In this part of the course, we're going to see how to price options in the binomial model. Before getting into the details of pricing options, however, I want to have an introductory module where we will raise some of the questions that we're going to consider. In the next series of modules, we'll actually study the 1-period binomial model. We'll follow that with the multi-period binomial model. We'll also discuss replicating strategies. In fact, that's how we're going to price options within the binomial model. We'll construct a replicating strategy that replicates the payoff of an option. And we'll use no-arbitrage pricing then to compute the fair value of the option. After that we'll discuss European and American options in the context of the, of the binomial model. We'll also discuss the Black-Scholes formula, and mention how it can be obtained by a convergence argument using the binomial model. Okay, but first of all in this module, I want to do an overview of some of the questions that we'll be considering. So here's an example of a binomial model, we're going to be working with the binomial quite a lot in this unit, and in the unit we cover next week. We're going to assume that the stock price starts off at the s of 0 equal to $100. And then in each period the stock price either goes up, so in this example it goes up to 107, or it goes down, it falls down to 93.46. And a factor, in fact in any period what happens is that the stock price either goes up by a factor of u, or it falls by a factor of d. And in fact we're going to be using the fact that d equals 1 over u in these, in these modules. And what this means, however, is that an up move followed by a down move, so this would be the security price at time 1. What it means, though, is that an up move, followed by a down move, gives you a price of ST plus 2 equals ST times u times d. But that, of course, is equal to ST times d times u, which is a down move followed by an up move. In other words, the stock price at time T plus 2 is the same if it had an up move followed by a down move, as if had a down move followed by an up move. Okay, and so it's recombining, an up move followed by a down move gives you the same price as a down move followed by an up move. And that's why we often call this a recombining tree, or lattice. Okay, so this is the binomial for the stock price, and in any period it goes up or it goes down, we've got a three period model here. So if the stock price goes up in every period, it ends up with a value of 122.5. If it goes down in every period, it ends up at a value of 81.63. We haven't discussed the probabilities of these moves. For now we'll assume that the probability of an up move is p in any one period. And so the probability of a down move is 1 minus p. And that these probabilities are the same at every node in the tree. So for example, down here the probability of going up to 100 is p. And the probability of going down to 87.34 is 1 minus p. Okay, and of course, we'd be assuming that 0 is less than p is less than 1. Okay, so that's the stock price, that's the security price, the risky security price. We're going to be figuring out how to price options on this stock. We also have another security in our model that's going to be called the risk free asset, or the cash account. Okay, we will assume that's available. And we'll assume the following. That $1 invested in the cash account at t equal 0 will be worth r to the power of t dollars at time t. So in other words, we're assuming a growth risk free rate of r per period, okay. And it's risk free because after t periods, we know exactly how much we'll have. We'll have r to the t dollars if we invested $1 in the cash account at T equals 0. So this is our binomial model. We've got the, the stock price, which is described by these dynamics here, a three period model of the stock price, and we also have our cash account over here. So now, we have some questions. One question that we'd like to answer is, how much is an option that pays the maximum of zero, and s3 minus 100 at t equals 3, worth? Well, let's go back and take a look. So, what is the maximum? So we have a maximum of 0 and S3 minus 100. Okay, so on each of these nodes, the maximum 0 and S3 minus 100, well that would be 0 there, it will be 0 here, it will be 7 here and it will be 22.5 here. So this our option payoff. And what we want to know is how much is this security worth? So the security is the security that pays 0 at this point. 0 at this point, 7 at this point, and 22.5 at this point. Okay, now, do we have enough information to answer this question? That's not clear. Another question is, should the price of the option depend on the utility functions of the buyer and seller? I'll discuss what, what I mean by utility function in a moment. But loosely speaking, I'm referring to how much value a buyer and seller might get from that security. Will the price depend on the true probability p of an up move on each period? Perhaps the price should be equal to this quantity here. So if you look at this, this is just the expected value of the option. So we're taking expectation of maximum 0, S3 minus 100. So that's the expected value. And this guy here is just the discount factor. So in fact this total quantity here could be called the actuarial fair value. Actuarial discounted fair value of the option. But is this how much the option should actually trade for in practice? I don't know, or at least, we don't know yet. We'll answer that question pretty soon. So here's another question, suppose you stand to lose a lot at date t equals 3, if the stock is worth 81.63. In other words, if you find yourself down here at date t equals 3, you're going to lose a lot of money. Similarly, maybe you start to earn a lot at date t equals 3 if the stock is worth 122.49. In other words if you're up here. So I've rounded the 0.49 to one decimal place, but if we're up here, we stand to make a lot of money, and if we're down here, we stand to lose a lot of money. So suppose you're in that situation, the question is, could you do something to eliminate this risk exposure? Is there some way to mitigate your risk, maybe even eliminate it? And we'll actually see that the answering this question is effectively the same as answering this question. And we'll be coming to that in later modules as well. Okay, so, just to address this particular question here where we say, should the price be equal to this amount? Let me give you some evidence for saying why the answer is no. The option price should not be equal to this quantity. All right, to do that we're going to come to a very famous example called the St. Petersburg Paradox, and the St. Petersburg Paradox considers the following game. A fair coin is tossed repeatedly until the first head appears. If the first head appears on the nth toss, then you receive 2 to the power of n dollars. The question is, how much would you be willing to pay in order to play this game? Now, you might want to pause the video at this moment and think about this for a couple of seconds and ask yourself, how much would you be willing to pay to play this game, if a friend came up to you and gave you this opportunity? Okay, I'm not sure how much I would be willing to pay to play this game, but it certainly wouldn't be very much. And yet, look at the following calculations. Let's compute the expected payoff of this game. So the expected payoff is the sum of the possible payoffs times the probability of those payoffs. So the probability of receiving a head on the nth toss, well, to get your first head rather on the nth toss, that means you must get n minus 1, tails, and then you get one head. And the probability of this event occurring is, well you, you get a tail with probability half, so you must get n minus 1 of them. So that's 1 over 2 to the n minus 1. And then you get your head with probability a half as well. And that's equal to 1 over 2 to the n. So the probability of getting your first head on the nth toss is equal to 1 over 2 to the n, which is that. Now remember, you get a payoff of 2 to the n dollars on the nth toss if that's where the first head appears. So your payoff at that point is 2 to the power of n. Well of course, the 2 to the n counts is with the 2 to the n here, and you're left computing a sum. I equals or n equals 1 to infinity of 1, and of course that's an infinite number. Okay, so we, we have calculated that the expected payoff of this game is infinity, but pretty clearly nobody would be willing to pay an infinite amount of money to play this game. Even assuming they had an infinite amount of money to begin with. So how much would you be willing to, to pay to play this game? Just to give you an idea, let me ask you this, would you pay $1,000 to play this game? In order to break even, or at least to show a profit, you would have to get, let's see. So, 2 to the power of 10 is equal to 1024, if I am correct. So, this means that in order to break even or to show a profit, if you paid $1,000 to play this game, you would have to get nine tails on your first nine tosses, and only after that point would you actually be assured of showing a profit. So I personally don't think I'd be willing to play, to pay $1,000 to play this game. I don't even pay, actually, a much smaller amount to play this game. So the point of this is to, is to, is to emphasize that the fair value of a security should not necessarily be its expected payoff. And in fact, Daniel Bernoulli, a famous mathematician, resolved this paradox by introducing a so-called utility function. The utility function has the following properties. It measures how much utility or benefit you're paying from x units of wealth. So u of x measures how much utility or benefit you obtain from x units of wealth. Different people of course, have different utility functions. The utility function should be increasing and concave. It should be increasing to reflect the fact that people prefer more money to less money. And concavity is there to model the fact that getting an extra dollar when your wealth is say, $1,000, gives you less additional benefit than getting $1 when your wealth is $0. In other words, going from $0 to $1 has more benefit than going from $1,000 to $1,001. And this idea is captured by using a concave utility function. So, Bernoulli suggested using log utility function, the log function is increasing and it's concave. So, this is an example of a concave function. It's like an inverted saucer. Look, it's increasing and it's concave. So the log utility function is what Bernoulli suggested. And if we did that with the St. Petersburg game, we find the following. The expected utility of the payoff is now the sum of the utility of the payoff. So it's now log of 2 to the n if the first heads occurs on the nth toss, times the probability of the first head occurring on the nth toss, which is 1 over 2 to the power of n. And if you recall, the log of 2 to the n, this is a property of logs, equals n times the log of 2, well, we get this quantity over here. And it's quite straightforward to show that this is, in fact, a finite number. So this is how Bernoulli resolved the St. Petersburg Paradox. He said that people don't compute values of gains by computing their fair value or their expected value, but instead everyone has a utility function and what they would compute is the expected utility of the payoff. And from there you can determine how much an individual would be willing to pay to play the game. Okay. So given this, you might think that all you need to do is to figure out the appropriate utility function of an individual and use it to compute the option price. Well, maybe, but whose utility function? The buyer's utility function, the seller's utility function, or maybe some other utility function in the, in the marketplace? We're going to come back to that question later. We'll talk about it very briefly later, but in fact, we're going to actually price the options using a much simpler method, which won't use utility functions at all.
