ECON 252: Financial Markets (2011)
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Financial Markets (2011)
ECON 252 (2011) - Lecture 17 - Options Markets
Chapter 1. Examples of Options Markets and Core Terms [00:00:00]
Professor Robert Shiller: All right. Good morning. The subject of today’s lecture is options. And I think, maybe I’d better first define what an option is, before I move to say anything about them. Because some of you may not have encountered them, because they’re not part of everyday life for most people, although they are, in a sense. I’ll get back to that. Let me just define terms here.
Professor Robert Shiller: So there’s two kinds of options. There’s a call and a put, OK? A call option is an option to buy something at a specified price, and the price is called the ”exercise price” or ”strike price.” Those are synonyms.
And a put option is the right to sell something at the specified exercise price. And it has another term it has to be specified, and that’s the exercise date. [addition: The exercise date applies to both call and put options.] OK.
Options go back thousands of years. It must have happened before we have any recorded records. If you’re thinking of buying something from someone, but you don’t want to put up the money today, you go to some lawyer, and say, write up a contract. I want to buy an option to buy this thing.
So, for example, if you are thinking of building a building on land that is owned now by a farmer, but you’re not ready to do it. You may be thinking about it. You can go to the farmer and say, I’d like to buy that corner of that acre there. I’d like to have the right to buy it. I’ll pay you now for the right. And you get a lawyer and you write up a contract. And that’s an option. You have an option to buy at the exercise price until the exercise date.
Now, in modern terminology, we have two kinds [addition: two kinds of options], American and European. It doesn’t refer to geography, those two terms. The terms refer, instead, to when you can exercise.
So, the American option is better than the European option for the buyer, because the American option can be exercised at any time until the exercise date. Whereas the European option can be exercised only on the exercise date. But you see the American option has to be better, or not worse than–I don’t know if it’s strictly better–but not worse than the European option because you have more options. [addition: The latter ”options” is used in the sense of flexibility.]
I think, we’ve defined what they are. Do you understand well enough what they are? They occur naturally in life. I remember, Avinash Dixit was writing about options and he said, well, when you’re dating someone and you know the person will marry you, you have an option that you can exercise at any time by agreeing to marry.
Now, one of the theorems in option theory is, you usually don’t want to exercise a call option early. [addition: This theorem abstracts from dividend-payments of the underlying asset.] And so, Dixit would say, well, maybe that’s why a lot of people have trouble getting married. They don’t want to exercise their option early.
What we’ll see is that options have option value. They give you a choice, and so there’s something there. When you exercise an option, that is, when you actually buy the thing, or in the case of a put, sell the thing, then you’re losing the choice. So, you’ve given up something. Of course, you have to also exercise eventually, if things are going to make sense.
Usually, when we talk about options, we’re talking now about options to buy a share of stock, or 100 shares of stock, and that’s the usual example. But they occur all over the place.
Let me mention some other examples of options. The usual story is the stock option. You go to your broker and you say, I’d like to buy an option to buy 100 shares of Microsoft. I don’t want to buy Microsoft, I want to buy an option to buy Microsoft, which happens to be cheaper, by the way, usually. Usually, it costs more money to actually buy the thing than to buy the right to buy the thing at another price. We’ll get back to that.
But in a sense, let’s think about this, stocks themselves are options in a sense, with a zero exercise price. Maybe, I’ll have to get back and explain, but what I mean is–let me get back and explain that in a minute.
But let me go ahead to other examples, mortgages. An ordinary home mortgage has an option characteristic to it, in the sense that, if the price of your home drops a lot, you can just walk away from the mortgage, and say, I’m out of here. It’s like not exercising an option. It’s analogous. Or I can choose to prepay a mortgage early, and that’s like exercising an option. So, option pricing gets into all sorts of things. OK.
Chapter 2. Purposes of Option Contracts [00:07:11]
I thought I should say something about the purposes of options, before I move on to try to discuss what their properties and pricing are, which is the main subject of this lecture. I can give two different justifications for options.
Why do we have options? Some people cynically think that options are just gambling vehicles. It’s another way to gamble. You can go to the casino, you can play poker, or you can buy options. Well, I think for some people that’s just what it is, they’re volatile risky investments that can make you a lot of money.
But I think, they have a basic purpose, or purposes. First of all, theoretical. If we were trying to design the ideal financial system, what would we do? Some people thought of ideal economic systems without reference to finance, like Karl Marx–I come back to him–the great communist, thought that we would have an ideal communist state and there’d be no financial markets. When they actually tried it and they tried to do it, I think they gradually realized that not having any financial markets makes our entrepreneurship, our management of enterprises, kind of blind. We can’t see where we’re going because there’s no prices. We don’t know what anything is worth. There was an old joke that the communist countries survived only because they had prices from capitalist countries to rely on. Otherwise, they don’t know anything about values or profits, right?
So, we need prices. Many people have written about this, but I mentioned, in 1964, Kenneth Arrow, who is an economic theorist, wrote a classic paper, in which he argued that, unless we have prices for all states of nature, there’s a sense in which the economic system is inefficient. You really need the price of everything, including the price of some possibility. In a sense, that’s what options are giving you, the existence of options is giving you.
So, Steven Ross, who used to teach here at Yale, a friend of mine, lives here in New Haven, in 1976, in The Quarterly Journal of Economics, wrote a classic paper about options, showing that, in a sense, they complete the state space. They create prices for everything that affects decision-making. I’m not going to get into the technicalities of the paper, but I wanted to start with a theoretical justification for options, so you’ll see why we’re doing this.
I don’t want this to come across as a lecture, how you can gamble in the options market. This is about making things work right for the economic system, improving human welfare. But a lot of people don’t get that. That’s why Karl Marx was so successful. It seems too abstract. What does this options market do for us?
Let me just go back to the example I started out with. You’re a construction firm, and you’re thinking of building something, a new supermarket, where people can buy their food. And you note that there’s a pair of expressways crossing somewhere, and you think, that’s the perfect place to build a mega supermarket, because everyone can get there by car. And there’s a lot of land, I can build a big parking lot, perfect. But before you think further, you go to buy an option on the land, right? So, you knock on the door at the farmhouse, and there’s a farmer with all these acres, and you say, I’m thinking of building a mega supermarket here. I’d like to buy an option on your farm.
You learn something at that moment. You might learn that the farmer says, I’ve already sold an option, so I can’t do it. You could try to talk to the person I sold it to, and see if you can buy it from him. Or the farmer might say, I’ve had three other offers, and I’m raising my price to some millions of millions of dollars. Then, you have second thoughts about doing it.
You see what I’m saying, that the price discovery is in there? It’s making things happen differently. You’re learning something. The farmer is learning something. You are learning something from the options market, and ultimately it decides where that supermarket will go. So, that’s the theoretical purpose of option.
I wanted to talk, also, about a behavioral purpose of options. It’s a little fuzzier about the actual benefits of options from this standpoint. The behavioral theory of options says that–very many different aspects of human behavior tie into options, but I would say it has something to do with attention anomalies and salience.
Psychologists talk about this, that people make mistakes very commonly in what they pay attention to, what strikes the fancy of their imagination. Salience is something psychologists also talk about. Salient events are events that tend to attract attention, tend to be remembered.
Now, when you think of options, a lot of options are what are called incentive options, OK? And when you get your first job, you may discover this. They’ll give you options to buy shares in the company you work for. Why do they do that? I think it’s because of certain human behavioral traits that I mention here, your attention and your salience.
It’s not necessarily very expensive for a company to give you options to buy shares in the company, but it puts you in a situation, where you start to pay attention to the value of the company. It becomes salient for you, and you start hoping that the price of the company will go up, because you have options to buy it, at a strike price. You hope that the company’s price per share goes above your strike price, because then your options are worth something. They’re in the money. So, it may change your motivation and your morale at work, or sense of identity with the company. All these sorts of things figure in. That’s why we have incentive options. They can also give you peace of mind.
Insurance is actually related to options in the sense that, when you buy insurance on your house, it’s like buying a put option on your house, although it may be not directly connected to the home’s value, right? When you buy an insurance policy in your house, and the house burns down, you collect on the insurance policy. Well, the price of your house fell to zero. If you had bought a put option on the house, it would do the same thing, right? You would have an option to sell it at a high price on something that’s now worthless. So, insurance is like options, and insurance gives you peace of mind.
So, people think in certain repetitive patterns, and one of them is, that I would like to not worry about something. So, I can get peace of mind, if I have a put option on something that I might otherwise worry about. All right, maybe that’s enough of an introduction, but I’ve given you both theoretical reasons for options and behavioral reasons. I think of them as basically inevitable. You may have people advising you not to bother with options markets. That might be right for you, in a sense, but I think that they’re always going to be with us, and so it’s something that we have to understand.
Chapter 3. Quoted Prices of Options and the Role of Derivatives Markets [00:17:11]
I have a newspaper clipping that I cut out. I’ve been teaching this course for over 20 years, so sometimes I don’t update my newspaper clippings. I have a newspaper clipping from the options page that I made in 2002, OK? So, that’s nine years ago. But I can’t update it anymore, because newspapers don’t print option prices anymore. So, I could go on some electronic trade account and get an updated option page. But why don’t we just stick with The Wall Street Journal? This is a clipping from The Wall Street Journal, April 2002, when they used to have an options page, OK?
I just picked America Online. I don’t know why. It’s an interesting company. You remember America Online, a web presence? It used to be bigger than it is now. And actually, in 2000, America Online merged with Time Warner, OK? So, we actually have two different rows corresponding–Forget Ace Limited, the second row says AOL.TW. That’s America Online Time Warner, the merged company, and then below that, they have America Online itself. These were options that were issued before the merger, and they apparently are being exercised in terms of the same AOL Time Warner stock.
AOL, by the way, was spun off by Time Warner last year, so they had a divorce. They were married in 2000. They were divorced in 2010. So you can get back to it, AOL options, now.
So, anyway, it shows the price of the share at $21.85 a share. So, you take any of these rows, and it shows you, for various strike prices, what the options prices are. So, let’s go to the top row. A strike price of 20, expiring in May of 2002, which is one month into the future. Remember, it’s April 2002 right now. The volume is the number of options that were traded yesterday, and the $2.55 up there is the price of a call option, the last price of the option to be traded yesterday. This is the morning paper. It’s reporting on yesterday morning’s prices [correction: yesterday’s prices at closing]. And then, there’s put options traded. A lot more puts were traded on that day. There were 2000 put options traded on that day in April 2002, and the last price of the put option was $0.85. For $0.85, you could buy the right to sell a share of AOL Time Warner at 20, OK? And similarly, you could buy the right to buy it at $20.00 for $2.55. So, these are different strike prices and different exercise dates. This one–I can reach it–is to buy it at, if it’s a call, $25.00 strike price, costs you $0.45 to buy that. But if you want to buy a put, it costs you $3.60.
And we want to try to understand these prices, OK? That’s the purpose here. So, let me say one thing more before I get into that. This is presented for the potential buyer, OK? These are options prices. There’s also the seller of the option. They’re called the writer of the option. I gave you an example before, when I talked about the farmer and you thinking of building a supermarket. So, you are the buyer of the option and the farmer is the writer of the option. The farmer is writing the option to you.
You could also consider buying an option from someone else, who’s not even the farmer, right? It could be some speculator. You don’t have to go to the farmer. You can go to somebody else and say, I’d like to buy an option on that farm over there. And someone would say, sure, I’ll sell you an option on it. And then I’m good for it. That means I have to go and buy it at whatever price from the farmer [addition: in case that the option is exercised]. Maybe that’s not such a good idea. He might sense my urgency to buy it.
But if it’s a stock, someone can write an option, who doesn’t even own the stock. And so, that’s called a naked seller of an option, OK? Neither the buyer nor the seller ever have to trade in the stock. This is a market by itself. You could buy an option, and then you could sell it as an option without ever exercising it. The writer could write an option, and then buy an option to cancel it out later, and then, essentially, get out of that contract.
So, the option becomes a market of its own, where prices of options start to look like an independent market, and this is called a derivatives market. There’s an underlying stock price, but this is a derivative of the stock price.
The first options exchange was the Chicago Board Options Exchange, which came in in 1973. Before that, options were traded, but they were traded through brokers and they didn’t have the same presence. You didn’t see all these options prices in newspapers. It’s when they opened the market for options, that the options trading became a big thing.
So, options markets are relatively new, if you consider ‘73 new. You weren’t born then. It’s not really that long ago. Since then, there are many more options exchanges, but CBOE is the first one. They’re now all over the world.
And we also have options on futures. And so, futures exchanges now routinely trade options on their futures contracts. So, that’s a derivative on a derivative, but it’s done.
Chapter 4. Call and Put Options and the Put-Call Parity [00:24:54]
So, let me draw a simple picture of option pricing. So, this is the stock price and this is the option price, OK? And I’m going to mark here, the exercise price. Let’s look at the exercise date, the last day. The option is about to expire, and this is your last chance to buy the stock. Then, it doesn’t matter, on that day, whether it’s an American or European option. They’re both the same on the last day.
What is the price of the option as a function of the stock price? Well, if the stock price is less than the exercise price, the option is worthless, right? It will not be exercised. You won’t exercise an option to buy it for more than you could just buy it on the market, right?
Student: You have to say ”call”.
Professor Robert Shiller: Did I not say call? Yes, I’ll put it up here. We’re talking about call options. Thank you.
But if it’s above the exercise price, this is a 45° angle, that’s a line with the slope of one, the option prices rises with the stock. In fact, it just equals the stock price minus the exercise price, right? So, this region, we say, is ”out of the money.” The option is out of the money, when its prices [clarification: the stock price], for a call, is less than the exercise price.
Here, it’s ”in the money.” I’ll put it up here, in the money. And then, on the exercise date, it will always equal the stock price minus the exercise price. So, it’s very simple.
Now, one confusion that’s often made: I gave the example of building a shopping center or a supermarket on a farm. Now, someone might think that you buy an option on it, so that you can think about it and make up your mind later. Well, in a sense, you could do that. But the thing is, you will exercise the option whether or not you build the shopping mall or this supermarket, if it’s in the money, right? Suppose, you changed your mind, and I don’t want to build the supermarket. But I’m sitting on an option that I bought, to buy his land for a price, which is less than the market price for it. Of course, I’ll buy it. So, you’re going to buy it, whether you build the shopping center or not.
You always exercise the option, if it’s in the money on the last day. That’s the assumption. I mean you could not, I suppose, if you like the farmer and you want to be a nice guy. I don’t know. But usually, what it is, is a non-linear relation between the stock price and the derivative. So, the derivative is a broken straight line function of the stock price. Whereas all the portfolios we construct, are linear. They’re straight lines. They don’t have a break in them. So, the option creates a break in the stock–and this is why Ross emphasized that options price something very different, that’s not priced in the regular–no portfolio shows you this broken straight line relation.
Now, I wanted to then talk about a put. What is a put? Let me erase, where it says in and out of the money. I’ll show it. I’ll do this with a dashed line, so that you’ll see which one is which–I’m leaving the call line up. With a put, a put is out of the money up here–I can’t really show it too well–if the stock price is above the exercise price, because you’re selling now. And it’s in the money, if the stock price is less than the exercise price–I didn’t draw that very well. That’s supposed to be a 45° line. That’s 45° angle, has a slope of -1, right? On the exercise date.
Now, it’s interesting that there’s a pretty simple pattern here between puts and calls. What if I buy one call and I short one put, all right? Or write a put, writing a put and shorting a put are the same thing, all right? What does that portfolio look like? Well, if I put that portfolio together, I want to have plus one call minus one put, all right? [addition: Both options have the same strike price and the same maturity date.] My portfolio relation to the stock price is going to look like that, right? It’s just going to be a straight line.
So, the value of my portfolio is equal to the stock price minus the exercise price. Simple as that. And my portfolio can be negative now, because I’ve shorted something. I can have a negative portfolio value. That’s very simple, can you see that?
This leads us to the put-call parity equation. If a put minus a call [correction: a call minus a put] is the same thing as the stock minus the exercise price, then the prices should add up too, right? So, put-call parity–there’s different ways of writing this. But it says that the stock price equals the call price minus put price plus exercise price on the last day, on the exercise day, right? [clarification: The two options involved have to have the same exercise price and the same maturity date.] It’s simple. This is put-call parity on the exercise date.
Now, let’s think about some day before the exercise date. [addition: The following argument only applies to European options.] Well, you know this is going to happen on the exercise date. So, at any day before the exercise date, the same thing should hold, except that we’ve got to make this the present value. Present discounted value of the exercise price. And also we have to add in, in case there is any dividends paid between now and the exercise date, plus the present discounted value of dividends paid between now and the exercise date. Because the stock gets that, and the option holders don’t, OK? So, that’s called the put-call parity relation.
And now I can cross out ”exercise date.” This should hold on all dates. Because if it didn’t hold, there would be an arbitrage, profit opportunity.
So, it should hold on this page, except for minor failures to hold. It should hold approximately on this page. And let me give you one example. See, if it holds. Let’s consider the one that I can reach. OK, oh, this is the stock price. So, what do I have? The biggest thing here is the strike price, exercise price. So, we want to do–we’ll do this line, $25.00 + $0.45 - $3.60, and I’m assuming there’s no dividend paid between now and May. It comes out very close to $21.85. I can’t do the arithmetic in my head. It may not hold exactly, because these prices may not all have been quoted at exactly the same time, and there’s some transactional costs that limit this. Do you see that?
So, because of the put-call parity relation, The Wall Street Journal didn’t even need bother to put the put prices in, because you can get one from the other. But they do put them in, just because people like to see them, and some people might be trying to profit from the put-call parity arbitrage. But for our purposes, we only have to do call pricing. Once we’ve got call pricing, we’ve got put prices. So, I just use the put-call parity relation and I get put prices.
Chapter 5. Boundaries on the Price of a Call Option [00:34:56]
So, now let’s think about how you would price puts [correction: calls]. The price of a put [correction: call], we know what it is on the exercise date, right? I’m going to forget the dashed lines. There’s no dashed lines here anymore. We’re just talking about call prices. All right, so this shows the price of a put on the last day–of a call on the last day.
Now, what about an earlier day? [clarification: The following argument about price bounds solely applies to call options. It also abstracts from dividend payments of the underlying stock.] Well, the price of a call can never be negative, right? So, the call price has to be above this line. It can never be worth less than the stock price minus the exercise price, even before the date. And also, it can’t be worth more than the stock price itself. I’ll draw a 45° line from the origin. That’s supposed to be parallels of that.
It’s obvious that the call price has to be above this broken straight line, but not too far above it. Above this broken straight line, representing the price as a function of the stock price on the last day. And the closer you get to the last day, the closer the options price will get to that curve.
So, on some day before the exercise date, the call option price will probably look something like that, right? It’s above the broken straight line because of option value. So, think of it this way, suppose an option is out of the money today–well, you can see out of the money options. For a call, this is out of the money, right? Because its stock price is $21.85, but I’ve got an option to buy it for $25.00. All right, that’s going to be worthless, unless the option price [correction: stock price] goes up before it expires.
So, it’s only worth something, because there’s a chance that it will be worth something on the exercise date. And what are people paying for that chance? $0.45, not much. Why are they paying so little? Well, you can say intuitively, it’s because it’s pretty far. $21.85 is pretty far from $25.00, and this option only has a month to go. What’s the chances that the price will go up that much? Well, there is a chance, but it’s not that big. So, I’m only willing to pay $0.45 to buy an option like that. So, we’re somewhere like here on that row that I’ve shown you.
The reason you don’t want to exercise an option early is, because, if you exercise it early, your value drops down to the broken straight line, right? It’s always worth more than the broken straight line indicates before the exercise date. So, if you want to get your money out, sell the option. Don’t exercise it early.
So, that’s why the distinction between European and American options is not as big or as important as you might think, at first. [clarification: American call options should indeed not be exercised early. However, there are circumstances under which it is optimal to exercise an American put option early.] So, we can just price European options, and then we can infer what other options would be, what put options would be worth.
Chapter 6. Pricing Options with the Binomial Asset Pricing Model [00:39:07]
Let’s now talk about pricing of options. And the main pricing equation that we’re going to use is the Black-Scholes Option Pricing equation. But, before that, I wanted to just give you a simple story of options pricing, just to give you some idea, how it works. And then I’m going to not actually derive the Black-Scholes formula, but I’m going to show it to you.
I’m going to tell you a simple story, just to give some intuitive feel about pricing of options. And to simplify the story, I’m going to tell a story about a world, in which there’s only two possible prices for the underlying stock. That makes it binomial. There’s only two things that can happen, and you can either be high or low, all right?
So, let me get my notation. I’m going to use S as the stock price, all right? I’m going to assume that the stock price, that’s today–this is also a simple world in that there’s only one day. The option expires tomorrow. There’s only one more price we’re going to see. So, the stock is either going to go up or down.
So, u is equal to one plus the fraction up that it goes up. u stands for up. And d is down, is one plus the fraction down. So, that means that stock price either becomes Su, which means it goes up by a fraction, multiple u, or is Sd, which means it goes down by a multiple d. And that’s all we know, OK?
But now we have a call option. Call C the price of the call. We’re going to try to derive what that is. But we know, from our broken straight line analysis, we know what Cu is, the price if the stock goes up. And we know what Cd is, it’s the price if down, OK? So, suppose the option has exercise price E, all right. Do you understand this world? Simple story.
Now, what I want to do is consider a portfolio of both the stock and the option, that is riskless. I’m going to buy a number of options equal to H. H is the hedge ratio, which is the number of shares purchased per option sold. I’m going to sell a call option to hedge the stock price, to reduce the risk of the stock price, OK? And so, hedge ratio is shares purchased over options. Each option is to buy one share, OK? So, what I’m going to do is, write one call and buy H shares. So, let me erase this and start over again. I’m on my way to deriving the options price for you–a little bit of math.
So, I’m going to write one call and buy H shares, OK. If the stock goes up, if we discover we’re in an up world next period, my portfolio is worth uHS minus Cu, right? Because the share price goes from S to uS, and I’ve got H shares, and I’ve written a call, so I have to pay Cu. If it’s down, then my portfolio is dHS minus Cd, OK? This is simple enough?
Now, what I want to do is eliminate all risk. So, that means I want to choose H, so that these two numbers are the same. And if I do that, I’ve got a riskless investment, all right? So, set these equal to each other. And that implies something about H. We can drive what H is, if I just put these two equal to each other and solve for H. And I get H=Cu-Cd/(u-d)S.
So, I’ve been able to put together a portfolio of the stock and the option that has zero risk. If I do this, if I hold this amount of shares in my portfolio, I’ve got a riskless portfolio. So, that means that the riskless portfolio has to earn the riskless rate, right? It’s the same thing as a riskless rate [correction: same thing as a riskless investment], so it has to earn that [clarification: earn the riskless rate]. If I can erase this now, I’m almost there, through option pricing.
The option pricing then says that, since I’ve derived what H is, the portfolio has to be worth (1+r) times what I put in, which is HS minus C. And that has to equal the value of the portfolio at the end, which is either uHS minus Cu, or dHS minus Cd, the same thing, OK?
So, I’ve already derived what H is, and I substituted into that, and I solved for C. So, substitute H in and solve for C, and we get the call price, OK? It’s a little bit complicated, but C=[(1+r-d/u-d)x(Cu/1+r)]+[(u-1-r/u-d)x(Cd/1+r)].
And I’ll put a box around that because that’s our option price formula, OK? Did you follow all that?
This is derived–this option price formula was derived from a no arbitrage condition. Arbitrage, in finance, means riskless profit opportunity. And the no arbitrage condition says, it’s never possible to make more than the riskless rate risklessly, all right?
If I could, suppose I had some way–suppose the riskless rate is 5%, and I can make 6% risklessly, then I will borrow at the riskless rate and put it into the 6% opportunity. And I’ll do that till kingdom come. There’s no limit to how much I’ll do that. I’ll do it forever. It’s too much of a profit opportunity to ever happen. One of the most powerful insights of theoretical finance is, that the no arbitrage condition should hold.
It’s like saying, there are no $10 bills on the pavement. When you walk down the street and you see a $10 bill lying there on the street, your first thought ought to be, are my eyes deceiving me? Because somebody else would have picked it up if it were there. How can it be there?
I once actually had that experience. I was walking down the street in New York. It was actually a $5 bill. It was just lying there in the street. And so, I reached down to pick it up, and then, suddenly, it disappeared. And it was people on one of the stoops of one of these New York townhouses playing a game. They’d tied a string to a $5 bill. And they would leave it on the street, and watch people reach for it, and they’d snatch it away. That’s the only time in my life I ever saw a $5 bill on the pavement. And so, it’s a pretty good assumption that, if you see one, it isn’t real.
And that’s all this is saying, that if the option price didn’t follow this formula, something would be wrong. And so, it had better follow this formula. Now, that is the basic core option theory.
Now, the interesting thing about this theory is, I didn’t use the probability of up and the probability of down. So somebody says, wait a minute, my whole intuition about options is: I’d buy an option, because it might be in the money. When I was just describing here, this is $0.45, I said, that’s not much, because it probably won’t exceed $25.00. It’s so far below it. So, it seems like the options should really be fundamentally tied to the probability of success. But it’s not here at all. There’s no probability in it. You saw me derive it. Was I tricking you? Well, I wasn’t. I don’t play tricks. This is absolutely right. You don’t need to know the probability that it’s in the money to price an option, because you can price it out of pure no-arbitrage conditions.
Chapter 7. The Black-Scholes Option Pricing Formula [00:51:02]
So, that leads me then to the famous formula for options pricing, the Black-Scholes Option Pricing formula, which looks completely different from that. But it’s a kindred, because it relies on the same theory. And there it is. This was derived in the late 70’s, or maybe early 70’s, by Fisher Black, who was at MIT at the time, I think, but later went to Goldman Sachs, and Myron Scholes, who is now in San Francisco, doing very well. I see him at our Chicago Mercantile Exchange meetings. Fisher Black passed away.
It doesn’t have the probability that the option is in the money, either, but it looks totally different from the formula that I wrote over there. The call price is equal to S·
where S is the share price, r the interest rate, T time to maturity, E the exercise price, and d1 as well as d2 are given on the slide:
And the N function is the cumulative normal distribution function.
I’m not going to derive all that, because it involves what’s called the calculus of variations. I don’t think most of you have learned that. In ordinary calculus, we have what’s called differentials, dy, dx, et cetera. Those are fixed numbers in ordinary calculus. In the mid 20th century, mathematicians, notably the Japanese mathematician Ito, developed a random version of calculus, where dx and dy are random variables. That’s called the stochastic calculus. But I’m not going to use that. I’m not going to derive this.
But you can see how to price an option using Black-Scholes. But Black-Scholes is derived, again, by the no-arbitrage condition and it doesn’t have the probability. Oh, the other variable that’s significant here is sigma, which is the standard deviation of the change in the stock price.
So, once we put that in, someone could say, well, probabilities are getting in through the back door, because this is really a probability weighted sum of the changes in stock prices. Well, probability is not really in here at all, but maybe there’s something like standard deviation, even in this equation. Because we had Cu and Cd, and that’d give you some sense of the variability. [clarification: In the binomial asset pricing model, u and d give you some sense of the variability of the underlying stock price, analogous to sigma in the Black-Scholes formula.]
I’m going to leave this equation just for you to look at. But what it does do is, it shows the option price as a nice curvilinear relationship, just like the one I drew by hand. Which then, as time to exercise goes down, as we get close to the exercise date, that curve eventually coincides with the broken straight line.
Now, I wanted to tell you about implied volatility. This equation can be used either of two ways. The most normal way to do it, to use this equation, is to get the price that you think is the right price for an option, to decide whether I’m paying too much or too little for an option.
So with this formula, I can plug in all the numbers. To use this formula, I have to know what the stock price is. That’s S. I have to know what the exercise price is. And I have to know what the time to maturity–these are all specified by the stock price and the contract. I have to know with the interest rate is. And if I also have some idea of the standard deviation of the change in the stock price, then I can get an option price.
Chapter 8. Implied Volatility - The VIX Index in Comparison to Actual Market Volatility [00:55:49]
But I can also turn it around. If I already know what the option is selling for in the market, I can infer what the implied sigma is, right? Because all the other numbers in the Black-Scholes formula are clear. They’re in the newspaper, or they’re in the option contract. There’s this one hard to pin down variable, what is the variability of the stock price?
And so, what people often use the Black-Scholes formula to is, to invert it and calculate the implied volatility of stock prices. So, when call option prices are high, why are they high relative to other times? Well, it must be that people think–I’m going back to the old interpretation, that the probability of exercise is high, right? If an out of the money call is valuable, it must be people think that sigma is high.
So, let’s actually solve for how high that is. I can’t actually solve this equation. I have to do it numerically. But I can calculate, for any call price, given the stock price, exercise price, time to maturity, and interest rate. I can calculate what volatility would imply that stock price. And so, that’s where we are with Black-Scholes.
So, implied volatility is the options market’s opinion as to how variable the stock market will be between now and the exercise date. So, one thing we can do is compute implied volatility. And I have that here on this chart here. What I have here, from 1986 to the present, with the blue line, is the VIX, V-I-X, which is computed now by the Chicago Board Options Exchange. When the CBOE was founded, they didn’t know how to do this. Black and Scholes invented their equation in response to the founding of the CBOE. And now, the CBOE publishes the VIX. You can get it on their website, and that’s where I got this, off their website cboe.com. [Clarification: The VIX is computed from a different formula involving more than one options price.]
And so, they have computed, based on the front month, the near options, what the options market thought the volatility of the stock market was. That’s the blue line. And you can see, it had a lot of changes through time. That means that options prices were revealing something about the volatility of the stock market.
Now, the blue line is from the Chicago Board Options Exchange. What I did, and I calculated this myself, the orange line is the standard deviation of actual stock prices over the preceding year, of monthly changes, annualized. That’s actual volatility. But it’s actual past volatility.
Let’s make it clear, what this is. What the VIX is, is [essentially, not exactly] the sigma in the Black-Scholes equation. But it is, in effect, the market’s expected standard deviation of stock prices. And to get it more precise, it’s the standard deviation of the S&P 500 Stock Price Index for one month, multiplied by the square root of 12, because they want to annualize it. It’s for the next month.
Why do they multiply it by √12? Well, that’s because, remember the square root rule. These stock prices are essentially independent of each other month to month, so the standard deviation of the sum of 12 months is going to be √12 times a standard deviation of one month.
And this is in percent per year. So, that means that the implied volatility in 1986 was 20%. And then, it shot way up to 60%, unimaginably quick. That might be the record high, I can’t quite tell from here. Remember, I told you the story of the 1987 stock market crash? The stock market fell over 22% in one day. Well, actually, on the S&P, it was only 20%, but a lot in one day.
It really spooked the options markets. So, the call option prices went way up, thinking that there’s some big volatility here. We don’t know which way it’ll be next. Maybe it will be up, maybe it will be down. It pushed the implied volatility, temporarily, up to a huge level. It came right back down.
My actual volatility, I calculated this for each date as the volatility of the market over the preceding year. Well, since I put October 1987 in my formula, I got a jump up in actual volatility, but not at all as big as the options market did.
See, the option market is looking ahead and I have no way to look ahead, other than to look at the options market. So, to get my actual volatility, I was obliged to look at volatility in the past, and it went up because of the 1987 volatility, but not so much. So what this means is, that, in 1987, people really panicked. They thought something is really going on in the stock market. They didn’t know what it was and they were really worried, and that’s why we see this spike in implied volatility.
There’s a couple other spikes that I’ve noted, the Asian financial crisis occurred in the mid 1990’s. Now, that is something that was primarily Asian, but it got people anxious over here as well. You know, Korea, Taiwan, Indonesia, Hong Kong, these countries had huge turmoil. But it came over here in the form of a sudden spike in expected volatility. People thought, things could really happen here. So, all the option prices got more valuable.
And then there’s this spike. This is the one that you remember. This is the financial crisis that occurred in the last few years. Notably, it peaks in the fall of 2008, which was the real crisis, when Lehman Brothers collapsed, and it created a crisis all over the world. There was a sharp and sudden terrible event. And you can see that actual volatility shot up to the highest since 1986, as well, at that time.
So, implied volatility, you can’t ask easily from this chart, whether it was right or wrong. People were responding to information, and the response felt its way into options prices. There’s no way to find out, ex post, whether they were right to be worried about that. But they were worried about these events, and it led to big jumps in options prices.
Now, I wanted to show the same chart going back even further, but I can’t do it with options prices, because I can show volatility earlier, but I can’t show implied volatility before around 1986, because the options markets weren’t developed yet.
But I computed an actual S&P Composite volatility. Well, in my chart title, I said S&P 500. The Standard and Poor 500 Stock Price Index technically starts in 1957, but I’ve got what they call the Standard and Poor Composite back to 1871.
And so, these are the actual moving standard deviations of stock prices, all the way back to the beginnings of the stock market in the U.S. Well, not the very beginnings, but the earliest that we can get consistent data for, on a monthly basis. And you can see, this goes back further than the other chart. You can see that the actual volatility of stock prices, except for one big event, called the Great Depression of the 1930s, has been remarkably stable, right? The volatility in the late 20th century, early 21st century, is just about exactly the same as the volatility in the 19th century. It’s interesting, how stable these patterns are.
There was this one really anomalous event that just sticks out, and that is the Great Depression. 1929 precedes it, it’s somewhere in here. But somehow people got really rattled by the 1929 stock market crash. And not just in the U.S. This is U.S. data, but you’ll find this all over the world. It led to a full decade of tremendous stock market volatility around the world, that has never been repeated since.
The recent financial crisis has the second highest volatility after the Great Depression. This isn’t long ago. This is well within your memories. Just a few years ago, we had another huge impact on volatility. And as you saw on the preceding slide, it had a big impact on implied volatility as well. So, I think that we had a near miss of another depression. It’s really scary what happened in this crisis.
Also shown here is the first oil crisis, which we talked about, in 1974, when oil prices had been locked into a pattern because of the stabilization done by the Texas Railroad Commission. But when that broke, and OPEC first flexed its muscles, it created a sense of new reality. And it caused fear, and it caused a big spike up in volatility of the stock market, but not quite as big as the current financial crisis.
So, this is an interesting chart to me. A lot of things I learned from this chart, and let me conclude with some thoughts about this. But what I learned from this chart is that, somehow, financial markets are very stable for a long time.
So, it would seem like it wouldn’t be that much of an extrapolation–when are you people going to retire? You’ve pick a retirement date yet? Well, let’s say a half century from now, ok? So, that would be 2060? So, you’re going to retire out here, all right? Your whole life is in here. What do you think volatility is going to do over that whole period? Well, judging from the plot, it’s probably pretty similar, right? That’s not much more history compared to what we’ve already seen. Probably just going to keep doing this. But there’s this risk of something like this happening again. And we saw a near miss here, but this plot encourages me to think that maybe outliers, or fat tails, or black swan events, are the big disruptors to economic theory.
Black-Scholes is not a black swan theory. It assumes normality of distributions, and so, it’s not always reliable. So, this leads me to think that option pricing theory–I presented a theory. The Black-Scholes theory is very elegant and a very useful tool, especially useful when things behave normally. But, I think, one always has to keep in the back of one’s mind, the risk of sudden major changes like we’ve seen here.
Chapter 9. The Potential for Options in the Housing Market [01:09:33]
So, let me just you give us some final thoughts about options. I launched this lecture by saying, they’re very important. And they affect our lives in many ways. I’ve been trying to campaign for the expansion of our financial markets.
Working with my colleagues and the Chicago Mercantile Exchange, we launched options, in 2006, on single-family homes in the United States. We were hoping that people would buy put options to protect themselves against home price declines, but the market never took off. We have, since, seen huge human suffering because of the failure of people to protect themselves against home price declines.
There were various noises that were made by people in power, that suggested that maybe something could be done. President Obama proposed something called Home Price Protection Program, and it sounded like an option, a put option program. But, actually, it was a much more subtle program than that. It was a program to incentivize mortgage originators to do workouts on mortgages, if the mortgages would default–if home prices were to fall. And nothing really happened with it. The President can’t get things started, either, always.
I’ve been proposing that mortgages should have put options on the house attached to them. When you buy a house, get a mortgage, you should automatically get a put option. I’ve got a new paper on that.
But these are kind of futuristic things at the moment. I’m just saying this at the end, just to try to impress on you, what I think is the real importance of options markets. People don’t manage risks well in the present world. Having options or insurance-like contracts of an expanded nature will help people manage their risks better, and it will make for a better world.
OK. I’ll see you on Monday.
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