 Okay, so the first thing that we're going to do today, we're going to work on a very, very simple basic model. It's called the binomial model, single step binomial model. But we're going to look at it from a perspective that you probably have not quite seen before, even if you have seen the binomial model. So what's the idea? The idea is as follows. Suppose I have an asset today, and I know what I have, and I know the following bit of information. I know that there are only going to be two possible values for this asset at some future date. I know what those two possible values are, and I know the probability of those two possible values. What I don't know is its current value. So the way I'll draw this is I'll sketch it by saying I have an asset today. I don't know what its current value is, but I do know what its two values are at the end of some time period. AU and AD are known, and P, this is what I'm going to call the branching probability, are also known. And the question is, what is a rational price for the initial value of that asset? So this is even before introducing anything into the, this is before introducing, you know, no arbitrage arguments or anything of that kind. It's just going back to basic economic theory. You have two random outcomes, or a random outcome taking on two possible values, and you want to know what is a reasonable value for that game, for that, you can think of it as a lottery or a game. So what's your, anyone have a reasonable answer as to what that initial value should be? Okay, so we might think that perhaps I'll put equals question mark. So if R is our discount rate that we're discounting future cash flows at, we have a discounting, and we're going to compute the expected value of that asset at time one. And of course, since there are only two possible values, the expected value of that asset at time one is simply P1 minus P in this way, okay? Okay, so that's one reasonable answer. Anyone have an alternative? So let's say this was a fair coin flip game. P is a half. And if it comes up heads, you win a dollar. If it comes up tails, you lose a dollar. According to this pricing rule, you would pay nothing to play that game, correct? Now the question is, would you actually do that as an agent, as an investor? Would you pay nothing to play that game? If I said, okay, right now, okay, you're going to pay nothing? Okay, let's flip a coin right now, and I'll give you a dollar if it comes up heads, otherwise you owe me a dollar. Do people actually behave that way? They don't. Why? They're risk averse, exactly. So people are risk averse agents. We're actually averse to losses. And I mean, we like gain, of course. That's what we want. And we need to account for somehow investors' risk aversion. What's the way that, what's the economic background for accounting for risk aversion? What do you normally do? How do you prefer one thing over another? How do you make your preferences? Use utility functions, right? So if you have two games, or let's not talk about two games, if I have some amount of wealth, or some random wealth, what I normally do in order to find out whether it's something I prefer or not, is you want to compute the expected value, not of the random outcome itself, but of the utility of that random outcome, okay? This is the expected utility of terminal wealth. Now, this itself does not tell me how to price anything. It simply tells me that, okay, if I'm going to compare two things, let's say I have random outcome one and random outcome two, and I want to order them, I want to know whether I prefer random outcome one to random outcome two, I would compute the expected utility of random outcome one, compute the expected utility of random outcome two, and whichever is larger, that's the one I prefer. So you'd say that this is, and this is sort of a curly less than or equal to sign, okay? So it means I prefer x2 to x1 if and only if the expectation of u of x1 is less than, and you can put it the less than or equal to sign there. There are different choices for that. So I would say I prefer, let me write this in sort of English language here, I prefer x2 to x1 if and only if expected utility of x2 is greater than, and then we have to talk a little bit about what are utility functions actually, right? It's greater than expected, I'm going to shorthand this utility of x1, okay? So this is a sort of standard base economic theory. Now what requirements must I have on u for it to be used as a utility function? What sort of a natural, what are natural things? So there's one that's obviously natural. Should it be, okay, can, if u, if this is a little x, can this be a utility function? And why is that? Yeah, you don't want to have more wealth and less utility, right, for one. So clearly you have to be increasing. Now it could be increasing like this, and again, just something wrong with that, right? I have more wealth in the same utility. I should always prefer more to less. So what about this? Okay, there's a problem with this one because if you take some average wealth in between these two points, my average utility is bigger than the actual utility of the average, and you don't want that either. So utilities have to satisfy the three, you know, the three standard tenants of utility theory, basically. Or the requirements that, okay, u is a mapping from r to r, and u of x is increasing, and it's concave, okay? Concavity is what's preventing you from having the linear combination being larger than its individual utility, okay? So utility functions typically look like this, okay? Now there are some other technical conditions such as the Ionata conditions out of 0 and a plus minus and a plus infinity, but we will shove those under the rug and just say that this is all that we're going to really require on our utility functions. Okay, a classic example of a utility function, and we're going to use this one thoroughly right now, is exponential utility. So this is, gamma is some strictly positive, strictly positive constant. It represents the risk aversion level, okay, or the level of risk aversion of that agent. If gamma, so let's see, why is that a reasonable thing? Well, if gamma is very large, what would this function look like? Well, it starts at 0, correct? x equals 0, 0. It increases, and it goes asymptotically to 1 over gamma, right? The vertical scale actually doesn't matter, right? That scaling, the fact that I put this divide by gamma there doesn't change anything, right? When you look at the preference scene, gamma will cancel, always, from the denominator. It's a constant. But it's used, you'll see why it's normalized in that way in a second. So the vertical scale is not so important. But if you changed, if we sort of removed this denominator term, if we just ignored that, and all we did is changed gamma, a large gamma would push this curve to 1 faster. Agree? Because I exponentially decay to 1 more quickly. And that means that my, you know, I'm very averse. Small changes, if I have a small change in my wealth here, that's going to cause that much change in my utility versus that much change in my utility of a larger, of an agent with a larger gamma. So the increase in utility by an increment in my wealth is larger for the larger gamma agent. So gamma represents the risk aversion of that agent. And this also explains to you why this also has connection to another concept about this marginal utility. Imagine your wealth just as an agent with this level of gamma. If I get one extra dollar, my utility increases by that amount. But if I'm more wealthy and I get an extra dollar, my utility only increases by a little bit. And that's how people behave to some extent, because you can, you know, think of the simple example. You're starving, you get a hamburger, you're going to eat it, right? You love it. You just had 10 hamburgers, you get another hamburger, what are you going to do? Well, you might eat it, but you can take it or leave it, right? It doesn't really add value to you. So the more wealthy you are, the less one, an increment of one unit is going to cause to your utility. And that also helps to explain this concavity issue. Okay, so gamma is this level of risk aversion. Now I said that we divide by gamma there. It doesn't actually affect the ordering of preferences. But the reason why it's there is that if you take the limit as gamma goes down to zero, what happens to you? It just becomes x, okay? And how do you see that? Well, you just do a Taylor expansion of the exponential, right? The exponential is even negative gamma x. That's approximately one minus gamma times x. And the ones cancel out. You're left with the gamma times x. The gamma's cancel and you end up with x. By the way, if I'm saying anything that doesn't make sense or I said it too quickly or anything of that nature, please put up your hand. Let me know because if I did, for you, likely someone else is also wondering what's going on. So please do ask questions. Okay, so this is one of the reasons is that then the utility basically becomes preferences with respect to this type of utility function in the limit as gamma goes to zero corresponds precisely to expectation of random payoff. U of x is x. So my preference in here is expected value of x1 is preferred to the expected value of x2. Okay? So that's the reason that it's normalized in that way. Okay, so what I'd like to do now is somehow let's go back to our original problem, right? This was a little sidetracked just back to basic economic theory. Let's go back to our original problem. How can you incorporate that idea about utility functions to help me figure out what is a rational price for a zero? What can you do? Well, when we talked about this, if you think about it for a second, what we talked about just now was comparing one game to another game. You said x1 compared to x2. x2 is preferred if the expected utility of x2 is bigger than the expected utility of x1. Here I just have a single game. So you need another alternative game. What's the natural alternative game for you to compare with? Is not play the game at all, right? So buy the asset for the price of a zero. So that means my initial wealth gets reduced by a zero. But one time step later, my wealth will be increased now again by AU or by AD. That's game one. Game two is I don't buy the game at all. All I do is I take my money, and what does do nothing in financial terms really mean? Yeah, it can mean, well, cash, but if there is a risk-free investment for me to do, I would take it and put it into the risk-free investment. So those are the two games that we're going to compare. And then we're going to say, well, I'm going to choose a zero such that I actually don't prefer game one to game two. They're equally preferable. I'm going to choose them so that their expected utilities are identical. Then it wouldn't matter if I played the game or I don't play the game in terms of utilities. And that's now an economically sound rational price to charge for that asset. And it will depend on the agent's level of risk aversion. And in general, that of course depends on the agent's utility function. We'll see here, for this specific utility function, you can carry out the calculations all the way through the end and get a very nice simple intuitive result. Okay? So let's do that. So that procedure, what I just described is called indifference pricing or certainty equivalent price. So you're going to find a zero such that I'm just going to write this sort of pneumonically and then we'll write it correctly in a second. Such that the expected utility of game one is equal to the expected utility of game two. And in game one, this is the do nothing game, okay? And this is the buy the asset game. And that will allow us to determine a zero. So let's go ahead and see. So what is the random variable x1 represents, actually? What is it equal to? Well, since we did nothing, it's just equal to however much money we had, little x, that's our initial wealth, grown at the risk-free rates because we're, let's put in the bank account, say that there is a bank account there, otherwise it will be zero. If this was a game that was played over one second, r would be zero, right? If it was something that was over a year, then r would be non-zero, it would be about 2% these days. Okay? So x1 would be that, and this is not a random variable at all. So in fact, the expectation on the left is not random. Okay, what's x2? Well, x2 is, first, I need to give up, I need to buy the asset. So I purchased it for A0 dollars. So this is how much money I have left in my hand. Right? I had $100, it cost a dollar, I have $99 in my hand. I put the $99 into the bank account. So this is going to grow at the risk-free rate. Then, at maturity, at that one time step later, I'm either going to get A0, sorry, AU, or I'm going to get AD with the two probabilities. Okay? Now, we can write this in a couple of different ways. I'm just going to write it for now with plus A1. A1 is our random variable that represents the asset's price at time 1. Okay? So it's AU with probability P, and AD with probability 1 minus P. Make sense? Okay, so now we simply compute those two expectations. So the left-hand side, we already know what that is. That's a constant. It's e to negative gamma, little x, 1 plus r, all over gamma. And, yeah, I have to be careful. Sometimes my gammas on the screen look like ours. Okay? And then the right-hand side, what are we going to have? Well, we have 1 minus expected, since the expectation operator is linear, we can just pull it to lie under here. Here, we'd have e to the minus gamma times that times 1 plus r minus gamma times A1. And this whole thing is being divided by gamma. So, of course, like we said before, gamma just in the denominator just cancels out. The 1, in fact, even just cancels out. The minus sign cancels out. And there's a couple of other nice cancellations. Well, all of this is non-random, right? That's not random. So that can just be pulled out. So that can be pulled out of everything. And, whoops. I don't know what's happening here. My pen. There we go. So we have e to the negative gamma, little x, 1 plus r. That's e to the negative gamma, little x minus A0, 1 plus r. And then you have simply the expected value of e to the negative gamma, A1. Now, here's the other nice thing. This cancels exactly with that term. X just gets killed. My initial wealth is irrelevant. And what we're solving for is A0. So I can simply put that exponential factor on the other side, take logarithms, and I'm done. A0 is minus 1 over 1 plus r, log. The expected value of e to the negative gamma, A1. That is my rational price. Now, I didn't specify anything specific about A1 here. In this analysis, I didn't assume that A1 took on only two values. It could actually be multiple values. It could be continuous, a continuous set of outcomes. This analysis doesn't depend on that fact at all. And yet, at the end of the day, it gives you something which depends on the level of the risk aversion of that agent. Now, you might want to try to understand how does this cooperate with the initial sort of natural guess that we had earlier. Well, if gamma goes to zero, you know that that's when the utility becomes linear and that represents a risk-neutral agent, because then they're just using expectations to order preferences. And what happens with this expectation as gamma goes to zero? So, assume you can interchange the limits and expectations. Well, if you can, you can do an asymptotic expansion here. And then this is 1 minus gamma times A1 plus little o gamma. Stuff that goes away as gamma goes to zero. You have the log of 1 minus gamma times a constant. The log of 1 plus a constant is approximately that constant. And you'll see that something will cancel. And I just realized I have one small mistake here. Does anyone tell me what it is? Gamma's missing in the bottom. There's a gamma down there. I mean, there's a gamma up in front of that A0, so it's got to have gamma down here. Okay, so let's take the limit as gamma goes to zero. And then we're going to take the other limit of the completely risk averse, the strongest risk aversion agent. Gamma going to plus infinity. And we'll see something interesting happening. So, gamma goes down to zero. We can say that this looks like log, expected value, 1 minus gamma A1 plus little o gamma. Okay, remember what little o gamma means? Basically means it's something, if I divide it by gamma and I take its limit, that's going to be constant, right? So, in other words, this is basically, or sorry, go to zero. So, that basically means that this term here is effectively of order gamma squared, or higher. So, for small gamma, it's negligible. Okay, now if you, now you can see that expectation just goes, it goes under the, you can write it like this, 1 minus gamma, expected value of A1 plus little o gamma. And then I use the following fact. So, by the way, I already used the fact that e to some power alpha for alpha small is 1 plus alpha plus little o alpha, right? I've already used that. That's what I used to go from there to there. And now I'm going to use the fact that if you take the log of 1 plus alpha for small alpha, that's equal to just alpha plus little o alpha. This comes from just Taylor expansion basically. So, if you don't remember your Taylor expansions for these things, go back and look at them. So, using this fact that I just wrote down, I can simplify that logarithm, right? Gamma is the thing that's small and it's multiplying by, I have to make one assumption now. I have to assume the expected value of A1 is finite. And in fact, I also have to assume that the expected value of the exponential of A1, e to the minus A1 is finite. And this actually puts requirements in the boundedness of A1, but we'll ignore those little extra details. So here we see we get gamma times expected A1. That's what's going to come out. And that gamma is going to be cancelled with this gamma. The minus sign will cancel that minus sign and we get exactly the result that we intuitively thought right at the beginning. Plus little o gamma. So it's something that's going to go to zero with it. And that's our rational price. Questions about that? Okay, what's going to happen as gamma goes to plus infinity as I become someone who is extremely risk-averse? So this is the risk-neutral limit. So we have the risk-neutral limit. We know that the price is, in fact, just discounted expectation of my random payoff. And this doesn't depend on whether that random payoff is just two states or not. It's just some random payoff that has a bound in this requirement. It has to have a finite expectation. What about as gamma goes to plus infinity? Forget about the equations for a second. Let's think about it from the perspective of just financial intuition. If I'm extremely risk-averse and I'm going to buy some random outcome, as risk-averse as I could possibly be, what price would I pay for it? No, I wouldn't pay zero. I mean, sure, I would pay zero, but I'd be willing to pay more. Of course, if someone gives me a random payoff, and assuming the random payoff is positive, they have another, exactly. I have a random payoff. I look at the worst possible outcome that's going to happen, and that's the most I'm willing to pay, because I'm so risk-averse that I'm putting all my financial intuition to tell me, I don't care about any positive, about any other good outcomes. I just care about what is the worst possible thing that can happen, and that's as much as I'm willing to pay. So what we should see come out of this is 1 over 1 plus R times the minimum value of that random outcome. Now, if the random outcome isn't bounded from below, then you wouldn't buy it. You'd actually require money to purchase it. You'd actually request money in addition to being exposed to that risk. So let's take a look at it just from the perspective of, actually, let's say A1 is a random variable bounded below by AD. So because we know that we need a bound in this requirement on it in order for this limit to work. So let me just add a sentence here. This is the risk-neutral agent, and now we're going to look at the infinitely risk-averse agent. So for the infinitely risk-averse agent, what we're going to do, given that financial intuition that we've got there, what we're going to try to do is take, sorry, there's a logarithm missing here. So that was our result that we had from above. I just copied it over. What I'm going to do is, okay, this is something that's bigger than AD. It's bigger than or equal to AD. So I'm going to multiply, I'm going to add and subtract AD up in that exponential but what I end up with is a random variable that's strictly positive or greater than or equal to zero, I should say, because if it's bounded, so I'm going to do this. I'm going to say subtract AD plus AD. I'm perfectly allowed to do that, aren't I? And here I'm going to put an extra requirement here. A1 is bounded from below by AD. So the reason for doing that is now, if you look at this random variable, AD is not random but when I subtract AD from A1, which is random, I get a random variable. This random variable is now greater than or equal to zero. And why does that help me? Because it helps me because now I can take the limit in which gamma goes to infinity in a nice simple way. So first off, I'm going to factor out this AD term here, write it as, put it as a product outside. So I'm just going to do that in front of your eyes here. So I get e to negative gamma AD like that, right? I have the log of a product of things. That's the same thing as the log of the sum, or sorry, the sum of the logs. And hopefully, well maybe I don't need to do too many steps here. You're a fourth-year student. It's easy enough. Take the log of the product. So that gives me the log of the sum. I'll have the log of this plus the log of that. You can see that the log of this is minus gamma AD. That cancels the minus one over gamma here. And what I'll get is AD discounted minus this logarithm term, right? This other logarithm term. Log expected value e to negative gamma A1 minus AD. Now this is a perfectly valid result for any gamma. I've actually done nothing here that depends on gamma. This is perfectly valid for any gamma. But I do know that this thing here is strictly positive. So as I take the limit, as gamma goes to infinity, I can be assured that this expectation is going to go to zero. Sorry, this, yeah, so we got, what did I do here? I have something positive. Yeah, I mean what I want to argue is that this ratio is going to zero, right? That's what I want. As gamma goes to infinity, yeah, that's fine. Oh yeah, yeah, yeah, that's fine. That's, it's a little bit, sorry. No, no, no, gamma goes to zero and this isn't one. This is actually zero. You have the log of zero divided by something that's going to, yeah. So we need to, I think I made a small faux pas here. Let me just step back and look at this again. And there's got to be a way to argue that asymptotically this is going to zero. And right now it's eluding me, so I will just remind you of how it works on the web, okay? It has to do basically with the rate of convergence, right? Of the logarithm of something versus gamma. Gamma is going to infinity faster than the logarithm of that exponential. That's what we have to argue. And right now, for some reason, it's just eluding me. So the end, the point is that what you should find is exactly that intuition, that financial intuition that we gave a second ago. Okay, before we go for our first break, let's recap what we've done here. Okay, and then talk a little bit about the behavior of this as a function of gamma. So what we did here is we were looking at this simple question of how to value that asset right now if we know what the random outcomes are and the probabilities of those outcomes. We don't know which ones will occur, but we know the probabilities and the potential outcomes. First naive guess is discounted expected values. Okay, fine. It's not grounded in sort of sound economic theory. What is? Use expected utility. We're going to prefer one random outcome over another if expected utilities of those random outcomes are ordered. With this exponential utility, we went through, we thought we're going to use exponential utility and we're going to use the idea of indifference pricing to figure out a price. So there, since preferences say look at expected utilities, we're going to say I'm indifferent between this random outcome and that random outcome if their expected utilities are equal. Then you just go through the little bit of calculation and we end up with our very nice result that it's a discounted value not of the expectation but of the logarithm of the expectation of the exponential. You notice that what you have there is sort of a utility distortion of your payoff and then you undistorted the utility distortion being the exponential and the removing of the distortion being the logarithm. This is actually a very general and generic feature that you find for almost any utility. You'll find the sort of distortion and undistortion of the random payoff. They're distorted by utilities. We then saw the risk neutral agent has this limiting price and the risk averse agent, believe me, it is going to be that and I just have to argue that this is zero. It is the worst case outcome. So what's bigger is this, how would you compare this with that expectation? Clearly the expectation is bigger, right? I have a random variable. I have its smallest possible value. I take an expected value. The expected value has got to be bigger unless there's a probability one event of that worst outcome occurring. That's the only way that you would get these two things being equal. So the risk neutral agent would be willing to pay a larger price than the most risk averse agent and that's natural and seems to make sense. So then the next natural question is, well, is this thing monotonically increasing or does it have a maximum? So what we have is if we draw that price as a function of gamma, we know that it has a nice limiting behavior here. It's 1 over 1 plus r times the expected value of a1 and we know what its limiting value is over there. It's 1 over 1 plus r, ad. So the question is, does it do that? Do you think it can do that? Can it do that? It's not a difficult thing to answer. It's not a difficult thing to check explicitly but it's going to be monotonically decreasing. So that's a good little exercise for you. It shows that that function is in fact monotonically decreasing in gamma. It's not at all hard. So this is what your indifference price would tell you. You're bound above by this risk-neutral price and you're bound below by the most risk averse agent, the worst outcome. Okay, so what we're going to do when we get back from the break to give you some foresight is we're going to now take this result as given. We're going to say, now there is a market out there and a is traded at that price. So we're no longer going to try to find a0. But what we are going to do is we're going to ask whether or not that market with a being traded and the bank account being traded, whether that market is free of arbitrage. And I have to define what that means and then we're going to see some consequences. And then we can come back and ask, is this price free of arbitrage? It seems reasonable, but is it actually free of arbitrage? Okay, so let's take a 10-minute break. Okay, so as I said before the break, what we'd like to do now is come back again to this basic question and say, now that we've got a way of answering what's the rational price to charge, what's the rational price for a0, we want to answer whether or not that price leads to something that avoids arbitrage. So we want to ask about the concept of no arbitrage and see if we can actually prove that there is no arbitrage in the economies of this type. And we'll do this in the context of, we'll do the proof at least in the context of just the two states. The multiple state requires some, well, it just requires something called a separating hyperplane theorem. I don't know if you guys know it very well or not, but okay. We'll leave that part out and if you want you can read it. So let's go back to this, so let's go to the concept of just no arbitrage and in particular, first of all, what is an arbitrage? Okay, so an arbitrage is in general what it is, let's talk about it first of all vaguely and then more specifically, in general what it is, it's a strategy of some kind, a trading strategy of some kind that is self-financing. So what does self-financing mean? It means I put some money in into the strategy and then I let the strategy run and any money that I need to keep the strategy going has to come from selling one asset and buying another asset. It cannot come from my pocket, I cannot throw more money into it. The money could come from a bank account, but that all is part of the strategy. But it cannot just be extra cash that's infused and I cannot extract cash from the strategy either. So that's what we mean by self-financing. So an arbitrage is some sort of self-financing strategy where I make a riskless profit. That's intuitively what you think of. So right off the bat we can ask, making a riskless profit from a self-financing strategy, is that really an arbitrage? You could call that an arbitrage but then does that lead to a concept that seems very useful? The answer is no, why? Because if I take $100 and I put in my bank account, I withdraw it a year later, I've made a profit with no risk. I don't put in the bank account, I put in say government Canadian bonds. That's the zero risk of default, essentially zero risk of default. I'll get interest from that. It's risk-free profit. That's not an arbitrage. So an arbitrage is going to be some sort of self-financing trading strategy that leads to riskless profit in excess of the risk-free rate. So you need that little extra caveat. So let's write that in quotations and then we'll formally try to write down a more mathematical definition that we can then use to actually prove whether things have arbitrage or they don't. So it's a riskless profit in excess of the risk-free rate through a self-financing strategy. So you made some sort of riskless profit in excess of the risk-free rate and using the self-financing strategy. That's what we'd like to formalize somehow. So here's how we're going to formalize it. We're going to say we've got a trading strategy and the value of that strategy is going to be self-financing. And in the current context, self-financing is a very simple thing. It just means I buy, there's only two assets, asset A and the bank account, I buy some of them and I hold it for one period and that's it. There's nothing to it. When there are multiple steps, it will be a little more involved. But in general, we have a value of a self-financing strategy which we will call, we'll denote by VT. And an arbitrage is a strategy such that first of all, it costs you nothing to set up, zero dollars. Now that sounds odd, how can it cost you nothing? Well, if you buy one thing and you sell another thing of equal value, that's a net of zero. So that's how you in general create something which has initial value zero. So this means it costs nothing. Second thing is there exists a T, a time, such that you have the following conditions. P, the probability. So I always denote probability measures and probabilities with this sort of blackboard P or a blackboard Q and we'll talk about the differences in a bit and I'll write out explicitly what this is in the second here. So the probability that that strategy is greater than or equal to zero is one. So what does that mean if you think about it financially? It means that you never lose any money, or at least you never lose with probability one. You don't lose with probability one. And in a discrete world, that means it never actually happens. In a continuous world, well, yes, it could happen, and sets of measures zero. So we're in a discrete world, so all it means is never happens. You never lose. B, the probability that you end up with a strictly positive value is also positive. It's not one, but it's positive. So this means if you think about it intuitively, there are some states of the world, there are some outcomes in which you actually do gain a profit. So you sometimes win here. So these are the three conditions that you need. Now, as I said, in our current setting, since there is only one time step, there exists a T such that can simply be replaced by the probability of V1 being bigger than or equal to zero equals one. The probability that V1 is bigger than zero is bigger than one. You can just change those events to T equals one is the only other time available right now. There's only one time step. Okay, so let's draw a couple of things and ask ourselves whether they are arbitrages or not. So here's the question. Suppose I built a strategy, somehow, not only how, I built a strategy in some economy, and that strategy happened to look like this. Its initial value is zero. It ends up being 10 there and minus one there. So I would ask, is a strategy that led to this value an arbitrage? The answer is no, clearly because you lost in at least in one scenario. So it violates part A. So this is not a strategy. This is not an arbitrage. What about this? Is this an arbitrage? Let me come back to that one. I'll let you sort of stew over it for a bit. What about this one? If I had a portfolio or a strategy that led to that value. So this is an arbitrage because, yes, I never lost. There are no states in which the value is negative, and there is at least one state in which I have a positive, strictly positive value. So this is an arbitrage. Okay, let's go back to the one in the middle now. Okay, perhaps, what if R is zero? So the answer is that just by looking at this alone, you cannot tell whether there's an arbitrage in the economy or not. That's in the economy or not. But you can look at this and say whether this itself is an arbitrage. And the answer is it is not an arbitrage. It violates the definition. It does not satisfy point one. It does not have initial cost zero. So it looks like it could be, sorry, I have something running on my machine back in the office and it's every now and again updates through the internet this thing. That's what that note is. So here, what was I going to say? So here it could be an arbitrage if I knew something else about the economy or I shouldn't say this is an arbitrage. I could create an arbitrage by using this strategy together with something else, but this by itself is not an arbitrage. So never confuse the two things. Do not confuse there exists an arbitrage in the economy with is this particular strategy an arbitrage? They're not interchangeable. The economy may admit an arbitrage, but your example is not one. And this particular one leads us to that case. So I'm going to say here that this is not an arbitrage, but now I'm going to add some more information. I'm going to say let's take, let's continue on and say suppose I had a strategy which led to this and as well you are allowed to trade. So I told you you can have this strategy. I can give you that strategy to work with. And now you're also allowed to trade in the money market account. Can you create an arbitrage? So I've changed the question now, right? It's not simply is the thing the strategy I gave you an arbitrage. It's can you create an arbitrage with this? And that's basically what we're going to try to answer in the general setting. But here we can see if R was zero or negative in fact. Negative interest rate by the way is not a violation of arbitrage. It's just the definition of the economy. There would be an arbitrage if you had negative interest rates and you had cash in the economy. If you had cash, you'd be able to violate. And we'll see how that works in a second. Let's just continue our thought with this example. So if that were zero, well what you could do is you could simply take the strategy and then short the money market account. What the shorting the money market account mean? It means borrow a dollar, right? And now when you combine that together, those two things together, or maybe I'll do it like this. When you combine those two things together, the outcome is cost me nothing, I get one, and I get zero. So I was able to transform the strategy that I gave you initially by shorting in the bank account to create a portfolio that is in fact an arbitrage portfolio. So what I would say, what you would say in this instance is that the strategy you gave me is not an arbitrage but the economy does admit an arbitrage. So this is not an ARB, the economy does admit an arbitrage. What do I mean by it does admit an arbitrage? I mean I can find an arbitrage, I can construct one. And that's the arbitrage, the one I just showed you. Take that initial strategy, short that thing, and you get this payoff. So what we're going to do now is we're going to go back to that question of that asset and we're going to ask what are the conditions on that asset such that the economy does not admit arbitrage? Because at this point we simply have a definition of what arbitrage is and now we want to go back to our basic economy which has the tradable asset A and the bank account and ask is there arbitrage in this economy? And under what conditions are there and what conditions must be satisfied to avoid it? So these are our two traded assets. And by the way, a word of caution. Am I allowed to consider that scenario? Asset A goes toward the up branch and the money market account goes toward the down branch. It's kind of a trick question. In general, yes, I could in principle consider that but what I mean whenever I draw diagrams of this nature is there's a single coin flip and that single coin flip tells me whether I go up in which case we follow that branch and that single coin flip, if it turned out to be head tells me that I might go down in which case they both follow the down branch. They go together. Sometimes instead of drawing it like that sometimes people find it easier to think of it as there are two outcomes and what you have at these two outcomes is a vector. You have a vector, A is your own one and either that vector changes to become A U and 1 plus R or it changes to become A D and 1 plus R. So those two diagrams, both diagrams are used to represent the same thing. Later on we'll discuss combinations where we can have the crisscrossing things happening but to do that we need two-dimensional trees. For now we're just sticking with the one-dimensional tree. Okay, so here's our question. We have these two traded assets and we want to figure out the conditions such that there are no arbitrage. So what must I do? Well, what you can do is try to create an arbitrage and then look at the condition under which you would be able to create it and what conditions you won't. Simple as that. So how would I attempt to create an arbitrage here? I would buy some number of units of this asset and some number of units of that asset and that's my portfolio because that's the only thing I can do in this one-period economy, right? Buy asset A, buy some of the money market account, hold it for one period and then see what happens. So if I did that, what's the net result? The net result is simply alpha times A0 plus beta. That's what it would be worth. I have to put that much money in. I get alpha times U plus 1 plus R times beta. I get alpha times AD plus 1 plus R times beta. Because once I hold alpha units, when I get to maturity, either I'm going to get alpha AU or I'll get alpha AD and if I put beta in the money market account or the bank account, once I get to maturity, I'll get beta times 1 plus R. So I just add them together and this is what I have. Those are the two assets. That's my portfolio choice. Now if this is to be an arbitrage, what must I do first? To make a potential arbitrage, I want this thing to be zero, right? The sum. Maybe I'll... So that's right here. For an arbitrage, I must have alpha A0 plus beta 0. That's my V0. That has to be zero. So in fact, this tells me what beta... I could either solve for beta or solve for alpha. This tells me since A is given, the only degrees of freedom are how many units of things to buy. So this tells me to create an arbitrage, however many units of AI buy, I must buy that many units of the money market. Understand? Does that make sense? Okay. So if I did that, then let's again draw this portfolio out down here. The net result after making that substitution, I would have zero. I'd have alpha. I'd have AU minus 1 plus R times A0. And I have alpha AD minus 1 plus R times A0. That's just by substituting beta equals minus alpha A0. In the end there. I end up with that. Okay. If this is to be an arbitrage, what must I have? I have to have either... There are sort of two cases that we could consider. So before considering the two cases, we might as well put an order on AU and AD. AU and AD were arbitrary, but I could assume AU is bigger than AD. And if it wasn't, I can just flip the tree. Agreed? So suppose AU is bigger than AD. So whatever's in the bracket is clearly, and the top is bigger than whatever's in the bracket in the bottom. It has to be. And now I can think of the situation, well, if I want there to be an arbitrage, I need to make sure that at least the bottom is zero. That would be the critical point. Right? Because I never lose. And since the top is bigger, that's got to be bigger than or equal to zero. In fact, strictly bigger than. If it was equal to zero, then I would have 0, 0, 0, and that's not an arbitrage. So I need bigger than zero in the bottom, bigger than or equal to zero in the bottom, and bigger than zero in the top in order for me to find an arbitrage. So this is one approach I could take. I could also take the other approach. I don't really want to find arbitrages. What I want to find is a condition which avoids arbitrage. Isn't that really what we're after? What conditions on my economy avoid arbitrage? So if I were to avoid arbitrage in this economy, what must I have instead of these two things? So this would create arbitrage. But to avoid arbitrage, what must I have? Think about it for a second. To avoid arbitrage, if you think, I'm going to write it down and then just tell me if this makes sense to you. If one is positive, then the other has to be negative. We know the top is bigger than the bottom, so certainly if they're both positive, that's an arbitrage, right off the bat. We can't have the top and the bottom being positive. So if the top is positive, the bottom has to be either zero or negative. If it's zero, then again, we have an arbitrage. So the downstate has to be negative if the top is positive and vice versa. So I can either have this or this is negative and that is positive. But that's avoided by stating that AU is bigger than 80. Because AU is bigger than 80, I will never have this last column. Okay, so that's our no arbitrage condition. To avoid arbitrage, we must have that. So what we've shown here is that if AU minus 1 plus R times A0 is bigger than zero and the stuff in the bottom, actually, no, we haven't quite. Look, there's that alpha coefficient there, right? So if alpha is positive, this would be true and if alpha is negative, then we'd require the other one. But so certainly, we avoid arbitrage if this is true. Do you agree? We rewrite that equation for a second. If you just look at it quickly and just rearrange it in your mind, you get this nice little condition. Now look at that in terms of a timeline here, in terms of an ordering. That means A0 grown at the risk-free rate is between AU and AD. Then there's no arbitrage. Now we can ask the question, did our previous results actually doesn't satisfy no arbitrage? So recall from indifference pricing, we had that. So the question is, does this line not bound? You know that this is bounded above by the expected value of A1. The expected value of A1 when you discount it is clearly in between AU and AD. The lower bound for this we saw was AD divided by 1 plus R. So once you multiply 1 plus R, you find that this gets, in the limit, it touches AD, but for any finite gamma, it will be bigger than. Do you agree? So this does satisfy, so for any finite gamma, that expression does or admits no arbitrage because it's in the bound. Are we happy with that? That's a good result, isn't it? The economic argument of using utility functions and then doing the difference pricing, which is very good sound economics, is consistent with the concept of no arbitrage. Doing that does not allow you to introduce an arbitrage into your economy unless you are an infinitely risk averse agent, but no agents are. Agents will always have a finite level of risk aversion. Okay, now this statement that I did here, I showed the implication in one direction. A, U, AD, and A0 ordered in this implies no arbitrage. Well, it's not difficult to show the other way as well, and I urge you to try it. No arbitrage implies this ordering, okay? So try that. So if you combine those two statements together, you actually get what's sometimes called, it's one of the fundamental no arbitrage results, says that there is no arbitrage in the economy if and only if, so IFF, okay, if and only if, AD is less than or equal to 1 plus R, A0 is less than or equal to AU, because this shows both directions. Whoops, I'm not really adding, I haven't changed anything here, I've simply just rewritten what we had above, but now I've made it an if and only if statement. So that is an exercise for you to show the other way. All right, so we're almost to where I'd like to go. Now we've got, so what have we done so far? What we've done is we started with a single asset and a bank account. We said that single asset has some random outcomes. Let's see if we can figure out whether the economy is free of arbitrage, let's figure out what's the reasonable, rational, economically sound price to charge or to sell at. We figured out what that is. Then we've gone and asked what about the rational, what about the concept of no arbitrage, what is that and is that original price consistent with it? The answer is yes. Now as I was saying that, it just made me remember or realize that when we talked about indifference pricing, we actually looked at it from the point of view of only a one-sided agent. That is, this agent who's figuring out their indifference price is buying the asset, is thinking about buying the asset. I purchased for A0, right? I purchased the A0 and then I get A1 at this time, T1. So here's a little exercise for you. Repeat this exercise for the person who is selling the asset. Their indifference price is not identical. It's very similar in form, but it's not the same. In fact, I'll tell you the result. So this was for the seller, this was sorry, for the buyer. For the seller, you can find that A0 should be plus 1 over 1 plus r, gamma log expectation e to the gamma A1. So the differences are signs here. You get positive and positive up in the exponential instead of negative and negative. And you can kind of think of that because, well, what happens with that agent? That agent's cash flows are basically flipped, aren't they? They're negative of each other. So this is why you get this result. And if you plot this same pattern of the price versus level of risk aversion, what do you think you're going to find? By the way, what's your guess? Gamma goes to zero limit. What should it be? Same as before, right? It's still discounted expectation. And you can try that. But now, for the seller, think about the infinitely risk averse seller. What price would they charge? They would charge no less than the maximum price, the maximum outcome. Because they are subject to making that payment to you if it happens. So they will be making sure that whatever they get, it's going to at least match that maximum. So now there's actually an upper bound, not a lower bound. And that upper bound is, let's call AU. So let's suppose, again, that A1, it's bounded below by AD. It's bounded above by AU. So here, this would be 1 over 1 plus R times AU. And you can easily show that that's also just nice and increasing. Simply increasing. Very distinct behavior from the seller, sorry, from the buyer. One is increasing and one is decreasing. Do you notice something sort of interesting about that? At what price would you actually be able to make a deal? Remember, as a seller, you're going to be following this curve here, and your risk aversion level is there, say, and as a buyer, your risk aversion level is there. Are you going to be able to make a deal? No. The seller is going to want to charge more than the risk-free, the risk-neutral price, right, the discounted expectation price, and the buyer is going to want to pay less than that, no matter what their risk aversion level is. There's only one point at which there's actually a deal made, and that is if all the agents are risk-neutral. And this is going to play a very interesting role. This risk-neutral agent is going to show up again, actually, very shortly. Okay. So at this point now, we're able to discuss, oh, and of course that seller's price also satisfies the no arbitrage requirements. Do you notice? In this limit, instead of moving toward AD, for the buyer, A0 moves toward AD as gamma gets larger. For the seller, A0 moves towards AU, but it satisfies that constraint for any level of risk aversion. So you're always within that. Okay, so now what we want to do is we want to say, now we've got a viable economy. We have one asset, one bank account. We want to try to add another asset into the mix. And that third asset, we'll call it B. So again, we have these two already there. We're going to throw in this new asset, B, and remember these branches, you can really just interpret this as, say, a vector in three-dimensional space. So when you move along one branch, they all move along that same branch. Okay? These are not separate branches. These are not independent branches, sorry. Okay, what we would like to ask now is if I want to value this new instrument, we've added it into the market. We've already figured out an A0, so let's, we don't need to go back and do the optimization thing or do the indifference pricing for A0. That's already traded. And I add this B into this mix. So here's my question, and is it true that B0 should be 1 over 1 plus R times gamma, the economically sound rational price, should it be negative? Let's think of the buyer. Is that true? What's your gut feeling? Who thinks yes? Who thinks no? And the rest of you? What do you think? Yeah? Okay. All right, so that's going in the right direction to some extent, but let's, but why is this wrong? Why can't I just use the same reasoning as before? When we were thinking about value in A0, what were we allowed to trade? What were the games that we were comparing? Do nothing, put money in the bank, or buy the asset and then sell it at the end, right? That's game one, game two. Now, when those two things are already traded in the market and I now add B into the mix, are my games the same? The comparing, the games that I'm comparing, are they the same? They're no longer the same, right? I have money in my hand. Those are not simply buy B or not buy B. The options are I have money in my hand. How much of A should I purchase? How much of the money market should I purchase? When I don't buy B, that's game one. Then game two, buy B. Now whatever's left, how much of asset A should I buy? How much of asset money market should I buy? That's a completely different comparison than what we compared before. So you could still, in principle, use the indifference pricing methodology and come up with an equilibrium or an indifference price. And that price, once you look at all of your spectrum of strategies, will correspond to what we're going to do in a minute, and it won't be equal to that. So that's what I want to pound in your head, is that the reason this is wrong is not because you must do something else. The reason it is wrong is because this result was obtained by comparing the only options that were available to you at the time of one asset being the money market and the other one being the asset you're pricing. Now the options that are in front of you are completely different. You can trade an asset A, you can trade in the money market, and then you purchase asset B. So those are completely different scenarios and strategies. And I urge those of you who are sort of keen to try to work out what this, try to work out through the indifference pricing methodology the results we're about to find, okay? But in class I'm not going to go through it, and if you want you can see me during office hours and I'll tell you about that. In class instead we're going to take another perspective that leads to the same result. And this perspective can only work now that we have A being traded and the money market being traded. It couldn't work when we didn't have those already traded in the market. So here's the idea. It's based on something similar to what we did in constructing arbitrages. What we'll do financially is take a portfolio built out of asset A and the money market account and make that portfolio have the same value as asset B in all outcomes. And if it has the same value of asset B in all outcomes then the value of that portfolio must be the price of asset B. Otherwise there's an arbitrage. That's the argument that I will use. So I'll write this out and we'll see how that works. So here's the idea. So we take alpha units, asset A, beta units of asset B, of the money market, MM, the bank account. I often use MM instead of bank account because it's not a bank account. It's really the money market. People sort of think of them as bank accounts sometimes. So I take this portfolio and as we saw before the value of that portfolio is alpha, A0 plus beta and it will be worth alpha AU plus 1 plus R times beta and alpha AD plus 1 plus R times beta. And here's what we're going to do. We're going to make this. We're going to choose alpha and beta such that we match the payoff of this new asset. There are two degrees of freedom here. So therefore there is a solution and in fact it is a unique solution unless for some reason these two equations are collinear. But you can actually quickly check. The only way that these two equations would be collinear is if AU equals AD and if AU equals AD what do we violate? We would have violated no arbitrage. We would have squeezed that region down to a point. And there's no way the discounted version of A0 could be within AU and AD if they squish to a point. So you would actually introduce an arbitrage. So these equations are in fact not collinear and they cannot be otherwise there's an arbitrage. That's the mouthful. Just get back to the simple linear system. You have two coefficients, this, this. You solve it. I'm not going to do the algebra for you. I think most of you have actually seen the algebra. I'll just write down the end result of that procedure. So you solve for these guys and now here's the no arbitrage argument. We say that since the portfolio replicates, this is called replication. The portfolio replicates the value of asset B at T equals 1. The portfolio must be worth, must be equal to asset B value at T equals 0. i.e. B0 must be equal to alpha A0 plus beta. Otherwise, there is an arbitrage. Okay? How would you construct that arbitrage? Let's do a simple example and in fact, let's make this into your quiz while we're at it. Let's do a very simple example. Let's suppose you have $10 becoming $20 or $5. The money market account is this. Okay? And I'm trying to think here, what do I want? Let's make this $30 and make that $0. Okay? So here's your little quiz, okay? You're going to do this now in groups of four. You're going to tell me first of all, what is this B0 equal to? Using this principle, okay? Yeah? Using this principle that we just talked about here, this replication idea, okay? Using that principle, find what B0 is. That's part one, using this replication idea, okay? And part two, suppose B0 was 25. Construct an arbitrage. It's a very simple example. I want to make this a little bit smaller so that it shows up on the whole screen. All of that. Can you see that? Yeah? That's basically everything you need to do right there in front of you. Okay? So work in groups of... Okay, so let's go through... I'll just go through the solution very quickly. As it was described there, you're going to pick alpha units of this, beta units of that. Your portfolio at the end, the two equations you're going to have are 20 alpha plus beta. You want that to be 30. You want 5 alpha plus beta. You want that to be 0. Well, that's a pretty easy equation to solve. Subtract the top equation, or the bottom equation from the top. You get 15 alpha equals 30. So alpha equals 2. Okay? And put 2 into the, say, the bottom equation. And you get beta equals minus 10. Okay? So this means you should buy two units of the asset A and short or sell 10 units of the money market account. And this will replicate the payoff. And then B0 has to be the initial price. So that's 10 times alpha plus beta. And in this case, that would give me $10, right? So hopefully we got that. Yeah? 20 minus 10 is 10. Good. Okay. And what about the... So that's part one. Part two. What about the arbitrage? Well, remember, for it to be an arbitrage, you need to make sure V0 equals 0. And you know... So you know that this price, the rational price is $10. The price it's being sold at is 25. Bought and sold is at 25. So if you can buy or sell at 25 and replicate at 10, what you should do is somehow you should be selling the asset, actually selling it somehow, and replicating the payoff. Right? So that's what you want to do. But you want to do it in such a way that you start with nothing in order for you to be a real arbitrage. So maybe I'll do this question... I'll do this two ways. First of all, the naive way, which is not... I shouldn't say the naive way. First of all, I'll do the way where we're forgetting about the formal definition of arbitrage and simply construct something that we believe sounds like an arbitrage. Okay? That's the first way of doing things. So that means sell the asset at that price by the replicating portfolio at this price. So what I would do is you'd sell B, you buy two of A, and sell ten of the money market account. Right? Because this is the replicating portfolio. Right? Two of A minus ten of the money market. So if I do that, what are my net cash flows? Well, I sell B, so that means I got $25. I purchased two units of A that cost me $20. I sell ten units of the money market. Actually, let me write this down. Right? I got 25 by selling B. I buy two units of A, so that's costing me two times ten. And I'm selling ten units of the money market account, so that's given me ten. Agree? So what does that add up to? So that's minus 20, minus 10, 15, right? And that should, that makes sense because 15 is the difference between the price it's been sold and bought at in the market and the actual no arbitrage price. Right? The value that is rational is $10. So the difference between those is 15 and that's what you gain here. And what will happen at the two endpoints? It should become zero by definition. Right? That's exactly the equation that we solved. That's how we found alpha and beta. We made these things zero. So this is sort of the first way of doing things. That's what you would have called an arbitrage. I got $15 now and I owe nothing. And that in any normal person's mind, if I have a strategy that's going to get me $15 now and I owe nothing in the future, sure, that's an arbitrage. But it doesn't conform to our formal definition. That's the only thing. So how could you potentially make this conform to our formal definition? It's one very simple thing. Buy 15 more units of the money market account. Spend that $15 profit. Buy 15 units of the money market account. That cost me 15 so I have nothing now and I get 15 back at the end. So what you do is I buy, this is one way. I mean there's many other ways. Let's say 15 more. And now what I'll have is I cost nothing and I get 15 back. Now this conforms to our definition of arbitrage. But like I said, it's okay for today I will take anything that looks like this as a valid arbitrage. But only today. No more after the day. All right, good. So let's slide back up here. So what we've discussed at this point is I've said if I introduce this third asset into the mix there's another approach rather than doing the whole indifference pricing to figure out what it's worth. And that is simply replicate that new asset's value by trading in the assets that already exist. And then I can value that new asset. And that's what we've done here. Now algebraically if you actually just solve the linear system there and directly find alpha and beta plug it back into this equation and do some simple arithmetic and I urge you to do this on your own what you find in general for arbitrary values you find the following expression for B0 something that looks quite surprising. You find this expression in general. If you just plug it in, do the algebra and it takes you a few lines to get here but it's not hard and it's a waste of time to do it in class. So just do it on your own, you'll get that. And that has a very suggestive form. It looks as if the price of B is the expected value of its payoff where the expectation, the probabilities are given by this expression Q. So that would be true if Q was in fact a probability. Right? So if I could, then I could interpret that then I could interpret this as 1 over 1 plus R expectation under a new probability measure of my random payoff. So the superscript Q there what that means is the probability of the event that B1 equals this is little Q probably B1 equals BD equals 1 minus Q. This is similar to in fact in the model, the original model, we had this probability P, a little P, right? The branching probability. So here I'm just defining a new probability measure. The probability of that event is little Q. The probability of the other event is 1 minus little Q. So we have a question mark over there because we are not sure whether Q is in fact a probability. So what is the requirement for Q to be a probability? Q needs to be between 0 and 1, right? Okay, so then that's, so the question is is Q between 0 and 1? Why? Yes, we need to invoke the concept of no arbitrage. The no arbitrage constraint, what is it? Do you remember? It's AD is smaller than 1 plus R times A0 is smaller than AU. So no ARB implies that. It's an if and only if statement in fact, right? We said the economy has no ARB if and only if this is true. But this expression implies that Q is between 0 and 1. How? How does it imply that? Well, if I just draw a little, sketch a little diagram here, you'll see it very easily. The same one I had before. That's AU, that's AD. This is A0 times 1 plus R. That's what the inequality tells me, right? They're ordered in this way. And clearly, the ratio of that distance, Q is the ratio of the distance above to the distance below, right? It's that ratio. That ratio is clearly smaller than 1. It's also clearly positive because we don't hit, it clearly doesn't equal 1 and it doesn't equal 0 because the inequality is strict. So in fact, Q is between 0 and 1 if there is no arbitrage. So we have no arbitrage implies Q is in 0, 1. What about Q in 0, 1? Does it imply no arbitrage? So these are if and only if statements that we're trying to connect. Well, if Q is between 0 and 1 and AU is bigger than AD, because that was our assumption, Q is between 0 and 1, then you can easily manipulate this expression above to see that it implies that one. It's not hard, so try it. So it's also easy to see that this implies the expression above. So Q being between 0 and 1, or I should say Q is between 0 and 1 if and only if there is no arbitrage. No arbitrage is showing its magic again. So Q is in the open interval 0, 1 if and only if there is no arbitrage. I was able to say that because we have the equivalence of Q being in the interval 0, 1 to this inequality being true and this inequality being true is equivalent to no arbitrage. Therefore, Q in this interval is equivalent to no arbitrage. Okay, so that is fantastic. So we could also add to this, so this tells me as well that if Q is between 0 and 1 if and only if there is no arbitrage, that means that the value B0 has to be equal to 1 over 1 plus R times the expectation under some new probability measure B1 if and only if there is no arbitrage. Or I should say let me modify that statement a little bit. Let's say there must exist a probability measure Q such that B0 equals this if and only if there is no arbitrage. These statements are the same. Was there anything special about B? Did I assume any structure? Did I assume that BU is smaller or bigger than BD? Did I assume that B0 was ordered in any way? I did not. There was no assumptions on B at all. So this statement is actually an extremely powerful one. It tells us that there must be a probability Q such that if I compute the expected future value of an asset price it equals its asset price now grown at the risk-free rate. I'm going to rewrite that in this way. Multiply by 1 plus R. I can interpret this equation as saying the future value of the asset take its expectation. It equals its current value grown at the risk-free rate but not using the original probabilities, using this new probability, this probability Q. And what the statement tells me is that there must be such a probability otherwise there is arbitrage in the market. That's a very interesting logical statement because this is a mathematical statement that simply says, okay, I have a probabilistic model. I'm going to find some probability and I'm going to choose that probability so that the expected future value of the asset equals its current value grown at the risk-free rate. And it so happens that if I can find such a probability then this economy that I have is free of arbitrage. That's a really kind of bizarre thing, right? We started with this concept of arbitrage as being construct strategies so that you make riskless profits. And what the consequence of that is going through the set of logical arguments is there must exist a probability measure under which the asset grows at the risk-free rate. That's another way of saying this. Under the measure Q, the asset grows at the risk-free rate. And that interpretation of this equation is why that measure Q is called risk-neutral. This measure Q is called the risk-neutral measure because all, remember nothing special about B, all traded assets, they have to be traded, right? Grow at the risk-free rate under the risk-neutral measure. This is one of the most fundamental results in derivative pricing theory. And we've proven it in the context of this simple binomial model with two states, okay? We've shown it there. It extends more generally to any number of steps, any number of states. So, in principle, I could have a model where here's the next generalization, next simplest one. Suppose I had assets that took on three values, right? AU, AM, and AD. And I gave some probabilities for those branches. Let's call them PU, PM. Sorry, let's put the probability here. PU, PM, and PD. What this theorem is telling us in its general setting is that I must be able to find some new probabilities, Q, such that A0 times 1 plus R has to be equal to QU times AU plus QM times AM plus QD times AD. And if I can't find Qs that do that, then that economy admits an arbitrage. I think it's one of the most powerful results. Yes. Sorry, where's the question coming from? I'm hearing a voice and I'm looking around. Okay, there we go. Yeah, yeah, yeah. So, these are probabilities, right? Yeah, so they have to sum to 1. They have to be positive. That's what I mean by probability. So they strictly, these are all, these are all greater than 0 and they all sum to 1. Okay, because they're probabilities. So this is sufficient and necessary in order for each Q to be in the interval 0, 1. And there's some to be 1. So that's, again, when I make a statement like this, there exists a Q. What I mean by that is that Q has to be, that script Q has to be a probability measure, a proper probability measure. It can't be, one of the Qs can't be negative 0.5, for example. If you did find that, there is an arbitrage. This theorem does not tell you how to construct the arbitrage. It simply tells you that there exists one. And then it's up to you to go through that replication type argument to try to construct arbitrages. So it's a sanity check to tell you whether you're right or wrong. And then you have to do something else to actually find the strategy. So maybe to illustrate this point in a slightly different setting, let's do an example and then we'll call it a day there, okay? So covered a good amount of material. Let's take a look at an example where an asset is worth 100. In this particular state of the world, it's worth nothing. In this particular state of the world, it's worth 50. And in this particular state of the world, it's worth 110. Okay, so this is a three-state world. You have these two traded assets. And the question that I'd like to ask you is, does there exist an arbitrage in this economy? Two approaches that we can take. One, go back to the definition of arbitrage, try to make a potential arbitrage and see if you can find such a portfolio. Or two, find a queue such that this expectation is equal to it's value now grown at the risk-free rate. And if the queues that come out of that are not in the interval 01, then there's an arbitrage. Okay, so let's do that. Let's do both. Let's go through both exercises. So let's call this, one is the direct method. So the direct method means choose alpha units. Let's call this, so this is asset A and this again is our money market. Okay, we're going to choose beta of the money market. And in order for us to have v0 equals 0, what must alpha and beta be chosen as? Clearly beta must be minus 100 alpha, do you agree? Otherwise v0 wouldn't be 0. And if it is, then I'll get 0 there and I get 10 times alpha here. I have 110 times alpha from A minus 100 alpha from the money market. So that's 10 times alpha. In the middle state here, I would get negative 50 times alpha. Do you agree? And in the bottom one, I would have minus 100 times alpha. Agree? So then the question is, does there exist an alpha? Because now we have the most general portfolio. Does there exist an alpha such that I satisfy the other two criteria for arbitrage? Remember what they are? Probability this equals 1. Probability this greater than 0 is greater than 0. So can I satisfy these things? So here's one approach. I can say, well, in order to ensure the first point, what must alpha be? Alpha's got to be 0. As one coefficient is positive, the other are negative. So alpha must be 0 in order to satisfy the first. But if alpha equals 0, so alpha equals 0, let's see. So if this is true, it implies alpha equals 0. But then that also implies that this is 0. I never get a profit because alpha equals 0, I get 0 in all nodes. Therefore, this economy does not admit an arbitrage. There is no portfolio for which I can construct an arbitrage. So I can say this economy is arbitrage free. Does that logic make sense to you? I attempt to make the arbitrage as best I can, and then I look at the conditions. Can I satisfy both at the same time? No. Therefore, there is no arbitrage. Okay, let's go with the risk-neutral probability method. So what must we do? We must be able to find a measure Q such that the discounted expectation are equal to its current value. So I need to satisfy this constraint. I need to find such a Q. And in principle, I also should do the same thing for the money market account. But the money market account is not going to help me. Look. Okay, R0 in this case, right? This is 0. This is 0. Money market, this is 1, this is 1. This just tells me 1 equals 1. Any Q will satisfy that. And you will always find that situation. The money market account never helps you identify the risk-neutral measure. But any other asset will. It will give partial information. Since we only have one asset choice here, we only have one equation to play with. So we've got 100 in this case. It's equal to Q, let's call it QU. It's going to be up to the up branch times 110. Plus, let's make the middle branch since the Q's have to sum to 1. Let's make the middle branch 1 minus QU minus QD. And so QU and QD will be the two up and down probabilities under that risk-neutral measure. And that was times 50, I believe, was it? Yeah. And oh, you know what? That's the silly choice on my part. I forgot. The fact that A is 0 in the down state means that I should probably choose QM as my free variable and not QD. It simplifies my work. Sorry. So I have QM times 50, and here I'll have 1 minus QU minus QM times 0. Okay, so then the question is, does there exist, do there exist two Q's, QU and QM, between 0 and 1 that satisfy this equation? Can you find one? There are many, don't you see? Right? There are quite a few. Look, let's solve for QM here. QM equals 100 minus 110 QU, all divided by 50, okay? Or, in other words, 2 minus 11 over 5 times QU. Agree? Okay, well, if I want all of the Q's to be between 0 and 1, I know that I want, in particular, I want this to be between 0 and 1, and if that's between 0 and 1, what is it going to imply about QM? Well, QM is decreasing in QU, so I can find the largest value of QM by looking at the smallest value of QU. You agree? Smallest value of QU is 0, but that would make QM 2. That would be an arbitrage. So, in fact, this lower bound is wrong. This lower bound has to be supplemented by the constraint on QM. So, we need to satisfy this constraint as well, and this implies 0 is less than 2 minus 11 over 5 QU is less than 1, and let's just... Maybe I'll multiply everything by 5, so we get 10 minus 11 QU is less than 1. I know I'm doing this in gory detail here, but I think in this first example it's worthwhile. So, we get QU is bigger than 10 over 11 from the upper... Sorry? What did I do here? Did I make a mistake? Oh, 5. Thank you. Yes, thank you. So, QU has got to be bigger than 5 over 11, and QU has to be less than 10 over 11 from that constraint. So, we actually found that even though, you know, for Q to be a probability, this is necessary, but for QU to be a probability, this is necessary, but for QM to also be a probability, we have a further constraint on QU. So, we found that QU is, in fact, 10 over 11 and 5 over 11, and then you can find the range for QM. I guess QM is between 0 and 1. Are we done? One more constraint. QB also has to be between 0 and 1. So, I'll leave that one out for you. You work on that. See if this imposes any further constraint on QU. It might shrink the region a little further. So, we started out QU can be between 0 and 1. When we make QM also between 0 and 1, we shrink the region. When we make QD also between 0 and 1, we may shrink the region even further, but it will be non-empty at the end of the day, and since it's non-empty, there is no arbitrage. Okay? All right.