All right. Let's start with our first set of slides. The first thing is going to be an overview of the course, just to tell you what this course is going to be about, and what it's not going to cover. What we want to do in this course is provide you enough background so that you can go on and learn more on your own or take other courses or apply it for your trading or for the job. Basically, give you enough material that you are comfortable enough to continue on your own, or you may want to do more studies, including graduate studies on this topic, work in industry, your own business, whatever you find this useful for. A little bit about the prerequisites in terms of what you need to know to successfully take advantage of this course, you need to know your calculus, especially now also probability and statistics based on calculus, be able to compute the expected values as integrals, for example, especially related to normal distribution. Also, it's helpful to have some basic knowledge of differential equations. It's not strictly mandatory, but it's going to be easier to keep up with the course if you know what differential equations are. It doesn't have to be much more than that. I will do it from the scratch in the course. But for example, we will have to know how to solve ordinary linear differential equation. For some problems it is also useful to be able to do a little bit of programming or at least be able to use Excel to compute formulas or maybe Visual Basic to program something or any other programming language. Let me give you a rough outline of the course in terms of what is going to be covered throughout the several weeks that we have. First, we are just going to make sure that we all know what the basic securities are and that's going to be stocks and bonds. Then we are going to introduce derivative securities in particular options. After we talk about basic uses of options and basic definitions, we are going to start pricing payoffs that are paid in the future, and that's the basic topic, the main topic of the course is how to price something that will be delivered in the future. We're going to start with the easiest possible thing to price, which is pricing payoffs, which are known, you know you will get a $100 in three months, what is the value of that today? Those are going to be deterministic pairs, deterministic in the sense that they are not random, they are non-stochastic. But the same principle that we will be using for pricing deterministic pairs is also going to be applied to random payoffs. That's why they're going to go through the deterministic pairs first. It's really just the usual present value computations, but we are going to try to make a connection with random payoffs in terms of the basic principle of pricing. Also related to that, we are going to define interest rates, spot interest rates, forward interest rates, and bond prices, bond yields and things like that. Then the main part of the course is going to be about pricing stochastic random payoffs. In particular, the main example is going to be pricing options, and how are we going to do that? There is going to be the main assumption, and basically for awhile, the only assumption we are going to make is, there is no arbitrage in the market, meaning that there is no way to make money, my profits without risk and without losses from zero investment. That's going to be the principle of no arbitrage. Under that assumption, at least in some of those, is going to be possible to uniquely define the price of payoffs, and those models are called complete market models. We are going to define that later. But in particular, our main model, which is going to be the Black-Scholes model, and the discrete version of the Black-Scholes model called Binomial Trees model. Those are going to be complete market models, whatever that means. But in particular, that's going to mean, there's going to mean that there is a unique price for each random payoff or random or non-random, and that makes it much easier to decide what the price is since there's going to be only one consistent with no arbitrage. That's not necessarily realistic, but it's the benchmark, it's the basic model to start from and keeps things simple, having one and only price for each payoff. Now in reality, you'd have a range of prices. You have a bid-ask spread, and people don't necessarily agree what a fair price is but for the most part of this course, we are going to concentrate on these models where there is a unique price. To actually do the Black-Scholes model, we need a little bit more serious mathematics. In particular, we will talk about the so-called stochastic calculus or Ito's Calculus and the main rule, which is called Ito's rule, introduce a Brownian motion process. These are all the mathematical tools that we need to understand exactly, the Black-Scholes model and the Black-Scholes-Merton type of pricing. Then, we're going to have an engineering approach to that, meaning, that we will learn how to use it. When you use standard calculus, you don't necessarily need to remember how to prove those theorems, you just need to know the rules of calculus to be able to apply it. That's what we're going to do with stochastic calculus. We are going to learn the main rules and how to apply them, in particular, to derive Black-Scholes formula and similar pricing rules. That's what you're going to do next. We are going to derive the Black-Scholes formula, variations on the Black-Scholes formula, for different options, different situations. One main part of the course is pricing things, the other part is hedging or risk management in terms of having to deliver a random pay off, how can you make sure that you have enough money to do that. There's going to be a notion of replication of random payoffs or hedging of random payoffs. That's the second main thing that we want to do in this course: pricing and hedging. We will do that for most of the course on examples, mostly using equities, meaning, stocks. Then, at the end of the course, we are also going to talk about fixed income derivatives and options in the bond market, the markets where fixed income instruments are traded, in particular, bonds. What is not going to be covered in the course? We are not going to go into impractical implementation issues, which are really what is difficult in the real world to do. Once you know this theory, to actually apply it, you have to decide what the parameters of your model are; that means either some statistical estimation or calibration, I'll talk a little bit about that, but very little. In this course, we are simply going to assume that we know those parameters, somehow they had been estimated or calibrated, and then we produce pricing formulas or hedging strategies. Now, in reality, of course, you have to decide on what those parameters are. We are also not going to do numerical methods. Mostly, we are going to look at examples in which you can actually find analytical formulas, though there is no need for numerical methods. But in many cases, you cannot actually find analytic and explicit solutions, so numerics is necessary. But that, we will not do in this course. Just maybe another final remark on why we are doing bonds at the very end. It turns out, even though bonds, you know what they will pay in the future, let's say, $100 three months from now for stocks, you don't know that; however, for if you're thinking of a Google stock, that's one point that is moving in time that you have to model. That's one single value that changes every day or every second, whatever, but it's a one-dimensional thing that is moving through time, one-dimensional stochastic process, and that's what the Black-Scholes-Merton model is going to model. Now, with bonds, let's say, US government treasury bonds. You have many bonds of the same type: you have a three-year bond, a five-year bond, a 10-year bond, in fact, the bonds of the same type with many maturities. In fact, what we are going to assume typically in a mathematical model that you have, infinitely many, continuously many maturities of the bonds of the same type. It's not just one point that you have to model, you have to model infinitely many points moving at the same time through time. It's a multidimensional, you have an infinite dimensional modeling problem. You have to make sure that modeling three-year bond makes sense versus the model for the five-year bonds so that there is no arbitrage in your model. It has to be completely consistent. This is why modeling bonds is actually harder because, simultaneously, you have to model bonds or many maturities in a correlated way. They have to be correlated in a way that makes sense. But this is jumping way to ahead, that's going to be the very end of the course. Right now, this is all for the overview.