Fixed income markets are enormous and in fact, they're bigger than equity markets. According to SIFMA, in Q3, 2012, the total outstanding amount of US bonds was $35.3 trillion. In comparison, the size of the US equity markets was only approximately $26 trillion in comparison. Fixed income derivatives markets are also enormous. They include interest rate and bond derivatives, credit derivatives, mortgage-backed securities, and asset-backed securities more generally. In this section, we're going to be focusing mainly on interest rate and bond derivatives. We're going to use binomial lattice models to understand the securities, the mechanics of these securities, and also how to price them using risk-neutral pricing. We're going to use binomial lattice models to do this. The slides and Excel spreadsheets should be sufficient, but Chapter 14 of the Luenberger text is also an excellent reference for the material in this section. We're going to use binomial lattice models as our vehicle for introducing the mechanics of the most important fixed income derivative securities, bond futures, and we'll also discuss forwards, caplets and caps, floorlets and floors, swaps and swaptions. Now, I should mention at this point that the interest rate underlying these securities are the so-called LIBOR rates. Whereas the interest rate underlying bond futures, for example, would be government rates. These are different interest rates, but we're not going to make that distinction in these modules. We're going to assume it's the same underlying rate and a reason for doing that is twofold; One, we just want to get familiar with the securities and the mechanics; and two, we want to get more practice with risk-neutral pricing. It would only distract us if we have to concentrate on different rates for different securities. We're also going to discuss the philosophy behind fixed income derivatives pricing. It's actually quite a bit different to the philosophy underlying the binomial model for stocks that we saw recently. We're going to use binomial lattice models as a vehicle for introducing: number one, the mechanics of the most important fixed income derivative securities; and number two, the philosophy behind fixed income derivatives pricing. Going to one, we'll mention that we are going to focus mainly on the securities, caplets and caps, floorlets and floors, swaps and swaptions. I should mention that the key interest rate underlying these fixed income derivatives are the so-called LIBOR rates. On the other hand, government risk-free rates are the interest rates underlying bond futures and bonds more generally. We're going to use binomial lattice models as a vehicle for introducing both the mechanics of the most important fixed income derivative securities, as well as the philosophy behind fixed income derivatives pricing. We'll talk more about the philosophy soon, but let's look at some of the most important fixed income derivative securities. We're going to talk about bond futures and also forwards. We'll also talk about caplets and caps, and floorlets than floors, and swaps and swaptions. Now I should mention at this point that LIBOR rates are the interest rates that underlie these securities here, whereas these securities typically have government rates underlying them. These are different interest rates, but we're not going to make that distinction here in these modules, we're going to assume that it's the same underlying rates. Our reasons for doing this are twofold; we want to focus mainly on the mechanics of these securities, how they work, and how they're priced. We're going to price them using risk-neutral pricing, and it would only distract us if we have to focus on different interest rates when we were pricing different securities. Now, fixed income models are inherently more complex than security models. The problem with fixed income models is that we need to model the evolution of the entire term structure of interest rates. For example, Let's come down here and let's see a little plot of the term structure of interest rates. We have time t here, and we have s_t here. s_t is going to stand for the spot interest rate at time t, and maybe you've got some function like this. This tells us what the term structure of interest rates look like. For example, if this is t1, then we have over there and that's s_t1. S_t1 is then the spot interest rate that applies to borrowing or lending at the time t1. When we want to build a fixed income model or a term-structure model, we need a model which will model how this entire curve moves through time. Note the distinction when we have a model for stock prices, we just need to model the evolution of a single stock, a scalar random variable. Here, we need to model the entire evolution of this term structure. Term-structure models are inherently more complicated. But actually, we're going to see there are some easy ways to get around this problem. One of the classical ways to get around this problem is to focus on what's called the short rate. The short rate or little t is the variable of interest in many fixed income models including binomial lattice models. It is the risk-free rate that applies between periods t and t plus 1. R of t is going to be the risk-free rate that applies from period t to period t plus one. It's a random process or is random. Remember, interest rates in the real world are random, and so the short rate will also be random. However, r_t is known to us by time t, so at time t, we know what we're going to get at time t plus 1 if we deposit $1 in the bank account at time t. r_t is random, to repeat it again, but it is known to us at time little t. We're going to use as I said, s_t to denote spot interest rates and r_t to denote the short rate. The short rate is the rate that applies for just a single period.