It's time to introduce mathematical models in the course. We have done as much as we could without mathematical models, but to actually price options we need them. And we will start with simpler models, discrete time models and then later we will do continuous time models and Black-Scholes. So let's move to the next slide. The simplest 1 period model that 1 could think of is, the simplest discrete time model is 1 period model or single period model. So we only have time 0 and time capital T. Which probably just called 1 time 0 and time 1. And there is maybe financially many stock prices s1 to sK. And some probabilities pi, such the probability that the stock price in the future is a si=pi. Okay, so this is just a generic description of 1 period model. Single-period model. I'm going to assume that we have the following assets in the model, we have a bank account which is denoted B and the times 0 is going to be normalized to 1 and then B a time 1 is going to be 1 times 1+r. So 1+r. The interest rate is r in this period. Initial wealth, I'm going to use capital X to denote the wealth of our portfolio strategies. I'm going to have 1 investor who is investing in the market and he starts with X0 amount of money, the X0 is usually going to be denoted by lower case x. That's the initial amount of money the investor has. And then the portfolio strategy consists in choosing how many shares of each asset to invest in. That denoted by delta i. Delta is the usual location for the number of shares used in an option theory. So at the end of the period, the wealth is going to be X(1), delta 0 just denotes how much money you put in the bank. You can think of this as the shares in the banks. It's going to be dealt with 0 times 1+r times B(1)=1+r. That's just how much initial money delta 0 and the bank owes to plus number of shares of the money in asset 1 times the value of the price of asset 1 time 1 plus and so on. A number of shares in us at 10 times the value asset and at time 1, okay? So there's something called the budget constraint or a self financing condition which simply says that there is no extra money coming in or out of my portfolio. I just start with X and that's it. And then all the money stays in and nothing comes in from outside. And the way to present this in this single period model is that initial amount of money you have has to be split between all the assets that are available. So there's going to be initially delta o, B(0) is just 1. But if it wasn't 1, you just multiply by B(0), there's the money in the bank initially plus money in the stock at time 0 plus money in stock 1 and plus money in the stock N, times 0. So just split Initial money X(0) into all the assets that are available. Let's call the budget constraints. I see. Next. Monetization, Profit loss or gain. So, portfolio strategy is simply going to be the difference of what you have a time 1 X(1) minus what you have at time 1 X(0). And then I'm going to need notation for discounted version of whatever process y we are looking at. So Y part of the bargain discounting, discounting is simply dividing by the bank accounts, right? That's what discounting really means. So we are going to wipe t over divided by the t that denoted by bar of t, okay? Monetization, changing price is in audit capital delta Si(1) of asset i. Is Si(1)-Si(0). That's a change in price over the period in asset i. We could decompose the the total gains, profits and losses into profits and losses in each asset. So in the bank account it's a profit of delta 0 times r and then in others, it's just how many shares times the change in the asset price. All right. And we can also write this way, we can write X(1) as X(0) + the profits and losses. This is just going to be convenient later on, now it's just extra notation. Also let me do you know about delta Si bar the discounted version of the profit loss. Si bar 1 -Si(0). I'm not discounting here because there is no discounting at time 0. Okay, so and G bar is going to be a number of shares times the discounted profit losses for all the assets added up together. Okay, and it's an easy exercise to show that discounted wealth in the future is this kind involved in the past plus discounted profit loss, okay. Again, this is going to be useful later, not so much now, but later on when we go to multi period and continuous time models. All right, that was a lot of notation. Let's go on. A multiplayer model will simply be starting with the initial stock price which is known today, then goes to some possible stock prices tomorrow with some probabilities. Then if it's up here goes to some other possible stock prices in day2 or period 2, if it goes here goes to some possible prices in period 2 here and so on. So this is a two period model. So that's how it would generally look like. Okay, Can go on. All right. So this is the same type of notation for a multi period model. I'm going to use t for denoting which period I'm at. Again, I'm going to start a bank account or the risk free assets at 1 and then it's going to be 1 plus r, maybe r depends on t where the period times the previous the money in the in the bank account in the previous time. Number of shares in asset i during the period t-1,t is denoted delta i(t). Okay, that's maybe it's not the best notation in the sense that this is already known and t -1 at the beginning of the period t-1. You decide how much shares, how many shares to hold in asset i. That's delta delta i(t). So during the tth period, okay, that's the notation we use. The wealth process by definition, the wealth that you have in your portfolio time t, given that you hold shares delta 0. Delta 1 delta n in the assets is simply the, you add up all the money you have in different assets, number of shares times the asset number of shares times the asset number of shares times the asset. You have a map, okay, now there is a self financing condition for each period, it says there is no extra money coming in and out each and in any of the periods. So the way to write that, is this the same thing x(t) s also equal when I change these teas in the number of shares to t plus 1, why? Because this is this is the money during the period t minus 1, to t then st, I'm going to change from these number of shares possibly to some other number of shares that there's a plus one plus one. Okay and this is this is the amount I have just before changing and this is the amount I have. Just after changing the number of shares in each other, so this just says at the moment, I'm changing from a certain number of shares in the assets to some other shares. It has to be the same amount because I don't I don't have extra money so the managers before changing my positions has to be equal to the total money I have. Just after changing the positions that's what it says okay, in symbols, this has to be able to this but it just means money. Just before changing my positions is the same as the money I have after changing the positions. All right, that's I will need this to show you some effects on the next slide, again, the same notation is before changing prices capital Delta during the period. The T games process is the sum of all the profits and losses in each assets so now I'm betting over time, I'm heading over S 12 t all the changes, profit losses in the bank account, profits, losses in stock one. So I look at the change in price at each period times the number of shares that I was holding in that period. All right, so this is this is the total gains and losses and again, it's easy algebra to check that the time T wealth is equal to time 0 wealth plus all the games gains or profits and losses up to time team. Okay, there is so again, this is not really that we will need this much with this three time models but it's going to help us understand better continuous time always. Namely, let me just kind of tell you about that here to preview so in continuous time models I want to pen here in continuous time models, I will have something like this, except is going to be equal to x0 plus integral. And then there is going to be some overall assets so but let's say there is only two assets, it's going to be integral 0 to t let's say delta 1, the s1 It's good. The s1(u) plus integral 0 to t delta 2 so I would say there's two stocks and forget about the bank account. The s2(u) is just the variable of integration, it's time from 0 to t okay it's not quite clear what this means right now, but I will explain that later when we get to continuous time all this. But it's really it's just is just a generalization this because integral in the limits are nothing else but sums infinitely many parts. So we are going to replace these sums by integral, okay and this I forgot the deltas also can change with time. So they also depend on new delta1 of u, that the s2(u), yeah, so I'm going to have something like this later. And the way to understand this maybe is to think about this, it's really just adding profits and losses over each in this case infinitesimally small intervals, short intervals and you just add all those profits and losses continuously. Here's a discreet version, this is going to be a continuous version so this is really what I'm doing this now. Not that I really need is in this generality I it's just kind of help to help the intuition for later on when we use this model. All right, so discounting for similarly as before, we will have just sit here to this discounted change in price definition is the same as before you you subtract discounted profit, is this kind of profit loss? This kind of price at t is going to price at the minus 1 so you define discounted against process as number of shares, times this delta changing discounted price. And you can also check that if you discount the wealth process, it's going to be initial wealth plus discounted against process. This may not be that obvious actually, but I'm going to let's go to the next slide, we'll see In one example how this works, okay? So that discounted wealth is able to initial wealth plus this kind of games, I'm calling them gains, but they might be losses too good. So let's see how this this last line works out, so if I have just one risky asset, one stock and and two periods let's check that expression, so what I want to show is that okay? Is that x of 2? Actually, I'm not going to show that this kind of version I'm going to show that not non this kind of version even, that is not completely obvious. I'm going to show that except two is equal to x plus G, to that's what I'm going to show. Which is equivalent to this last line here, so that's what I want to do let's do that, this is just reminding you how we define the changing price. So GF two by definition is going to be the delta's during the first interval, so this is in the bank accounts delta zero in the first interval times the change in bank price in the first interval plus delta 0 and during the 2nd interval. So money in the bank during the second, several times the change in bank account during the second period plus. And these are the changes in stocks, a number of shares and start during the first period, change in price during the first period, number of shares and start during the second period, changing price during the second period. So what I need to do is actually use that self financing conditions. So self-financing condition, I'm going to use it in this form remember, it's that if I have delta one baskets of 1 + delta or the other assets of one or the other assets of one, it's the same as if I move deltas to the next period to 2. Okay, that was the self-financing condition before changing and after changing deltas, I have the same amount of money. So if I use that these two guys delta one, the 0 by 1 delta 101 as one is there is one here, right? So I'm going to replace them with these 2 guys, these 2 here, I'm going to replace with these 2 and they actually have them here is that they're going to cancel, this one is going to cancel this one that the 0 2- times -b 1. And this one is going to cancel this one, that one or two with the- times- s of 1 okay, so what is not cancelled is delta 0 2 B 2 Delta 1 or 2 SF 2. And then from here- delta 01 B of 0 and delta one A 10 with a- sign so I should do the algebra I did quickly now. But you will if you do this slowly you can check that you get this but this is nothing else. This by definition is access to money at a time to in the bank plus money time to in the stock that's X2 and this is X 0. Money initially in the bank and money initially the stock and the minor science our X2-60 year which is the same as except to being X 0 + 2. So this is what they wanted to show this is just one simple example all other expressions that we had can be shown in the same way. Okay that and let me now so we did very this very generally discrete time models, any type. In fact, in this course we are pretty much going to work with only 1 discrete time model so what is the simply as possible discrete time model for a stock? Which is interesting by interesting I mean there is an enormous, well the simplest possible 1 is having 2 possible values in the future for the stock at each moment okay? If I have only one possible, there's no randomness if I know where it's going, that's really the bank account, that's the risk free asset and there is no random, is there? So if I want to make it a stock, if I might want to make it risky, I have to have at least two and that's the simplest possible thing. So this is what the so called Binomial three binomial for having 2 values at each point or a cock strokes Rubenstein model. Is at each point in time the stock will go to one possible up value and one possible down value and cox trust and Rubenstein at all professors of finance. So this is called cox regression model they introduced this model, it's also called CR model for the initials of extras and Rubenstein. They in fact introduced this model historically after black Scholes in fact, a number of years in the eighties and black shorts was 1973. Even though this is mathematically much simpler model in fact, it converges, we will see, we'll talk about it later, it converges when the period goes to 0, it converges to the black Scholes. So they introduced a simple model historically after the more complicated model but that's how, it happened. And actually it's also very helpful, not only simpler for teaching, it's also a way to implement America created lectures model, but we'll talk about it later. All right, so the model formally is consists in stock at the future time being = stock at the previous time, time's U factor you and or stock at the future time is = the stock of the previous time times the down factor D. And in principle, we could introduce probabilities for going up and probabilities for going down so small P is the probability of going up 1- P. Is the probability of going down and there is a condition here that up factor has to be larger than 1+R and 1+ R. Has to be larger than D if you think about it a little bit, this is actually no arbitrage condition this simply means that the bank is sometimes better than stock. If the stock goes down the bank is better and if the stock goes up the bank is worse otherwise we would have arbitrage intuitively why? Well, because if the stock was always worse than the bank, you would just sell short infinite amount of stock or as much as you can and put all the money in the bank and make arbitrage money. If the stock was always better than the bank then you would just borrow as much as you came from the bank, put in the stock and make arbitrage. So to have no arbitrage, the bank has to be better than the stock if the stock goes down, but worse than the stock if the stock goes up. So that's this condition, we will see it later again all right so graphically by nominal three model is start with the norms value of the stock as the problems the BP goes to S times you again with the problems of BP. Goes to estimate a few times you as you square so each time you go up you multiply by you each time you go down, which is with probability 1-P you multiply by D. And one feature one and one consequence of this is that going up and down is the same as going down and up to multiply by UD or D you you get the same thing. So we call this recombining three up and down move brings you to the same value as the down and up moves so it looks like this. That's going to be the benchmark, discrete time model we are going to be pricing options okay, that's it for this set of slides thank you.
