The Trillion Dollar Equation

- This single

equation

spawned four multi-billion

dollar

industries and transformed the way everyone approaches risk. Do you think most people are aware of the size, scale and usefulness of derivatives? - No. No idea. - But in essence, this

equation

comes from physics, from the discovery of atoms, from understanding how heat is transferred and how to beat the casino at blackjack. So perhaps it shouldn't be surprising that some of the best who managed to beat the stock market were not veteran traders, but physicists, scientists and mathematicians. In 1988, a math professor named Jim Simons created the Medallion Investment Fund, and every year for the next 30 years, the Medallion fund generated higher returns than the market average, and not just by a little, but it earned a performance of 66% annually. .

At that growth rate, $100 invested in 1988 would be worth $8.4 billion today. This made Jim Simons easily the richest mathematician of all time. But being good at math does not guarantee success in the financial markets. Ask Isaac Newton. In 1720 Newton was 77 years old and wealthy. He had made a lot of money working as a professor at Cambridge for decades and had a side job as a master of the Royal Mint. His net worth was £30,000, the equivalent of $6 million today. Now, to grow his fortune, Newton invested in stocks. One of his big bets was on the South Sea Company.

His business was shipping enslaved Africans across the Atlantic. Business was booming and the stock price rose rapidly. By April 1720, the value of Newton's shares had doubled. He then sold his shares. But the stock price continued to rise, and in June, Newton bought back and continued buying shares even as the price peaked. When the price started to drop, Newton didn't sell. He bought more shares thinking he was buying the dip. But there was no recovery and he eventually lost about a third of his wealth. When asked why he didn't see it coming, Newton replied: "I can calculate the movements of celestial bodies, but not the madness of people." So what did Simons get right and what did Newton get wrong?

Well, for starters, Simons was able to stand on the shoulders of giants. The pioneer in using mathematics to model financial markets was Louis Bachelier, born in 1870. His parents died when he was 18 and he had to take over his father's wine business. He sold the business a few years later and moved to Paris to study physics, but he needed a job to support himself and his family and found one on the Paris Stock Exchange. And inside him was Newton's "folly madness" in its rawest form. Hundreds of traders shouting prices, making hand signals and making deals. What caught Bachelier's interest were contracts known as options.

The first known options were purchased around 600 BC. C. by the Greek philosopher Thales of Miletus. He believed that the next summer would produce a bumper crop of olives. To make money with this idea, he could have bought olive presses, which, if he was right, would be in great demand, but he did not have enough money to buy the machines. So he went to all the existing olive press owners and paid them a little money to secure the option to rent their presses in the summer for a certain price. When the harvest came, Tales was right, there were so many olives that the price of renting a press skyrocketed.

Thales paid the press owners the pre-agreed price, then rented the machines at a higher price and pocketed the difference. Thales had executed the first known call option. A call option gives you the right, but not the obligation, to buy something at a later date for a fixed price known as the strike price. He can also buy a put option, which gives him the right, but not the obligation, to sell something at a later date for the strike price. Put options are useful if he expects the price to go down. Call options are useful if he expects the price to rise.

For example, let's say Apple's current stock price is one hundred

dollar

s, but you expect it to go up. You could buy a call option for $10 that gives you the right, but not the obligation, to buy Apple stock in one year for one hundred dollars. That is the strike price. Just a little side note: American options can be exercised on any date up to the expiration date, while European options must be exercised on the expiration date. To keep things simple, we'll stick to the European options. So, if in one year Apple's stock price has risen to $130, you can use the option to buy shares for a hundred dollars and then immediately sell them for $130.

After taking into account the $10 you paid for the option, you made a profit of $20. Alternatively, if in one year stock prices fell to $70, you simply would not use the option and would have lost the $10 you paid for it. So the profit and loss diagram looks like this. If the stock price ends up below the strike price, you lose what you paid for the option. But if the stock price is greater than the strike price, then you earn that difference minus the cost of the option. There are at least three advantages of the options. One is that it limits its disadvantages.

If you had bought the stock instead of the option and it had gone down to $70, you would have lost $30. And in theory, you could have lost a hundred if the stock went to zero. The second benefit is that options provide leverage. If you bought the stock and it went up to $130, then your investment grew by 30%. But if you had bought the option, you only had to contribute $10. So your $20 profit is actually a 200% return on investment. On the negative side, if you had owned the shares, your investment would have only dropped 30%, while with the option you would lose 100%.

So with options trading, there is the potential for much larger profits, but also much larger losses. The third benefit is that you can use options as a hedge. - I think the original motivation for options was to find a way to reduce risk. And then, of course, once people decided they wanted to buy insurance, that meant there were other people who wanted to sell it or make a profit, and that's how markets are created. - So options can be an incredibly useful investment tool, but what Bachelier saw on the trading floor was chaos, especially when it came to the pricing of stock options.

Although they had been around for hundreds of years, no one had found a good way to price them. Traders would simply negotiate to reach an agreement on what the price should be. - Given the option to buy or sell something in the future, it seems like a very amorphous type of trading. Therefore, determining the prices of these rather strange objects has been a challenge that has plagued various economists and businessmen for centuries. -Now, Bachelier, already interested in probability, thought that there had to be a mathematical solution to this problem and proposed this as a doctoral topic to his advisor Henri Poincaré.

Investigating the mathematics of finance wasn't really something people did back then, but to Bachelier's surprise, Poincaré agreed. To accurately value an option, you first need to know what happens to stock prices over time. The price of a stock is basically set by a tug of war between buyers and sellers. When more people want to buy a stock, the price goes up. When more people want to sell a stock, the price drops. But the number of buyers and sellers can be influenced by almost anything, such as weather, politics, new competitors, innovation, etc. Bachelier then realized that it is practically impossible to predict all of these factors accurately.

So the best you can do is assume that at any point in time the stock price has the same probability of going up or down, and therefore in the long run, stock prices follow a random walk, moving up and down. down as if his next move was determined by the toss of a coin. - Randomness is a hallmark of an efficient market. Efficient economists often say that you cannot make money from trading. - The idea that you shouldn't be able to buy an asset and immediately sell it for a profit is known as the Efficient Market Hypothesis. - The more people try to make money by predicting the stock market and then trading on those predictions, the less predictable those prices become.

If you and I could predict tomorrow's stock market, we would. We would start trading today in stocks that we thought would rise tomorrow. Well, if we did that, instead of going up tomorrow, they would go up now as we buy more and more shares. So the very act of predicting actually affects the quality of future outcomes. And so, in a fully efficient market, tomorrow's prices cannot have any predictive power. If they had, we would have taken advantage of it today. - This is a Galton board. It has rows of pegs arranged in a triangle shape and about 6,000 little ball bearings that I can fit through the pegs.

Now, every time a ball hits a peg, there is a 50% chance that it will go left or right. Each ball then follows a random path as it passes through these pegs, making it basically impossible to predict the trajectory of any individual ball. But if I turn it around, what you can see is that all the balls together always create a predictable pattern. That is, a collection of random walks creates a normal distribution. It is centered in the middle because the number of paths a ball could take to get here is the greatest. And the further you go, the fewer paths a ball can take to get there.

If you want to end up here, well, the ball would have to go left, left, left, down. So there's only one way to get here, but to get to the middle, there are thousands of paths a ball could take. Now, Bachelier believed that the price of a stock is like a ball passing through a Galton board. Each additional layer of pegs represents a time step. So after a short time, the stock price might only rise or fall a little, but after more time, a wider range of prices is possible. According to Bachelier, the expected future price of a stock is described by a normal distribution, centered on the current price that extends over time.

Bachelier realized that he had rediscovered the exact equation that describes how heat radiates from high-temperature regions to low-temperature regions. This was first discovered by Joseph Fourier in 1822. That is why Bachelier called his discovery the radiation of probabilities. Because he was writing about finance, the physics community didn't pay attention to him, but the mathematics of the random walk would go on to solve a nearly century-old physics mystery. In 1827, Scottish botanist Robert Brown was observing pollen grains under a microscope and noticed that particles suspended in water on the microscope slide moved randomly. Since he didn't know if he had anything to do with pollen being living material, he tested non-organic particles like lava dust and meteorite rocks.

He again saw them move in the same way. Brown then discovered that any particle, if small enough, exhibited this random motion, which became known as Brownian motion. But the cause remains a mystery. Eighty years later, in 1905, Einstein discovered the answer. Over the past two hundred years, the idea that gases and liquids were made of molecules became increasingly popular. But not everyone was convinced that molecules were real in a physical sense. Only that the theory explained many observations. The idea led Einstein to hypothesize that Brownian motion is caused by billions of molecules hitting the particle from all directions, at every instant.

Occasionally, they will hit more on one side than the other, and the particle will jump momentarily. To derive the mathematics, Einstein assumed that as an observer we cannot see or predict these collisions with certainty. So at any time we must assume that the particle is just as likely to move in one direction as another. So, just like stock prices, microscopic particles move like a ball falling down a Galton chart, the expected location of a particle being described by a normal distribution, which broadens over time. Therefore, even in completely calm waters, microscopic particles spread. This is diffusion.

Solving the mystery of Brownian motion. Einstein had found definitive proof that atoms and molecules exist. Of course, he had no idea that Bachelier had discovered the riderandom five years earlier. By the time Bachelier finished his PhD, he had finally discovered a mathematical way to price an option. Remember that with a call option, if the future price of a stock is less than the strike price, you will lose the premium paid for the option. But if the stock price is higher than the strike price, you pocket that difference and make a net profit if the stock has risen more than you paid for the option.

So the probability that an option buyer will make a profit is the probability that the price will increase by more than the price paid for it, which is the green shaded area. And the probability that the seller makes money is simply the probability that the price stays low enough that the buyer doesn't make more than he paid for it. This is the area shaded in red. By multiplying the gain or loss by the probability of each outcome, Bachelier calculated the expected return of an option. Now how much should it cost? If the price of an option is too high, no one will want to buy it.

On the other hand, if the price is too low, everyone will want to buy it. Bachelier argued that fair price is what equalizes the expected return for buyers and sellers. Both parties should win or lose the same amount. That was Bachelier's idea about how to accurately value an option. By the time Bachelier finished his thesis, he had beaten Einstein to the invention of the random walk and solved the problem that had eluded options traders for hundreds of years. But no one noticed. Physicists were not interested and merchants were unprepared. The key that was missing was a way to make a lot of money.

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So if you want to try it for yourself, click the link in the description and thanks again to Eight Sleep for sponsoring this part of the video. In the 1950s, a young physics graduate, Ed Thorpe, was doing his PhD in Los Angeles, but a few hours' drive away, Las Vegas was quickly becoming the gambling capital of the world, and Thorpe saw a way to make a fortune He headed to Las Vegas and sat at the blackjack table, back then the dealer only used a deck of cards, so Thorpe could make a mental note of all the cards that had been played as he saw them.

This allowed him to determine if he had the advantage. He would bet a larger portion of his bankroll when the odds were in his favor and less when they were not. He had invented card counting. This is a remarkable innovation, considering that blackjack has been around in various forms for hundreds of years, and for a time this made him a lot of money. But casinos caught on to his strategy and added more decks of cards to the game to reduce the benefit of card counting. Then Thorpe took his winnings to what he called the biggest casino on Earth: the stock market.

He created a hedge fund that would earn a 20% return each year for 20 years, the best return ever seen at the time. And he did it by transferring the skills he honed at the blackjack table to the stock market. Thorpe pioneered a type of hedging, a way of protecting against losses through balancing or netting transactions. - Thorpe did it mathematically. He looked at the odds of winning and losing and decided that, under certain conditions, you can tilt the odds in your favor by using certain patterns to place bets. -Suppose Bob sells Alice a call option on a stock and say that the stock has gone up, so he is now in the money for Alice.

Well, now for every additional dollar the stock price rises, Bob will lose 1 dollar, but he can eliminate this risk if he owns one unit of stock. So, if the price goes up, he would lose $1 on the option, but he would get that $1 back on the stock. And if the stock falls out of the money again for Alice, he sells it so he doesn't risk losing money either. This is called dynamic coverage. It means that Bob can make profits with minimal risk due to the fluctuation of stock prices. A pi hedge portfolio will at any time offset the V option with a certain amount of stock delta. - It basically means that I can sell you something without having to take the opposite side of the deal.

And the way to think about it is that I have synthetically manufactured an option for you. I created it from scratch doing dynamic operations. Dynamic coverage. -As we saw with Bob's delta example, the number of shares he has to hold changes depending on current prices. Mathematically, it represents how much the current option price changes with a change in the stock price. But Thorpe was dissatisfied with Bachelier's pricing options model. He means, on the one hand, that stock prices are not completely random. They can rise over time if business is going well or fall if it is not.

Bachelier's model ignored this. Thorpe then devised a more precise model for pricing options that took this trend into account. - In fact, I discovered what this model was in mid-1967, and decided that I would use it for myself and then keep it a secret from my own investors. Basically, the idea was to make a lot of money for everyone. -His strategy was that if the option was cheap, according to his model, he would buy it. If it was overvalued, sell it short, i.e. bet against it. And that way, more often than not, he would end up on the winning side of the exchange.

This lasted until 1973. That year, Fischer Black and Myron Scholes came up with an equation that changed the industry. Robert Merton independently published his own version of it, which was based on the mathematics of stochastic calculus, for which he is also credited. - I thought I would have the field to myself, but unfortunately, Fischer Black and Myron Scholes published the idea and did a better job with the model than I did because they had very strict mathematics behind their derivation. - Just like Bachelier, they thought. that option prices should offer a fair bet for both buyers and sellers, but his approach was entirely new.

They said that if it was possible to build a risk-free portfolio of options and stocks just as Thorpe was doing with his delta hedging, then in an efficient market, a fair market, this portfolio should return nothing more than the risk-free rate. risk. what the same money would earn if it were invested in the safest asset: US treasury bonds. The assumption was that if no additional risk is taken, then it should not be possible to receive any additional return. To describe how stock prices change over time, Black, Scholes and Merton used an improved version of Bachelier's model, as did Thorpe.

This says that at any time we expect the stock price to move randomly, in addition to a general upward or downward trend, drift. By combining these two equations, Black, Scholes, and Merton created the most famous equation in finance. It relates the price of any type of contract to any asset, stocks, bonds, whatever. The same year they published this equation, the Chicago Board Options Exchange was founded. Why is that equation so important? In terms of finances, how did that change the game? - Well, because when you solve that partial differential equation, you get an explicit formula for the option price based on a set of these input parameters.

And for the first time, you now have an explicit expression where you enter the parameters and this number appears so people can use it to trade. - This led to one of the fastest industry adoptions of an academic idea in all of the social sciences. - In just a couple of years, Wall Street adopted the Black Scholes formula as a benchmark for options trading. The exchange-traded options market has exploded and is now a multi-billion dollar industry; Volume in this market has doubled approximately every five years. This then is the financial equivalent of Moore's Law. There are other businesses that have grown just as quickly, such as the credit default swaps market, the OTC derivatives market and the securitized debt market.

These are all multi-billion dollar industries that in one way or another make use of Black Scholes Merton's option pricing idea. - This opened up a whole new way to protect against anything, and not just for hedge funds. Today, virtually all large companies, governments, and even individual investors use options to protect against their own specific risks. Let's say you run an airline and you're worried that a rise in oil prices will affect your profits. Well, using the Black Scholes Merton equation, there is a way to hedge that risk accurately and efficiently. You value an option to buy something that follows the price of oil, and that option will pay off if oil prices rise, and that will help you offset the higher fuel cost you have to pay.

Therefore, Black Scholes Merton can help reduce risk, but can also provide leverage. - An ongoing battle between bullish traders and hedge fund short sellers who have bet against the stock, GameStop shares, are now up around 700%. - Well, GameStop is a really interesting example for all sorts of reasons, but options figured prominently in that example because a small group of users on this Reddit subchannel r/wallstreetbets decided that hedge fund managers who were selling shorting the stock and betting that the company would go out of business had to be punished. And then they bought shares of GameStop to try to raise the price.

It turns out that buying stocks wasn't enough, because with a dollar in cash, you can buy a dollar in cash, but with a dollar in cash, you can buy options that affected a lot more than a dollar in stock. maybe in some cases $10 or $20 in stocks for a dollar in options. And so there is natural leverage built into these securities. And so the combination of buying stocks and options caused prices to rise very quickly. And what that did was cause these hedge fund managers to lose a lot of money quickly. - How big is this derivatives market?

How big is this whole area emerging from Black Scholes Merton? - There are estimates of how big the derivatives markets are and first, let's be clear about what a derivative is. A derivative is a financial security whose value is derived from another financial security. So an option is an example of a derivative. Overall, the size of global derivatives markets is on the order of several hundred

trillion

dollars. -How does that compare to the size of the underlying securities on which they are based? - They are multiples of the underlying values. - I just have to interrupt because it seems a little crazy to me that you have more money at stake in things that are based on the thing than on the thing itself. - That's how it is. -Then tell me if that makes any sense. - Because what options allow you to do is take the underlying thing and turn it into 5, 10, 20, 50 things.

So these pieces of paper that we call options and derivatives basically allow us to create many, many different versions of the underlying asset, versions that individuals find more acceptable due to their own risk-reward preferences. - Does this make markets and the global economy more stable, less stable or no effect? - The three. It turns out then that, in normal times, these markets are a very important source of liquidity and, therefore, stability. During abnormal times, that is, when there are periods of stress in the market, all of these securities can go in one direction, usually down, and when they fall together, that creates a really big market drop.

So in those circumstances, derivatives markets can exacerbate these types of market disruptions. - In 1997, Merton and Scholes received the Nobel Prize in Economics. BlackHe was recognized for his contributions, but sadly had passed away just two years earlier. - We were going to make a lot of money with options, but now Black and Shoals have told everyone what the secret is. - Now that the options pricing formula is available to everyone, hedge funds would have to figure out better ways to find market inefficiencies. Enter Jim Simons. Before having any exposure to the stock market, Simons was a mathematician.

His work on Riemannian geometry was fundamental to many areas of mathematics and physics, including knot theory, quantum field theory, and quantum computing. Chern Simon's theory laid the mathematical foundation for string theory. In 1976, the American Mathematical Society awarded him the Oswald Veblen Prize in geometry. But at the peak of his academic career, Simons sought a new challenge. When he founded Renaissance Technologies in 1978, his strategy was to use machine learning to find patterns in the stock market. Patterns provide opportunities to make money. - The real thing was collecting a huge amount of data and we had to get it by hand at first, we would go to the Federal Reserve and copy interest rate histories and things like that because they didn't exist on computers. . -His reasoning was that the market is too complex for anyone to make predictions with certainty.

But Simons had worked for the US Institute for Defense Analysis during the Cold War, breaking Russian codes by extracting patterns from large amounts of data. Simons was convinced that a similar approach could beat the market. He then used his academic contacts to hire a group of the best scientists he could find. -What then was his employment criteria? If they didn't know anything about finance, what were you looking for in them? Someone with a PhD in physics and who had been free for five years and had written some good papers and was obviously a smart guy in astronomy, mathematics or statistics.

Someone who had done science and done it well. - It is not surprising that mathematicians and physicists are dedicated to this field. First of all, finance pays a lot better than, you know, being an assistant math professor. And for several mathematicians, the beauty of option pricing is equally compelling as anything else they do in their professions. - One of them was Leonard Baum, a pioneer of hidden Markov models. Just as Einstein realized that, although we cannot directly observe atoms, we can infer their existence through their effect on pollen grains, hidden Markov models aim to find factors that are not directly observable, but that have a effect on what we can observe.

And shortly after, Renaissance launched its now-famous Medallion fund. Using hidden Markov models and other data-driven strategies, the Medallion fund became the highest-performing mutual fund of all time. This led to Bradford Cornell of UCLA, in his article Medallion Fund: The Ultimate Counterexample? conclude that perhaps the efficient market hypothesis itself is wrong. - In 1988, I published an article testing it, the US StockMarket, and what I found was that the hypothesis is false. In fact, you can reject the hypothesis contained in the data. And that's why there is predictability in the stock market. - So it is possible to beat the market, that's what you're saying. - It is possible to beat the market if you have the right models, the right training, the resources, the computational power, etc., etc., yes. - The people who have found the patterns in the stock market, and the randomness, have often been physicists and mathematicians, but their impact has gone beyond simply making them rich.

By modeling market dynamics, they provided new insights into risk and opened up entirely new markets. They have determined what the exact price of derivatives should be and, in doing so, have helped eliminate market inefficiencies. Ironically, if we are ever able to discover all the patterns in the stock market, knowing what they are will allow us to eliminate them. Then we will finally have a perfectly efficient market where all price movements are truly random.