Options Trading Lecture Master Class From Yale Economics Professor

TITLE: Options Trading Lecture - Master Class from Yale Economics professor CHANNEL: Rush Street Capital DATE: 2026-04-14 ---TRANSCRIPT--- This subject of today’s lecture is options. And I want to I think maybe I better first define what an option is before I move to just saying 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 and I’ll get back to that. Let me just define terms here.

So, there’s two kinds of options. There’s a call and a put. 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 that has to be specified and that’s the exercise date.

Options go back thousands of years. It must have happened before we have any recorded records. If you want to buy something 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 American and European. And 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 a European option can be exercised only on the exercise date. But you see the American option has to be better or not worse than a European option because you have more options.

I think we’ve defined 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 which 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. And so Dixit was saying 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, it means 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.

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.

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.

But in a sense, let’s think about this. Stocks themselves are options in a sense with a zero exercise price.

Mortgages ordinary home mortgage has an option characteristic to it in the sense that if the price of your home drops a lot and you can just walk away from the mortgage and say I’m out of here. It’s like not exercising an option. Or I can choose to prepay a mortgage early. And that’s like exercising an option. The option pricing gets into all sorts of things.

I thought that 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 instruments that can make you a lot of money. But I think they have a basic purpose.

First of all theoretical. If we’re 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, the great communist thought that we would have an ideal communist state and there’d be no financial markets. When they actually tried it 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 cuz 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.

In 1964 Kenneth Arrow who is an economic theorist wrote a classic paper in which he argued that we don’t 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.

Stephen Ross who used to teach here at Yale, friend of mine, 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 that 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.

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. 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. But before you think further, you go to buy an option on the land. 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.” Or the farmer might say, “I’ve had three other offers and I’m raising my price.” And 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 options.

The behavioral theory of options says that 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 their fancy or their imagination. Salient events are events that tend to attract attention, tend to be remembered.

When you think of options, a lot of options are what are called incentive options. 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. 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 cuz then your options are worth something, they’re in the money. So, it may change your motivation and your morale at work, your sense of identity with the company.

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. When you buy an insurance policy on 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. So, insurance is like options. And insurance gives you peace of mind.

I think of them as basically inevitable. You may have people advising you not to bother with options markets. And 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.

I have a newspaper clipping from the options page that I made in 2002. And so, that’s 9 years ago, but I can’t update it anymore cuz newspapers don’t print option prices anymore.

This is a clipping from the Wall Street Journal April 2002 when they used to have an options page. And I just picked America Online. And actually in 2000, America Online merged with Time Warner.

Under every row it shows the price of the share at $21.85 a share. So, you can take any of these rows and it shows you for various strike prices what the options prices are.

Let’s go to the top row. A strike price of 20 expiring in May of 2002, which is like 1 month into the future. 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 option to be traded yesterday. And then there’s put options traded. A lot more puts were traded on that day. There were 2,000 put options traded on that day in April 2002. And the last price of the put option was 85 cents.

So, for 85 cents, you could buy the right to sell a share of AOL Time Warner at $20. And similarly, you could buy the right to buy it at $20 for $2.55. So, these are different strike prices and different exercise dates.

This is presented for the potential buyer. There’s also the seller of the option. They are called the writer of the option.

You could also consider buying an option from someone else who’s not even the farmer. It could be some speculator. 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.

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 before as an option without ever exercising it. 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 because there’s an underlying stock price but this is a derivative of the stock price.

They were first traded 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.

Since then, there are many more options exchanges. 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.

So, let me draw a simple picture of option pricing. This is the stock price. And this is the option price. 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. 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. We’re talking about call options.

But if it’s above the exercise price this is a 45° angle. That’s a line with a slope of one. The option price rises with the stock price. In fact, it just equals the stock price minus the exercise price.

So, this region we say is out of the money. The option is out of the money when its price for a call is less than the exercise price. Here, it’s in the money. And then, on the exercise date, it will always equal the stock price minus the exercise price.

Now, what one confusion that’s often made. I gave the example of building a shopping center on a farm. Someone might think that you buy an option on it so that you can think about it and make up your mind later. But the thing is you will exercise the option whether or not you build the shopping mall or the supermarket if it’s in the money. You always exercise the option if it’s in the money on the last day.

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 don’t have a break in them. So, the option creates a break. And this is why Ross emphasized that options price something very different.

Now, I wanted to then talk about a put. With a put, a put is out of the money if the stock price is above the exercise price. And it’s in the money if the stock price is less than the exercise price. That’s a 45° line with a slope of minus one.

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? What does that portfolio look like? It’s just going to be a straight line. My portfolio is equal to the stock price minus the exercise price.

This leads us to the put-call parity equation. If a put minus a call is the same thing as the stock minus the exercise price, then the prices should add up, too. Put-call parity: the stock price equals the call price minus put price plus exercise price. On the last day, on the exercise day. This is put-call parity on the exercise day.

Now, let’s think about someday before the exercise date. On any day before the exercise date, the same thing should hold except that we’ve got to make this the present discounted value of the exercise price. Plus the present discounted value of dividends paid between now and the exercise date. Cuz the stock gets that and the option holders don’t. So, that’s called the put-call parity relation. And this should hold on all dates because if it didn’t hold, there would be an arbitrage profit opportunity.

Let me give you one example. $25 plus 45 cents minus $3.60. And I’m assuming there’s no dividend paid between now and May. Comes out very close to 21.85. It may not hold exactly because these prices may not all have been quoted at exactly the same time and there’s some transactions costs.

Because of put-call parity relation, the Wall Street Journal didn’t even need to bother to put the put prices in because you can get one from the other. But for our purposes, we only have to do call pricing. Once we’ve got call pricing, we’ve got put prices cuz I just use the put-call parity relation.

Now, the price of a call on the last day. We know what it is. What about an earlier day? Well, the price of a call can never be negative. It can never be worth less than the stock price minus the exercise price before the date. And also, it can’t be worth more than the stock price itself.

The call price has to be above the broken straight line, but not too far above it. And the closer you get to the last day, the closer the option’s price will get to that curve.

Suppose an option is out of the money today. 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? 45 cents. Why are they paying so little? It’s because $21.85 is pretty far from $25. And this option only has a month to go.

But 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. 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.

Let’s now talk about pricing of options. 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. 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.

Let me get my notation. I’m going to use S as the stock price. And 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 1 plus the fraction up. And D is 1 plus the fraction down. So, the stock price either becomes SU or SD. And that’s all we know. But now we have a call option called C the price of the call.

We know what CU is the price if the stock goes up. And we know what CD is the price if down.

Suppose the option has exercise price E. 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 shares equal to H. H is the hedge ratio which is the number of shares purchased per option. So, I’m going to sell a call option to hedge the stock price.

So, I’m going to write one call and buy H shares. If the stock goes up, my portfolio is worth UHS minus C sub U. If it’s down then my portfolio is DHS minus C sub D.

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. If I do that, I’ve got a riskless investment.

Set these equal to each other. And that implies H equals CU minus CD all over (U minus D) times S.

So, I’ve been able to put together a portfolio of the stock and the option that has zero risk. So, that means that the riskless portfolio has to earn the riskless rate.

The portfolio has to be worth one plus the interest rate times what I put in which is HS minus C and that has to equal UHS minus CU or DHS minus CD, the same thing.

Substitute H in and solve for C. And we get the call option price has to equal (1+R-D)/(U-D) times CU/(1+R) plus (U-1-R)/(U-D) times CD/(1+R).

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. If I could, 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.

So, 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. I once actually had that experience. I was walking down the street in New York and I saw a $5 bill just lying there on the street. And so, I reached down to pick it up and then suddenly it disappeared. It was people on the stoop of one of these New York townhouses playing a game where they had 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.

Now, the interesting thing about this theory is I didn’t use the probability of up and the probability of down. Somebody said, “Wait a minute. My whole intuition about options is I buy an option because it might be in the money.” 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 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 condition.

So, that leads me then to the famous Black-Scholes option pricing formula. Which looks completely different from that, but it’s kindred because it relies on the same theory.

This was derived in the early ’70s by Fisher Black, who was at MIT at the time but later went to Goldman Sachs. And Myron Scholes, who is now in San Francisco. Fisher Black passed away.

So, it doesn’t have the probability that the option is in the money, either. The call price is equal to the share price, S, times N of D1, minus E to the minus R times T times the exercise price times N of D2. 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. In ordinary calculus, we have differentials, dy, dx, etc. Those are fixed numbers. 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.

The other variable that’s significant here is sigma, which is the standard deviation of the change in the stock price. And once we put that in, someone could say probabilities are getting in through the back door because this is really a probability-weighted sum of the changes in stock prices.

This equation shows the option price as a nice curvilinear relationship. Which then as time to exercise goes down, 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 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.

But I can also turn it around if I already know what the option is selling for in the market, and I can infer what the implied sigma is. 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?

So, what people often use the Black-Scholes formula for is to invert it and calculate the implied volatility of stock prices. When call option prices are high, it must be that people think sigma is high.

Implied volatility is the options market’s opinion as to how variable the stock market will be between now and the exercise date.

I have a chart from 1986 to the present with the blue line the VIX, which is computed now by the Chicago Board Options Exchange. Black and Scholes invented their equation in response to the founding of the CBOE. And now the CBOE publishes the VIX.

The VIX is the sigma in the Black-Scholes equation. It is the market’s expected standard deviation of the S&P 500 stock price index for 1 month multiplied by the square root of 12 because they want to annualize it.

Why do they multiply by the square root of 12? Because of the square root rule. Stock prices are essentially independent month to month. So, the standard deviation of the sum of 12 months is going to be square root of 12 times the standard deviation of 1 month.

The implied volatility in 1986 was 20%. And then it shot way up to 60% unimaginably. Remember the 1987 stock market crash? Stock market fell over 22% in 1 day. It really spooked the options market. So, the call option prices went way up thinking that there’s some big volatility here. It pushed the implied volatility temporarily up to a huge level. It came right back down.

My actual volatility for each date was the volatility of the market over the preceding year. Since I put October 1987 in my formula, I got a jump up in actual volatility. But not as big as the options market did. The option market is looking ahead.

There’s a couple other spikes. The Asian financial crisis occurred in the mid 1990s. Korea, Taiwan, Indonesia, Hong Kong had huge turmoil, but it came over here in the form of a sudden spike in expected volatility.

And then the financial crisis peaks in the fall of 2008 when Lehman Brothers collapsed. The recent financial crisis has the second highest volatility after the Great Depression.

I also showed a chart going back further with actual S&P composite volatility back to 1871. You can see that the actual volatility of stock prices has been except for one big event called the Great Depression remarkably stable. The volatility in the late 20th century, early 21st century is just about exactly the same as the volatility in the 19th century.

There was this one really anomalous event the Great Depression. 1929 precedes it. Somehow people got really rattled by the 1929 stock market crash. It led to a full decade of tremendous stock market volatility around the world that has never been repeated since.

Black-Scholes is not a black swan theory. It assumes normality of distributions. And so, it’s not always reliable. One always has to keep in the back of one’s mind the risk of sudden major changes.

I launched this lecture by saying options are very important. 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.

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.

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.