Hi. Let's start with the second week of the course and what we are going to talk about first is pricing deterministic payoffs. Payoffs for which we know exactly how much they will pay and we also know which times in the future they will be paying at. This is usually called present value computation since an MBA course, but we want to point out here the principles that we will be using also when pricing random payoffs like options. The first thing we want to do is we're going to assume there is a risk-free asset in the market. In fact, the only asset we are going to be looking at is borrowing and lending into a bank account. The bank account with a deterministic known interest rate so when you put the money in the bank, you know exactly how much you will get after a certain amount of time. The bank would quote you an interest rate and it's really a matter of convention or agreement, what kind of interest rate the bank quotes you. If you understand the definition of the interest rate then you can compute how much one dollar will be worth in whatever time from now. Let's say the annual interest rate is denoted small r like here. If we invest one dollar today, so I'm going to call it present value for today, then the future value after one year is 1 plus r dollars. That's simply by definition of the interest rate. After T years, you can have different definitions, different understandings of how the interest rate will be computed. For example so-called simple interests, one dollar becomes 1 plus T times r dollars. But that's not usually the way banks quote interest rates. More frequently or pretty much always, banks would quote interest rates as compounded at certain time intervals. The simplest case would be compounded just once a year, which remains that after one year, the bank would add to your capital the interest and then after the second year, it would compute also interests on the initial capital and the interests of the first year. After one year, if you have 1 plus r, then it would compute after two years also interests, not just on one dollar but on 1 plus r dollars, including the interests that you received for the first year. It would be 1 plus r times 1 plus r after two years, so 1 plus r squared when T is two. Or whatever T is, you would just keep multiplying by 1 plus r depending on how many years have passed because you are computing also interest on interest. Compounding interest, but only once a year. Now in practice, typically banks would quote interest compounded more frequently than once a year. Typically, I'd say quarterly. In general, if the interest is compounded n times a year and we are looking at how much money we will have future value after n compounding periods, then the general formula for how much one dollar would be worth n periods from now, is 1 plus r over n to the power m. R would be called the annualized nominal interest rate and you divide it by the number of compounding frequencies, the number of times that the interest rate is compounded per year, then you put to the power of the number of periods that corresponds to the future time that you are looking at. Again, it's a matter of agreement. You just have to know what the definition is. In this case, how many times a year the interest rate will be compounded, and how many periods we are looking at. That's the general formula. Let's move to the next slide. Typically you compare different interest rates relative to one year. You can always look at the so-called effective annual interest rate which I denote here by r prime. If you compound n times a year, then after one year you will have 1 plus r over n to the power of n, because there's n periods in a year. That would be the actual amount of money that your one dollar becomes after one year. Then you define the effective annual interest rate as the number r prime such that 1 plus r prime is equal to this 1 plus r over n to the n. An example, quarterly compounding at a nominal annual rate of eight percent, it really means just that after one year you have 1 plus 8 percent over 4 to the 4. That's how much you have after one year, which happens to be 1.0824, which means that the effective annual interest rate is 8.24 percent. That's how much more money you will have, how much interest you'll receive on the dollar during one year. In this course, at least when we get to the continuous time models, it's going to be more elegant mathematically to do continuous compounding. Imagining that the bank compounds interests all the time continuously. As soon as you get interest on your dollar, then you get also interest on interest, interest on interest on interest, but continuously, every millisecond, whatever. That just means mathematically in a stylized way that you just take a limit when n goes to infinity here for your future value of one dollar and for the money that you will have after one year, if the interest rate is compounded continuously. This limit is known as the exponential function. The limit of 1 plus r over n to the n when n goes to infinity is the exponential function. This is just exponential function of r. That's just by definition. If I tell you that I'm giving you interest, which is continuously compounded, by that I just simply mean that for each dollar after one year, you will have e to the r dollars after T years. You just put this to the power of T, which is really multiplying by T in the exponent, which is also multiplying by T in the exponent here. After T years, it simply means that if r is continuously compounded interest rate, you will have e to the rT dollars for each dollar that you initially invested. These are, again, just the conventions definitions. You just have to know which interest rate a bank is quoting. For the continuously compounded rate, is it going to be more what you get after one year relative if you're looking at the same value of the rate, but it's not continuously compounded, but let's say quarterly compounded? Well, it's going to be more. The bank has compounded your interest rate more frequently so you will get interest on interest more frequently, so it should be more. Indeed, if we compute for our eight percent, suppose now this is continuously compounded rate, then e to the r happens to be 1.0833, which means that after one year you get effective interest of 8.33 percent, which indeed is higher that 8.24 percent. That's the highest you can get as frequently as you can compound it. This is just different definitions of interest rates. Now we want to reverse this procedure. I'm going to look at how much something that I will get in the future and I want to know how much this is worth today. What is the value, what is the price of this deterministic payoff which I will get in the future. What is it today? This is the typical present value computations and notions, but I'm going to make it a bit more complicated than it has to be. Because I'm going to already apply the principle which we will be using for pricing random payoffs. In order to extend the logic from the domestic payoffs to random payoffs in this present value idea, I will make a little bit more complicated than it usually is. I'm going to use what you could call the law of one price. What is the law of one price? The law of one price says that if you can create the same amount of pay off in the future in two different ways, then those payoffs, since they're going to be the same payoffs, they have to have the same price. No matter which way you create certain payoff in the future as long as it's the same payoff, it has to have the same price. Or here I'm saying it a bit differently, if two cash flows, so two sequences of payments have delivered the same payments in the future also at the same times, they have the same price today, they have the same value today. This sounds logical and it is, because otherwise you could sell the more expensive one and buy the cheaper one and make arbitrage. It's a natural thing to accept. Then thinking about this law of one price, I'm going to define the price of payoff of X of T dollars, whether it's random or deterministic. If I know that I can have X of T dollars at time T by investing X of 0 today, then today's value, today's price of X of T should be X of 0. The point here is maybe I can create X of T by buying a bond or maybe later we will see buying an option. But maybe I can also create it just by trading, let's say in the underlying stock and the bank account. No matter which way, I can create a payoff, if I know for one of these ways that I have to invest X of 0, then that should be the price of that payoff. This is basically the law 1 price. If I can create the payoff X of T, maybe by trading an options, maybe by trading stocks, bonds, whatever, whichever way I can create it, the values should be the same. It doesn't matter the price should be the same no matter how I create it. If I know that I can create it starting by X of 0 investing X of 0, then I know X of 0 should be its price today at time zero. Here in this set of slides, we are talking about deterministic X of T. So X of T say $105, and suppose I know that I have a bank which allows me to lend money at five percent and borrow money at the same rate of five percent. That means that after one year, I can get $105 by investing $100 today, so the price will be $100. If the rate is five percent, the price of $105 one year from now it should be $100 today because by investing $100, I can have $105 in the future. That's how we're going to define the present value, denoted PV of X of T. It's really just the amount that I have to have today if I put it in the bank, I'm going to have X of T in the future. Formally this is the definition. If the rate is compounded n times a year and I have m periods, so the X of T will be paid m periods from now, compounded m periods from now. Then the price today, what we also called present value, is just going to be the future value X of T divided by 1 plus r/n to the m, why? Because if I multiply this means that X of 0 times 1 plus r/n to the m is X of T. Which means that if I invest X of 0 today in the bank, I'm going to have that times this 1 plus r/n to the m factor, which is X of T. I'm going to have X of T at time capital T and m days from now. It's simply the present value is dividing the future deterministic value X of T by these compounding factors. By this definition, because that's how we get how much we have to invest today to have X of T in the future. That's the definition of the present value, a little bit more complicated than it would really have to be but because we are going to be using the same definition for options. When we divide by this, we are multiplying by 1/1 plus r/n to the m and that's called a discount factor because the price today is going to be lower than the value tomorrow. In the continuous case, you're discount even divide by e^rT, which is the same as multiplying by e to the minus rT. In the continuously compounded rate case, the discount factor is e to the minus rT, and so that's the definition of present value. It's easy to extend this to the present value of a sequence of payments called the cashflow. It's just going to be a sum of the present values of each payment. If I have payments today of X of 0, tomorrow X of 1, after two periods X of 2, after n periods X of n, and these are exactly the compounding periods intervals, and then the present value of a sequence of payments like that is simply the sum of the present values. I have to discount each one of them relative to the period in which it is paid, so here is to the power 1, to the power 2, to the power n. That's the present value of a sequence of payments, so cashflow. It's simply the sum of the present values. There is one special case which is convenient for applications. If I look at X of 0 equal to 0, so imagine I'm not getting anything today, but then I get the same amount in the future at every period. So X I's are all equal to X for I 1,2 and so on. Then we have a formula because have a geometric summing so-called geometric series. This is the present value and there is a formula for this sum and the formula is here. This is convenient. We are going to have examples in the following slides with loans. If I'm paying each month X dollars for my, let's say house loan for my mortgage, and I know that the interest rates, annualized rate, nominal A's are, and it's compounded n times a year, and my loan is going to be paid over m periods. Then I know everything to be able to compute the present value of those payments. That's the formula.
