Welcome to lecture one. The focus would be on interest rate instruments. You are going to have seven lectures. In the first lecture, we will be focusing on various different interest rates. At anytime, you saw from interest rate, they're very very first thing would come in mind would be actually borrowing and lending rates. I'm having two very, very simple questions here. If you borrow $1 today, how much would you have to pay in future? Definitely you pay more than $1, that's definitely the case. Now the question is that, how much? Then that depends on the borrowing rate. Now let's say now you lending $1 today, how much would you ask for a year from now? Again, definitely more than a $1, depending on the lending rate. Now, let's just put it in a very, very simple term. If you lending $1 today, And this is today, we call it time 0. And let's say this is a year from now this call it t=1 for a year from now. And let's assume the borrowing rate is 10%. Then definitely what you're asking for, if you lending one today, you asking for the $1.10 or $1.5, so a $1.1. Now, on the other hand, if this would have been 20% then definitely that would have been $1.20 or $1.2. Now, do not for get that when we talking about this rate, this rate is kind of annualized. That means, for example, if you would have been today, which is t=0 for today, you lending $1. And let's say this is six months from now, let's call it 1 over 2 years, which means six months. And let's say that lending rate was 20%, just for assumption. What you would have earned would be actually $1 + the $1 multiplied by this 20% divided by 2 because this is for just six months, which is half a year. This would be .10, that means it would be $1.10. Just make sure that it's clear that you need to adjust depending on the holding period or the lending period or the borrowing period. Now, we're not going to look at this from a different angle. Now let's say, how much you are willing to pay today to receive $1 in future? This is actually what we show as Pt,T. There are names for this, some people call it zero-coupon bond, some people call it T-bond, and T has to do with these maturity. And the reason for zero-coupon bond is because in between your not receiving anything, you just paying it now to receive something exactly at time T which we call it maturity. Later we will be talking about this. Now, it turns out that is very, very important to know this curve, to know how to find these curve, and then remember, the environment is changing. That means for example last year, rate was very different from this year rate. Because further reserved in increasing the rate and every typically six, three months, they visit, they getting together. There's a meeting, they decide if they have to increase the rate or not. And depending on how the increasing then the shape of the curve is changing throughout the time. Now, as I said, the amount that you are willing to pay today to receive a dollar and future time T is what we call as Pt,T, the interest for us is to have this curve. Let me show you I have few examples on that curve. Now for example here I'm showing you Pt,T going at 30 years, when I'm writing here Pt,T, I want to make sure that you understand. If I'm a starting from 0, that means I'm assuming t is 0. That means I'm really actually plotting P0T. But the typical we call Pt,T. But for t, which is equal to 0, for example at this point. If t is not equal to 0, then definitely would have been, for example, a point here. Now, I'm showing you two curves. I want to make sure that it's clear, that in a low rate environment, the curve is flatter than the high rate environment. And it typically makes sense because if the rate is higher, then You are pang less today to receive a dollar from future. And one thing is important to recognize. For example, if I go on 15 and I'll come up on this curve. This number which would come here is roughly about say 60 cents. Is how much you're willing to pay today to receive $1, 15 years from now. On the other hand, in a high environment. These 20 roughly 22 cents, is how much you're willing to pay today to receive $1 15 years from now. Look at the difference between the two. That means a high, because otherwise you would've put your money somewhere else because you know that you can earn more from it. That's how the curve, actually this curve works. This curve your seeing today but for what you are willing to pay today to receive a dollar from now. Now, if you have 0 interest rate, the curve would be absolutely flat. That means if we go out to a dollar. And if the rate is super, super high, you will see behavior like this, that you actually would go almost to 0 very, very rapidly. This is super high environment. This is 0 rate environment, 0 rate environment. Like for example, Japan for many years. Actually it was almost zero redevelopment, because that care was almost flat, almost flat. Now, why this curve is so important? This is, I'm having at the title here. Why it's so important to have this curve? As you recall, I would say in a first series, which we focus on derivatives, pricings, options pricing. What we focus on the derivatives value depend on single underlying asset, something like single name equity, either if you remember if we focus on apple and you could easily apply it. Of course, through various supermarkets or the same thing could be on foreign exchange or commodities. Now, those models we cannot apply to interest rates derivatives, and one simple reason for it is because in interest rate environment, as you will see. The various different points on the curve. That means if you are discussing something with maturity of two years or is it something with maturity of five years, a highly highly correlated to each other and in a way you cannot really look at them differently. I mean, it's as if the evolution of the entire yield curve is something you need to be able to capture. When it comes to single name equities, there are many of them. For example, if you're looking at Apple versus say Whole Foods. They almost no correlation between the two and what happens to Apple may not have any impact at all on what's happening to Whole Foods stocks. Now, the models that we are working on that they tried to capture the evolution of the entire yield curve. This is what is what we calling as term structure models, just to make sure that if you hear term structure models that means these are the models that they're trying to capture the evolution of the entire curve. And I explained you shortly what we mean by those curves, because it doesn't mean necessary just PTT, but everything gets price of the PTT, which we're going to get to it in the next lectures. Now, how to calculate actually this PTT, we said why is so important? Because a lot of things would get price off it or what we're doing is typically the way to calculate this. We assuming some model. Then I'm seeing a model. I typically mean some stochastic model. And we use highly liquid instrument as calibration instrument to actually calculate these PTT at time T for any T in the future. Now, what are those instruments? If you remember, we did the calibration already, we'll do exact same thing here, but the question is, what kind of instrument we will be using? For example, for Apple, what we did was we went to the options market for Apple and we look at the calls and puts just an Apple. Nothing else to be able to get evolution for the stock price. Assuming that Apple following some stochastic process, and that's exactly what we did. We did it for both pricing. We did it for both model calibration. Here, actually there are very different instruments that they can be linked to. Actually, this zero coupon prices. Zero coupon bond prices. Those instruments are LIBOR rates, I'm going to go through it, a very liquid. Swap rates and they tradeable as well. Futures, I mean, you've heard of them for example, one of the most liquids ones are, Dollar futures, caps and floors. These are options type and options on swap rates, which we call it actually swaptions. And the techniques can actually be used if you do remember for options pricing exact same thing technique even though I would not be covering but the exact same technique can be utilized an employee to price swaptions as well. Then my agenda for this lecture and the next six lectures through interest rate instruments, starts with defining various interest rates and interest rate instruments. I'm looking into some of the very very liquid ones and see how they are linked to P(t,T). I told you already about the importance of P(t,T), but the question is when I go to LIBOR market, when I go to swap market. How do I see that these rates gets linked to the so called 0 coupon bond curve? I closely look into their cross correlation, why? Why is so important? The reason is because I want to show it to you that when you going through let's say LIBOR rates with very different maturities or swap rates with very different terms. Actually if you just look at one against each other days, they could be highly highly correlated depending on the environment. And that means if you really heavily trade one side, it going to impact the other ones as well. Now we assume a model, you will see will discuss various different models. But we will be focusing on just two of them and then assuming that a model for evolution of an interest rate. There are limited different interest rate. But again we will be focusing on just one of them and we perform model calibration.