Welcome to lecture six. In the first five lectures, we focused on interest rates and interest rate instruments. We discussed a cross-correlation. We did data driven analysis. Now in the next few lectures, lecture six and seven, we will be focusing on or transitioning from data-driven analysis to model calibration. The model under there could be tons of models. We can consider. There are various different frameworks actually for valuing and pricing interest rate instruments and derivatives. I'm having a list here, but the list section is more than this but these four are definitely the major ones. The short rate models that simply means focusing as I'm going to go through at focusing on evolution of short rate, the Heath-Jarrow-Morton model, for short HJM focusing on evolution of forward rates, Libor Market Models or LMM focusing on evolution of directly evolution of the Libor rates and Swap Market Models, for short SMM they focusing on swap rate evolution. Now, the short-rate models as I said earlier we assumed some stochastic process for evolution of instantaneous short rate. We went through instantaneous short rates. Then what we will do is the zero-coupon bonds and other derivatives are being priced of that model. In HJM they assume stochastic process or evolution of instantaneous forward rate. We also went through instantaneous forward rate as well. Then zero-coupon bonds under HJM they're absolutely straightforward. You've seen already the expression for this. Other instruments that are being price of that model. The other two models when I said that earlier market models unlike short rate models and HJM where the rates are not observable because you cannot really actually go and purchase instantaneous short rate. This doesn't make any sense. But when it comes to it, for Libor and Swap rates, they both observable and they both actually traded instruments. The focus is directly focusing on finding a process for the evolution of those prices. In market models, we assume Libor Swap rates are evolving according to some a stochastic process. In LMM the assumption is a stochastic process evolution of Libor rates and in SMM we assume some stochastic process for evolution of short rates. I'm sorry swap rates. Now, in lectures six and seven, our focus would be on short rate models. The reason for this is because they're so very popular even though some people believe that they are not suitable enough for all fixed income instruments. Having said this, because they're so simple and the ease of implementation still remain pretty popular especially in the situation that the level of rates are not the shape is of primary importance to us. Now, as I said earlier, the focus is on evolution of interest rates in terms of instantaneous short rate which I write it as the art of S at time S. You do remember we discussed the instantaneous short rates and what we will do is actually we would be coming up with a few different stochastic processes assuming that the short rate evolves according to that process. Now, the model under consideration if we assume the model under consideration is short-rate models, then the zero-coupon bond prices would be priced according to this equation which is simply conditional expectation of this integral. And as you see, this R we are assuming that this R is evolving through time according to some stochastic process. Now, if you assume the sum of the stochastic process, then what we can do is actually we can find this expectation. The model that we are considering at this point is called the Vasicek model. It is the very very first model used for short rates. It is exactly equivalent of kind of, not exactly because that should be was before Black Merton Scholes. But like Black Merton Scholes the Vasicek is the simplest model for evolution of short rates. In the Vasicek model, the assumption is we're assuming that or instantaneous short rate evolves according to the following stochastic process, or it has following a stochastic differential equation. In this equation, as you noticed, there is a mean reversion component. This is what I put in the box. And I'm calling this mean reversion because we have a k here or kappa which is the mean reversion rate. And then the theta is what we call the long term mean of the short rate. That means the assumption is if you have the theta here, if you're over here that means you are going to mean revert to back here or when you're down here you're going to go back to this level. This is the so-called long-term mean rate and then kappa would dictate the speed of convergence for us. The bigger the kappa, the fastest you would converge to that level. Then we have the mean reversion component. Of course we have the diffusion component. Now, the solution actually exist for this stochastic differential equation and is given by this equation. And as you see for the process we already have theta, we already have kappa, we already have sigma, and also we need to have the starting point and is a four parameter model. Is R naught the initial where we're starting the initial instantaneous short rate? Theta, kappa, and sigma. Is a four parameter model. As I said the solution exists. It's for this will again take it for granted the solution exists for the zero-coupon bond prices and is given according to this equation, where if you have both a and b which are given over here, b and a they both are given here. And maybe taking this brand granted we don't go through the solution, and I'm actually through the calibration. I have the implementation of this and I'm going to walk you through what we exactly using the b and a to get the close form solution for PTT the zero-coupon bond prices. Now, it's a very simple observation, is we noticing that actually PTT is the analytical solution is time homogeneous. Because as you seeing everything is given as a function of t minus t. And that simply means either you having PTT or you having p say t plus tau. T plus tau does not make any difference whatsoever is time homogeneous because the difference it goes by the difference between these two. That's one simple observation which I'm stating it here. In case if you are interested in knowing, given the Vasicek model what evolution for the price of a zero-coupon bond itself is given according to this equation against without any proof. Given that the rt is equal to kappa theta minus rt dt plus sigma dwt the 0 coupon bond under this model is evolution actually follows the following sd. We would not be focusing on this one in case, I'm just giving it to you to know. That actually looks pretty straightforward and that's exactly the b that we had in the previous slide. What we do in calibration is pretty straightforward. Is a four parameter model. As I said earlier, is a four parameter model. It's important to notice that we are having just simply four parameters and one factor that means we are assuming that from time zero to going up to 30 years we just having one factor. By one factor means as opposed to if I write this form one more time assuming that there is just one driver for the entirety of going at 30 years. There are cases that actually maybe we want to have two or three different factors. That means two or three different Brownian motions running the process. But as I said for simplicity I'm just assuming that there is one factor module. Our focus would be on one factor model. Having said this we understand due to lack of number of factors or number of parameters. We may actually not do a very very good job if we're trying to capture the behavior going up to 30 years. But for illustrative purposes I'm just focusing on one factor here. Now, I already mentioned that one there is no reason to go through this. What I am doing is I am now starting the focusing on calibration. The calibration instruments. I'm having LIBOR rates, we having swap rates. I'm assuming I'm giving you four and half years of data on LIBOR and swap rates. For calibration purposes am picking just two days completely randomly I picked. Actually I do not know why this is not exactly correct. This is supposed to be December 14th, 2017. My apologies. I have to correct this. Then in the tables I'm getting the snapshots for December 14th, 2017. Another snapshot for October 11th, 2018 for one, two, three, six, twelve months maturity LIBOR rates. Then I'm doing calibration on these two days differently. For the swap rates, I'm assuming again on the exact same day December 14th, 2017 two, three, five, seven, 10, 15, 30 years swap rates that term for the swap and the same as snapshot for October 11th, 2018. Objective function I believe I already mentioned this. I'm assuming the objective function to be the following objective function. I'm assuming relative difference between market and model for LIBOR relative difference, the score of that and then simply I would sum this one up. Then you could come up with various different objective functions. Here the assumption is I'm looking at the score or relative difference. Model minus market over the market on LIBOR and unswap this separately. Now, the results set on getting are this; for December 14th, 2017, what we seeing this is after calibration. I'm going to go through the code. You will be seeing exactly how it works. This is the zero-coupon curve from Vasicek model calibrated to LIBOR and swap rates. Let's see how they've done. The blue line that you're seeing is actually the market and the red ones are the model. As you see when we do the calibration in some cases we've seen that it's missing a bit. In some cases is overpriced, in some cases it's under price, but it's nice you go through it. For the swap rates, again we are seeing that for these short maturities or short-term they're below it but when it goes to high they're above it. Again is going through it. But we've seen that is actually missing a bit. As I said one factor is not enough. Having said that there still is relatively doing a good job. Let's go to the next one which is for October 11th. The same scenario here is a slightly tighter here. We doing a better job on that day and one more time again we see that we are doing a good job for 23 and five and seven we are missing, on 10 we are doing an excellent job, on 30 we are doing a very nice job. Again, we missing on 15. I mean in a way is a trade-off. We can pass through all of those. But we noticing that on the very short run and the long one we doing a very nice job. In the middle we missing it. This is the curve coming out of it. In the last slide what I'm doing is I'm actually showing you the two against each other. At the October 11th, 2018 versus December 14th, 2017. This is the 2017 version. This is a 2018 version and this is very consistent to lecture one. If you remember lecture one I had to care, I said that low rate environment, a high rate environment. As you seeing this in 2018, the interest rate went higher. Federal Reserve increased the rates. As you see of course the curves goes lower than the other one which settlers the lower. Almost assuming zero of course occurs with come over here. If there is no discount as you start increasing it would start going down and down more. This has to do exactly depends on the level of the rate. Now I'm going to go through the court but first I'm going to go through lectures seven because the cause conduct combined I want you to see actually both of them. There is a drawback to Vasicek model that in lecture seven we'll be focusing on and then see what's the alternative to that model is which we will be discussing it in lecture seven. Thank you so much.
