we would try to compute in different models. And this is just a discrete time version. It is again discount only up to the maturity of the option. And here you discount from the majority of the option to the maturity of the bonds. Yeah, this is the type of thing that we want to compute.