March 18, 2024
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Gresham College Lectures

Option Pricing Theory Explained - Raghavendra Rau

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Gresham College Lectures

Option Pricing Theory Explained - Raghavendra Rau

Mar 18, 2024

Gresham College

We often change our minds after we decide to do something.

In finance and business though, if you think you might like to change your mind you will have to pay your counterparty so that your right to change your mind is agreed in advance.

But how much is the right to change your mind worth?

Option pricing is the art of determining the value of this right.

This lecture was recorded by Raghavendra Rau on 26th February 2024 at Barnard's Inn Hall, London

The transcript and downloadable versions of the lecture are available from the Gresham College website:

https://www.gresham.ac.uk/watch-now/option-pricing

Gresham College has offered free public lectures for over 400 years, thanks to the generosity of our supporters. There are currently over 2,500 lectures free to access. We believe that everyone should have the opportunity to learn from some of the greatest minds. To support Gresham's mission, please consider making a donation: https://gresham.ac.uk/support/

Website: https://gresham.ac.uk

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We often change our minds after we decide to do something.

In finance and business though, if you think you might like to change your mind you will have to pay your counterparty so that your right to change your mind is agreed in advance.

But how much is the right to change your mind worth?

Option pricing is the art of determining the value of this right.

This lecture was recorded by Raghavendra Rau on 26th February 2024 at Barnard's Inn Hall, London

The transcript and downloadable versions of the lecture are available from the Gresham College website:

https://www.gresham.ac.uk/watch-now/option-pricing

Gresham College has offered free public lectures for over 400 years, thanks to the generosity of our supporters. There are currently over 2,500 lectures free to access. We believe that everyone should have the opportunity to learn from some of the greatest minds. To support Gresham's mission, please consider making a donation: https://gresham.ac.uk/support/

Website: https://gresham.ac.uk

Twitter: https://twitter.com/greshamcollege

Facebook: https://facebook.com/greshamcollege

Instagram: https://instagram.com/greshamcollege

So this is a series of lectures I'm offering this year, which is on the big ideas of finance. There actually only six ideas in finance, but um, there only five of them have won Nobel Prizes. So this is one of the five of the six ideas which actually have won a Nobel Prize. The Gresham College lecture that you're listening to right now is giving you knowledge and insight from one of the world's leading academic experts making it takes a lot of time, but because we want to encourage a love of learning, we think it's well worth it. We never make you pay for lectures, although donations are needed. All we ask in return is this. Send a link to this lecture to someone you think would benefit. And if you haven't already, click the follow or subscribe button from wherever you are listening right now. Now, let's get back to the lecture. And the interesting thing about this particular idea is this is probably the most difficult of all the ideas of finance. Um, in fact, the people who originally came up with this needed the help of a physics PhD to actually understand their own formula. That's why we have physics. PhDs who typically go on Wall Street are referred to as rocket because you need the same level of math to understand these formula as you take to put a rocket on the moon. So that's what I'm gonna do in 15 minutes today, right? Get you to this, get you to the level of a rocket scientist. Okay. Alright. I did, I see some skeptical face in the audience. Don't worry, you will be there. Okay, as I mentioned as, uh, most of these are actually taken from my textbook. So this is chapter, I believe, six, uh, five in the textbook. So, and it's, uh, gives each of these ideas. So what are these six ideas of finance? Well, the first idea, which we covered in the lecture last year was on the concept of net present value. That how do we make a decision? We only ever make a decision when we get back more than we invest. That's the basic idea of net present value. But that's slightly more complicated than that. But that's the one idea of finance, which has not won our Nobel Prize because if you think about it, it's so obvious I'm not gonna invest unless I get back more than what I invest. That's a, that's pretty basic, right? So no one's caught a Nobel Prize for that idea, but when you want to figure that out, you need a discount rate. That's the second big idea of finance. And that indeed did win a Nobel Prize. The Markowitz and Sharp caught a Nobel Prize for this in 1990. The third idea, which we discussed last time was Capital Structured Theory. And two more people with the Glenn Miller got a Nobel Prize for this. This, as I said, is a fourth idea of finance, and that's option pricing theory and two people s sch and Merton got a Nobel Prize for this in 1997. There are two more ideas and we'll cover this in May and June respectively. So I'm gonna hold off on that, but let me go back to the first three and talk about how it relates to the fourth idea. So the first idea is if you think about finance in general, what is finance all about? It's about promises, right? The idea is somebody comes to you and says, gimme some money today and I'll give you a lot of money tomorrow, right? That's literally everything in finance. Gimme money and I'll give you back more money. The problem is the promises are always in the future. Gimme money now and I'll give you money in the future. So how much is that promise in the future worth? That's the question which finance answers, right? The first thing is you have to convert the promises in the future to a price today. That's, that's when you can compare giving up a hundred pounds today in return for getting 500 pounds in the future. Is that worth it? I don't know until I can bring the 500 pounds to today. So that's the first idea of finance, net present value, right? So the fundamental concept in finance, and so we used everywhere, we, every person in finance uses this formula all the time, right? And the general formula is the present value. Something is a discounted value of all cash flows in the future, okay? But if you look at the formula itself, the formula involves a discount rate. That means, in other words, if someone says, I'll give you 500 pounds one year from now, what's the value today? I need to figure out what is the interest rate they're actually offering me. So where is that interest rate coming from? That's the second idea of finance. So someone's coming to you and saying, please invest in my company. And I say, okay, if I don't invest in you, what is my next best opportunity? What do I do with my money if I don't invest in you? So I'm looking for the opportunity cost of investing in you, right? So that means that is looking at an opportunity which has the same level of risk as the opportunity I'm being offered right now. But what is risk? Well, in this particular case, Markowitz and Sharp got a Nobel Prize for showing that all rational people will end up holding only one portfolio, the market portfolio. And so the idea is risk is measured by its core variance with what people are already holding the market portfolio. And our risk measure is a standardized core variance. That's the beta. And the CAPM says the expected return risk-free rate plus the beta time the marketers premium. So that was the second lecture. The third lecture was about how much should you borrow? So, and we showed last time then a perfect world, which according to Mo Glenn Ann Miller, Chicago economists, so their perfect world is a government, a place with no government and no taxes, right? So no taxes, no lawyers. It's a wonderful world for, for these economists. So in that case, it didn't matter what you did, if you used debt or equity, you had, it was exactly the same thing In a not so perfect world where there are taxes involved, the idea is you minimize your taxes by borrow as much as you can. Borrow more enables you to reduce your interest rate, okay? But firms don't borrow as much as they can. And the answer is because the third imperfection, there are lawyers as well. And if you borrow too much money, you go bankrupt. The lawyers take a chunk of the money, you don't want that to happen. So firms have an optimal trade off structure, right? So now let's talk about how these three ideas relate to the fourth idea. Option pricing is not about promises, right? We said finance is all about promises, but option pricing is about what we call contingent promises. What's a contingent promise? It's a promise that depends on something else. Let's take an example. So somebody says, I'll give you some huge amount of money in the future in return for some money today. So what happens if you change your mind after making the investment? So I give you the money and then afterwards I say, wait, it didn't work out. I want my money back, right? Can you do that? What is the value of being able to change your mind? Or can you wait to make up your mind till you get more information? All of this is part of option pricing. Let's take an example to really drive this home, right? Take for example, booking a hotel room in London, right? So this is, I don't actually have anything to do with the town Hall hotel. I just pulled it off randomly from a website here. But you notice they have the same room has two different prices, 200, two pounds, 50, and 225. Why would the same room on the same night have two different prices? Well, the first one is non-refundable. The second one is free cancellation by February 25th, and this is much more expensive than the first one. So why is it much more expensive? Because the second one gives you the right to change your mind all the way up to February 25th. In fact, that is an option to cancel and that's why this is more expensive. If you don't change your mind, you end up paying 2 25, right? So why is the cancelable room more expensive than the non cancelable room? The answer is because now the hotel is taking a risk that you might change your mind. If you book the non cancelable room, you take the risk. If something happens, you can't get to the hotel, you lose your money. But here the hotel is taking the risk at the last minute. If you cancel, the hotel is has an unsold room. So they control for that by charging you a a higher price for the same room. That's the value of the option to cancel. But these options exist everywhere. For example, let's take Brexit. The problem with Brexit was not so much Brexit itself is because businesses didn't know what would happen, right? So Brexits happen, but what does it mean for the terms of trade between Britain and the European Union? If you don't know this, right, do you invest? The answer is in many cases they don't because you don't know whether you'll have to expand, whether you'll have to reduce, whether you can invest. So all of these is basically an option. You preserve the option by refusing to invest, right? If you invest, you're locked into place. I mean, then you have to reverse your investment. But if you don't invest, you have this possibility here. Okay? So why do you need a contingent investment to understand that? Let's take an example, right? So this is an example from Cambridge. So every year in Cambridge in the summer, they have the, what they call the Cambridge Shakespeare Festival. And you know, it's very, it's a lovely experience. You go buy a ticket, you sit in the gardens of the colleges and then you watch, uh, Shakespeare play like the mid someone night Dream, Romeo and Juliet, whatever, right? And it's a lovely experience, but there's one problem and the problem is the weather, right? So what is the issue here? Well, one possibility is if you don't buy the ticket early, you may not get a ticket, right? But if you buy the ticket early, what happens? So if you buy the ticket, now you are guaranteed a ticket. The con of course is you get the to get there, it pours down cats and dogs. You're sitting there in the rain watching Shakespeare, even in a Cambridge college garden. This is not a pleasant experience. Okay? So that's the big corner over there. The other possibility you can do is reserve the ticket now, but only pay for it on the day. By the way, this is also very common. So for example, when you make a restaurant reservation, many restaurants will ask you for your credit card upfront, right? So what you do is if you don't show up, they're going to charge your credit card anyway. So that's what's happening here. You're reserving the ticket, but you're only choosing to pay for it on the particular day. So if you don't show up, your card gets charged anyway. Okay, fine. So what's the pro? Well, the big pro is you don't have to pay for the ticket right now, but you do have to pay for it on the day of the play. Okay? So what's the con? First con is you will be charged on the day of the performance, and the second con is you're still subject to the weather, okay? But there's a third option. The third option is reserve the ticket now, but only pay for it on the day if the weather is nice. If the weather is not nice, you don't want the ticket, right? That was a perfect way out. You're basically saying I'm only going to get the ticket if the weather is nice, otherwise no point right? Now what's a pro here? You don't have to pay for the ticket now, but the con is essentially, now the festival is bearing the risk. So if you don't show up, if it's a rainy day, it is set up everything. The players are there, all the, you know, costs are being born, but there's nobody in the audience, right? So they pick up a lot of costs, but there's nothing to offset them. So they charge you an upfront fee. So to put this in thing, that's called a forward contract, and this one here is called an option, right? Okay, so the question now becomes what are these contracts? So these are two contracts I talked about forwards and futures. Well, a forward contract and a futures contract is essentially the same thing. What it is is a time shifted contract but no choice. So for example, if you reserve a Shakespeare play, all you have done is I will pay for it, not today, but at the day of the performance. So I have shifted the actual payment time till the day of the performance, but I don't have a choice if I don't show up, my card is gonna be charged anyway, right? There's absolutely no choice there. A futures contract is exactly the same thing, except it's traded on an exchange. So it's a standardized time shifted contract. The third one, the option is a time shifted contract, but with a choice. If the weather is nice, I buy the ticket, the weather is not nice, I don't buy the ticket, right? So the differences between the three. The question is how do you value these contracts? That's the presence of option pricing theory. That's what option pricing theory is all about. Okay, first question, right? So suppose you say, I wanna buy the ticket. Now let's say what we call the spot price. That means I go to the market, I go to the ticket office and I put down money to buy the ticket. How much does it cost? 20 pounds. By the way, I don't know that the price is actually going to be 20 pounds in in July this year and just making up that number, right? So don't quote me on that and said, Hey Professor Rao said the ticket for 20 pounds, I want it could be more, it could be less. I have no idea. But anyway, let's take the first second choice, which is reserve the ticket, now pay for it in August. What I want to do is figure out how much will the ticket office charge you for that? Will they charge you it all? All they say fine, 20 pounds, no problem. It's like, I mean I, I would love that honestly, because now I'm don't have to actually spend the 20 pounds. Now I only spend it on the day off and yet I guarantee getting a ticket. So it turns out interestingly that the choice number three, buy the ticket only if the weather is nice. That would be the option price. The question is, what is that option price? So you have these three things, buy today cost you 20 pounds, bite on the day off. What is the price bite on the day off? If the weather is nice, what is the cost? Fine, how do we price first? The second choice. The second choice is you only buy it, but there's no choice. And everything in option pricing depends on one fundamental concept. That concept is simple. No free lunch, nobody's giving you something for free. So how do we prove anything in option pricing? The answer is very simple. You prove that if the price was anything else, you'd get a free lunch. And since free lunches are not possible, this must cannot be a price. That's the idea behind most of option pricing. Now, what is a free lunch? A free lunch for a finance person is anything that means I've been going to get some money in the future, but I never spend any of my own money. I never take my wallet out, I never take any money outta my pocket, but somebody gives me money for free. That's a wonderful thing to have a free lunch. Unfortunately, as we see in option pricing, we are gonna show there's only one possible price at which there's no free lunch. That's the entirety of option pricing. No free lunch. Okay, so what is the price of the forward? Remember this? Choice number two was I'm reserving the ticket now, but on the day of the play I get charged, right? So I'm gonna assume that if you put your money in the bank right now, that it'll give you a risk free rate of 5%. That means the money's guaranteed you're not going to lose the money, you're just putting in the bank at 5%. Okay, fine. The time to August is about six months, right? So from here to August, about half a year. So the first formula is going to just say the forward price is 20 pounds times one plus the interest rate to the PowerPoint five. Now that formula should be familiar to you if you've seen the first lecture. This is the sim straight formula, which we use already. The future value of anything is a present value times one plus the interest rate to the power time, right? To give you an example, if you put a hundred dollars into the bank at 5% at the end of one year, you end up with $105, 5% interest, a hundred dollars. That's exactly what this is. A hundred dollars times one plus 5% to the power one is 105. That's all I'm doing here, okay? So the question becomes, okay, how do I know that 2050 is indeed the right price? To understand that? I'm going to say suppose the price is anything else, how do I get a free lunch? Okay, fine. Suppose for example the price is 21 pounds, can we get a free lunch? The answer will be yes. Let's prove it. Okay, so remember the theoretical price, the price I said it should be is 2050, and 2050 is low compared to the actual price, which is 21. Okay? However, so the actual price on the other hand is 21, which is expensive compared to the theoretical price. Now thank you guys all for coming out on a day like this, which is, you know, beautiful day outside, you could do a lot of other things. So I'm gonna give you a piece of advice which guarantees you will always make money on the financial markets, okay? That entire idea is worth the entire cost of coming here. The the thing is, if you wanna make money, buy low, sell high, I guarantee you will always make money <laugh>. Now of course the lecture is free and the advice is useless, so no free lunch, right? But anyway, same idea. So in this particular case, what I'm saying is the actual f the actual forward is 21 pounds is expensive. So what should we do? We should sell it, right? The expense, sell the expensive stuff, buy the cheap stuff. So in this case, you're selling a forward. If you're sell the forward, you're basically committed to selling a ticket to somebody else for 21 pounds, right? In August, fine, where do you get the ticket from? Well, you buy it today for 20 pounds. But remember freelance says you never take money out of your pocket. So you need the 20 pounds. Where do you get the 20 pounds from answer, you borrow it from the bank, the bank charges you 5% interest, you know that. So at the end of six months you have to repay 20 pounds and 50 pence to the bank, but you have sold a ticket for 21 pounds in advance. So the buyer will give you 21 pounds in advance and voila, you have a free lunch of 50 pe, right? Very easy, okay? Now suppose the price is less than the theoretical price. Remember the theoretical price is 2050, the actual price is 2010, theoretical price is too high, the actual price is too low. So how do we get a free lunch? The answer is buy the cheap stuff. So in this particular case, buy the forward, right? So you've committed to buying a ticket for 20 pounds and 10 pence in August, right? You bought the forward from somebody else. Now what you have to do to, to make sure you have a freelance is sell the ticket, but you don't have a ticket, right? You're just seeing this free launch. Where do you get the ticket from? Well, you know, you look around and you have some friends, right? And then you say, okay, you know what guys, would you mind giving me your ticket? It just, just let me just hold that ticket and you know, if you have some nice friends, they give you the ticket and you sell it, right? Remember he doesn't need it or they don't need it till August, so it's okay, right? They, you're going to give it back to them in August. So you sold the ticket, but what's the price of the ticket today? 20 pounds. So you sell it for 20 pounds, what do you do with that? 20 pounds. Put it in the bank, right? So you put that money in the bank. So it is called selling the ticket short. You borrowed a ticket from somebody and sold it for 20 pounds. Put that 20 pounds in the bank, you get 2050. Once you have that 2050, you pay 2010 for the ticket from the other person who's buying, who's selling the ticket for you. So you return the ticket, you've got the ticket back, it's not the same ticket as that was lent, but that's not a problem, it's a ticket. You just give it right back and everybody's happy, especially you because you have a free lunch of 40 pence. Okay? So the answer is anything except for 2010 is going to be, uh, except for 2050 is going to give you a free lunch. Either too high or too low. You will always end up with a free lunch and everything in finance says that's not possible. You can get fancier than that, right? So what usually um, uh, finance people do is, okay, that's too simple, let's make it more complicated. So what they do is they change that formula to write something which is 20 times the exponential function times rt, right? What is this exponential function? Why are they doing this? It's actually just a fancy way of saying the same thing. How do we get that fancy way? I mean, where do we go from here to this exponential function? The answer is, well think about if you get interest once a year, right? You're paying a 10% a year, what's the number you're gonna look for? You use that formula, in this case a hundred pounds times 10% interest, we'll give you 110, okay? Now if you get interest twice a year, that means 5% every six months, what are you left with after one year? Is it 110? No, it'll be more than 110 because in the first six months you get five, five pounds interest and you then the second half you get interest on the interest. So in fact, what this will end up being, you take the 10% divided by two and square it because you're getting interest twice a year. Essentially it's 101 plus 5% is a little more than 110, it's 25 pence more. Okay? What about interest? Every quarter? That means two and a half percent every quarter. Is it the same as 110 again, no. This time you take present value times one plus R by four to the power four because you're have four quarters in a year. So essentially it's a little more than that's 110.38, right? So essentially the more frequently you compound, the higher the interest rate you're going to get. Does this go on forever? The answer is no. If you compound it every year, it's 10%. If you compound every month, it's 10.47% compound every hour you get 10.51% and so on. Notice something interesting here. As you compound more and more frequently, the effective interest rate keeps going up, but the rate of increase keeps slowing down. Essentially what that means is if you compound every millisecond, every billionth of a second, you're gonna converge to an exponential function. That's essentially what's going on here, right? So why do I introduce this? It's because option people say you can never have a free lunch regardless of how small an interval of time it is. That means, in other words, if you have one year, right? So you say, okay, I can borrow for a hundred for one year and I get back 110, somebody else says, well I can borrow for six months and then compound it every six months. He shouldn't be able to have a free lunch either. Shouldn't get a free lunch with compounding every quarter, shouldn't get a free lunch for compounding every day, every second, every a thousandth of a second. You never ever have a free lunch and this will tell you that you can never have a free lunch. Okay? So that's the basic story here. But regardless, the price of the forward can be either, it's just the future value of the spot price at the risk free rate. So you can write that formula, you can write that formula, they're exactly the same thing, okay? But why the risk free rate, right? To explain that, let's think about who might want to buy a forward, right? Let's think about a farmer, right? So I'm a farmer, I am, you know, growing crops or whatever, but I need money, right? I need to borrow money for my crops for to buy seeds, to repair my tractor, whatever. I'm not, I'm not quite sure because I'm not a farmer, right? I'm more of a city slicker. So I really don't know what they do in the farms there, but in principle, I'm kind of assuming that's what they do. The problem is the bank is going to say, okay, show me a stable source of income. I'm not gonna lend you money unless I know that you can pay it back. So how do you assure yourself of that state of money? The answer is you sell your crops forward. That means you say, I'm going to lock in the price for my crops, which I'm gonna harvest in August. That means I have a stable income source now and then I can go to the bank and the bank will lend me money against my stable source of income in the future. That's the basic idea. But a lot of people don't need this. For example, I might choose to buy a contract on wheat even though I have no intention of taking delivery of the wheat. What I want is, I am committing to buy wheat at say a hundred dollars per bushel, but if the price of wheat goes up, I don't actually want to take delivery of the wheat. What I wanna do is sell a forward contract in the other direction. So basically I buy wheat at a hundred and I sell wheat at 110. I really don't care about the wheat. I'm just making money on the way the forward prices and the future price changes, right? That's what speculators do. So a lot of speculation is all about buying a forward or selling a forward waiting for the price to go up or down, and then selling a forward in the other direction. Commit to buy later on you commit to sell. Both will end on the same day so they cancel each other out, right? But this creates a problem. The problem is suppose the price of the ticket. You're committed. Remember that you have committed to buying it at the forward price, which was 20.51, but now the price of the ticket drops to five pounds, right? You're committed to buy it at 2051. So what are you really tempted to do? The answer is disappear. Change your name. May I call your bank, say I'm sorry this, change your address, put your, you know, get call the bank, say I need a new credit card. The old credit card was stolen, just make up something and they try to get, get hold of you. You're like, I'm sorry, nobody here by that name, right? You're done. Right? So obviously they need to stop that, right? That's a risk you take in the forward contract that your counterparty disappears on. You remember one group of people here, the guys who are selling are very happy because they're selling at 2051 when the actual ticket price is only five pounds. So one group wins, the other group will lose. It's always a zero sum gain. Right? Now, how do they solve this problem? The answer is they use standardized contracts. So for example, you can't, if you want to exchange pounds for dollars, you can only exchange 62,500 pounds per contract for, you can't exchange 70,000 pounds, you can't exchange a hundred thousand pounds. So if you want to exchange a hundred thousand pounds, you can't, you have to do either 62,500 or 125,002 contracts, one contract, right? These are standardized contracts. Or if you want to do wheat, you can have a thousand bushels of wheat, not 1,500, not 1,100, not 900. And they have to be wheat of a specific quality specific color, specific moisture content, right? That idea is, I don't really need to know of anything about the wheat. I know I'm getting a standardized off the shelf contract, so it increases liquidity. I'm buy and sell with up and I know I'm gonna get exactly the same thing every time. Of course this creates problem. The first problem it creates is that remember, no one's actually maybe taking delivery of the underlying product, right? So if I, if I buy for example, wheat, I don't need the wheat, I'm just taking advantage of the changes in price for the wheat. Okay? So that means where the wheat, the wheat is lying in the, in the, in a warehouse and everybody's buying and selling contracts. So somebody very resourceful, discovered in nickel for example, turns out that JP Morgan had a whole bunch of nickel contracts, two and a half ton bags of nickel, and then somebody walking into the store said, wait a minute, this doesn't actually feel like nickel. So they opened it, somebody had actually stolen all the nickel and replaced your bags of stones, right? And nobody knew because you know, everybody's buying and selling the nickel without actually checking that there was nickel. I mean, all you're looking is, is the market price. Nickel price goes up, you exchange sell contracts in the opposite direction, nickel goes down, you sell by contract in the opposite direction, but nobody's actually checking the nickel. So they discovered in this case that you know, a lot of the contracts the nickel had gone, it was just stones that people were buying and selling. So they had to check every warehouse. The problem is nickel is not magnetic. So how do you check that these big bags are full of nickel, right? And not stones? The answer is kick them, right? So the warehouse operators were advised to wear steel toed, uh, steel toe cap boots for safety. Uh, if the rule of thumb, if it hurts when you kick it, it's probably nickel. And so they go around kicking every bag just to make sure you actually have the nickel there. That's one way of one problem with standardized contracts, you actually have to check these contracts over time. But the other thing they do is, for example, it turns out that coffee nickel doesn't wear, right? I mean, it doesn't get stale, it doesn't sound, but coffee does get stale. So for a long time, what traders were doing is that if you have coffee, which is going stale in a warehouse, you take the coffee out. So write a contract, deliver the coffee, you take the coffee out and then immediately put it right back in the store. So they rewrite the contract with a new date, the coffee is fresh instead of actually being, you know, really old coffee, right? So that's another way in which you can cheat the financial markets. Of course that's been stopped too, but there are always new ways to do this. Now the second thing you do is they settle the contract daily. What does that mean? Let's say I have bought a forward contract to buy the, uh, Shakespeare ticket for 20 pounds, okay? And, but now what happens is, um, I can't just buy a contract. I have to deposit a sum of money in what's called a margin account, about 10 pounds. So what happens? I have bought the, I'm buying the Shakespeare ticket for 20 pounds. Let's say he's selling me the Shakespeare ticket for 20 pounds, right? So that's the thing. Now both of us have deposited 10 pounds in the, with the exchange. So tomorrow the ticket price changes to 19 pounds. I have agreed to pay 20. So I'm worse off. I'm tempted to run. What the exchange does is takes one pound out of my account and gives it to him, puts it in his account. Why? Because now it rewrites the contract to say, I have agreed to buy at 19, not at 20. Basically, I've settled a contract and written a new contract, and now his account contains 11 pounds, my account contains nine. So every day this marketing to market process continues and you know, eventually basically you end up with, um, you know, if the margin account drops too low, you have to top it up. You have to give more money to keep the account normal, okay? And this work works pretty well except on one occasion. So this occasion happened in March, 2022, and what happened here is nickel again, nickel seems to be a particularly interesting metal, right? So, but anyway, turns out that nickel, um, it stays usually about between 10 and 20,000 pounds or dollars per ton and fluctuates no more than about a few hundred dollars a day. So they set the margin account based on that, right? The only problem is March, 2022, what happened was Russia invaded Ukraine. And so people thought the supplies of nickel were going, we get cut off. So what happened? The price of nickel, it started climbing. How much did it climb? Well, it turns out it had jumped the previous day to about 48,000. Remember, it never went above 20,000 per day. In one day it went to 48,000.

Beyond that, in four minutes from 5 42 to 6:00 AM it passed a hundred thousand dollars per ton. So what does that mean? What that means is if you have shorted nickel, if you have sold nickel at$20,000 and the price is now a hundred thousand dollars, the exchange will say pay up. Right? Gimme $80,000 because you know, now you are gonna write a new contract at a hundred thousand. And so most people, yeah, they called, gave the margin calls except for one guy, right? This guy was a Chinese gentleman called Ang Guang and he had sold nickel. How much nickel do you ask? Right? Well, a million dollars, $2 million. Wild guess what do you guys think?$3 billion worth of nickel, right? And so they went to him and they said, and they were terrified. The bankers, right? Because you know the famous story, if you lose a thousand pounds, it's your problem, but if you lose 3 billion pounds, it's a bank's problem, right? So these bankers were terrified they'll lose their jobs. So they went to him and they said, please, can you pay up the margin account? He said, I'm not gonna do it. And he said, well, you know, you are. This is tough luck. They said, well, you know, how do we recover this? He said, well, you know what, I bought all the stuff from my holding companies in Malaysia, in Indonesia. Go for it. You can try to sue me, but I'm based in China, so the Chinese government, I know everybody there, you try to sue me, tough luck, good luck recovering everything. So what do they do? They said, you know what we're gonna do is we are going to shut down the entire market for five days. You're going to pretend all the trades that happened in the middle never happened. And you can imagine people like Goldman were very upset because they had bet that prices would go up and they had made a lot of money. They, unfortunately all that trades were canceled. And then they said, they came to him and they said, okay, the price has dropped to about 50,000, would you please settle your debt? He said, oh, I'm not willing to do it. And they said, okay, how much would you be willing to take a loss? He says, well, if the price goes back to 30,000, I'll be willing to do it. And they said, okay. So they shut down the market and they keep opening it every day and they would say, oh shit, the price is too high. And then drop it and they keep closing and opening the market until eventually the price dropped back over to about $30,000 and then they opened it. This guy closed out his account and basically didn't lose that much money. But of course LME was sued by all the people on the other side because they said, wait, we made a ton of money, you just sold out our profits to a bunch of Chinese speculators. So, but that's a totally different story. Okay, so now let's go to the third part. How do you price an option, right? Remember, your option is buy the ticket in August only if the weather is nice, otherwise you don't want it. Okay? So same thing. Bank rate 5% time to August is about six months, that's about half a year. How do I solve this? Well, it's going to be, I'm going to use something called a binomial model. What is a binomial model? It says only two possibilities. Binomial is two. So a binomial model is only two possibilities for the final ticket price. Okay, I'm gonna simple, that's a very simple way of thinking about this and I'm gonna look at three different things. So what do I have here is basically I have an underlying asset, the ticket, I have an option on the ticket that is a right to buy the ticket in August if the weather is good. And the third part I have is a bank account. I have these three things. So what I know the value I'm going to get for my bank account, I know the price of the ticket today. The question is how do I use those two things to figure out the price of the third thing, which is the option? So first thing I can do is to combine the asset, the ticket, and the option to cut up a bank account. Second of possibility is expanded tree easily. I'll keep that as one B, or I can create a combination of the asset and a riskless bank account, which replicates the option. That's a dynamic trading strategy. I won't talk about it today. And finally, you can also add the option to riskless bank account to create the ticket. So three possibilities, I'm only going to look at the first one. So let's start with this. The ticket is worth 20 pounds right now in August, if the weather is nice, people will love to go to the show. They'll pay 22 pounds for the ticket. If the weather's horrible, it'd be worth 18, right? Very simple. That's what it looks like. So at this point you arrange for the ticket office that you will buy the ticket in August for 21 only if the weather is nice. Okay? So what is the value of that option today? That's what you want to find out. Now before we get there though, let's see what it is worth in August, not today, in August. So remember that you have the right to buy the ticket for 21 in August. So if the ticket price is 22, the value option will be one, right? Because you have the right to buy something which is worth 22 pounds and you only going to pay 21 pounds for it. So you make one pound profit. In contrast, if the weather's horrible, the ticket price is 18, but you have the right to buy it for 21. Will anyone buy the ticket for 21 if the market price is 18? Right? Obviously the answer is no. Right? So people with that right will throw it away. The word to the option is zero. The question I wanna ask you guys is, but what we want to find out is this option, right? So the weather is nice, it's worth one pound. If the weather's horrible, it's worth zero. Okay? That's, that's what we have right now. So what I would like you to do is use that and tell me what do you think the price charge for the option today Depends on, go ahead. You can use your phone, use that and tell me what you think. It depends on, we know what it means. One at the end, if the weather's nice, it depends. Every value's one pound. If the weather's awful, the value is zero. The question is, what is the value today? What does that depend on? What do you think? It depends on the weather. Okay, of course it depends on the weather, but more important, you don't know today what the weather is going to be like. So what do you know today that might tell you the probability of good weather, right? It sounds better than just the weather itself. Of course it depends on the weather, but the weather, probability of the weather today. Well that's what I wanna find out. So there we go. Probability of the weather, expectation about the weather, the forecast, weather, prognosis, all of these are exactly correct, right? Okay, I'm gonna tell you, no, it doesn't depend on any of these, okay? That was why black and Shaws got the Nobel Prize because they said everybody else is thinking it depends on the probability of the weather. The answer is it doesn't. Let me prove that to you. Okay? So I'm gonna close that off. Come back in here. So the question is, remember the risk is being born by the ticket office. I have sold you this ticket at 21 pounds only if the weather is nice. So the ticket office is bearing the risk of this option. It sold you. So that means the ticket office doesn't, by the way, the ticket office, the seller of the option does not have the right to change its mind. Only the buyer of the option has the right to change their mind. The ticket office can't say, oh, I'm sorry, it's nice weather. I'm not gonna sell you the ticket. They have to sell you the ticket at 21. You don't, they don't have a choice, they don't have an option. Only the buyer has an option. Okay? So the idea is the ticket of is selling the option to you. So what it does is it buys some of its own tickets in the secondary market, StubHub or whatever. Why? Let me tell you that. Okay? So remember this is what we are trying to solve. So how many tickets does the ticket office sell? Just enough to make that portfolio riskless. How does that mean? How make the portfolio riskless? Well, let's suppose it sells you one option and buys delta ticket. Delta is something I wanna solve for. It's a, it's a variable I want to solve for. Okay? So that means the value of portfolio in good weather will be 22 delta. Because you have Delta tickets, each ticket cost 22 to 22 delta, but you've sold an option, the other person will exercise the option and you lose one pound. Because remember they can buy it for 21 when the ticket price is actually 22, right? So 22 delta minus one. That's the value on that side. Okay? What about if the ticket price goes down? Well it's 18 delta because the other person not gonna do anything. It's just 18 kind of delta. So the other person is thrown the option away. You're left with this. The value today by the way, is 20 delta minus the option value, which you don't know you're trying to solve for. Okay, fine. When is the ticket office not care about the weather at all? When 18 delta is equal to 22 delta minus one or delta is a quarter. What does that mean? What that means is I'm effectively saying I'm selling you one option and buying one fourth of a ticket. Wait a minute, what does that one fourth of a ticket mean, right? How can you buy one fourth of a ticket? Answer is easy. You sell four options and buy one ticket. So the ratio has to be the same. It's not that you're selling buying one fourth of a ticket, but fine. In that case, what's the value of the portfolio? 22 times one, fourth you get 4.5, right? You make money on the tickets, which you're bought back because now you can sell them at 22, but you lose money on the option that's 4.5. Otherwise it's 4.5. What is the value of 4.5 for sure receive six months from now? Same formula. Put it back. It's worth 4.371. If you want to get a little more fancy, it's worth future value times either the power minus rt, which is 4.367. Pretty similar, right? But we also know that the value of the portfolio today is 20 delta minus the option. So we know that 20 delta minus the option 20 times one fourth because you're one fourth is delta is 4.367, which mean the option value is 0.63. Just the difference between those, all I've done is I've combined the ticket with the option to create a riskless bank account. It doesn't matter whether the weather's good or bad, I get the same amount of money in either scenario. So I mean I can now use a riskless rate to bring it back. Okay? Okay. At this point you're saying, wait a minute, what about the probability? The nice thing about this calculation was probabilities did not matter. I never used the probabilities. I don't care whether it's good weather or bad weather, I just adjust it so that I get the same number. Whether the weather's good or bad, I don't care about priorities. But the beauty of that is I can use any priorities and it'll work 20%, 80%, it'll work 80, 75, 25, it'll work. Any probabilities will give you the same answer because priorities don't matter. And in particular, I can use one set of probabilities that everybody is risk neutral. What does risk neutrality mean? Well, let's take an example. Suppose you have a thousand dollars, you have two choices. Choice one, invest in a sure thing, it's a government bond, safe bond risk-free interest rate is 5%. So you get $50 for sure. That means you end up with 1 0 5, 0. The other possibility is you toss a coin. If a coin comes up heads, you get a 50% chance of earning 25%. So you get 1, 2, 5, 0, you are $250, 1, 2, 5, 0, but you have a 50% chance of losing 15%. So your a thousand becomes eight 50. Okay? So the expected value of this is still the same as 1 0, 5, 0. So the question is, would you go for the sure thing or for the gamble? Let's take a look. So you have a sure thing, government bond, fi gamble, 50% chance of losing one 50, 50% chance of earning two 50, or you're indifferent between the gamble or the sure thing. What do you think you would do? Now this is a very normal answer. Most people would either pick this or this, but I'm gonna pretend that everybody is this way, right? That basically risk neutral people don't care about the probability, they're not risk averse. These people here are extremely risk averse. These people here are love risk. These people here don't give a damn one way or the other, right? I was trying to think of a way not to say, but I forget it, right? Okay, so why? Why? What's the beauty of risk neutrality? The beauty of risk neutrality is because these guys don't care about risk. All they want is the bank interest rate for everything. Every account, every transaction they do, if they get the bank interest rate, they're fine, right? Because they don't care about risk one way or the other. It doesn't mean anything to them. So the beauty of that is, remember, this is what it looks like. When will this guy not care about good weather or bad weather? They say you have a choice. Take your 20 pound and put in the bank at 5% or if there's good weather, it'll go 22 it bad weather, it's worth 18. When will the risk neutral person not care about it? Well, when 20 invested at the risk free rate for half a period is equal to 22 times the property, the weather is nice plus 18 times the property, one minus the weather is not nice, okay? These things are not probabilities, they are made up probabilities, they're things which only a risk neutral person would do. And I'm pretending everybody in this room is risk neutral, which you are not. Okay? Okay, so why do I need this? I can also do this with that. That's fine, right? That's a future value of 20. That's the expected value of 20. And these guys don't care between one or the other. Okay? Why do I do this? Well, because now I can take that same number and apply it to the option value. So I know the probability risk, neutral probability of the weather being good is 0.65, not the actual property. Nobody knows the actual probability. So what's the value of the option? 6, 5, 2, 3 times 1, 3, 4, 7. Seven times zero, discounted back. That's 6 33, right? So expected value the option, get the value back to today. That's pretty much it. Alright, now why did I do that? The answer is because I want to expand the three. So this was only one period, but now suppose I won't go one more period. Remember this is going up by 10% or going down by 10%. I say, okay, next period it goes up by another 10% goes down by another 10%. So I've expanded the three to two periods, not one, but the ties stay the same because the going up and the going down rate is exactly the same, 10% up, 10% down. So that means the, if I have the right to buy the option, uh, if I have the right to buy the ticket for 21, the ticket is worth 24.2. The option is worth 3.2, 24.2 minus 21 ticket is worth 19.8 in the market, I'm have the right to buy for 21. Useless. Throw it away. The same thing here, but you have the probabilities. So I can bring those probabilities back by saying the price of the option is zero times 6, 5 2, 3 plus zero times 3, 4, 7, 7. Bring it back to today. That's zero. What about that one there? Well, 6 5 2, 3 times C 0.2 3, 4, 7, 7 times zero. Bring it back to today. That's 2 0 7 3. And then go back one more period. That's the price of the option today, 1.28, right? So just work backwards through the tree. Why am I doing this? Well, think about a tree that looks like this, right? You start, how many ways are there to get to a only one goes up down. What? Let's do a two pound three. How many ways are there to get to C? Only one. You have to go up twice. How many ways you get to E one? You have to go down twice. How about to D? Well, two ways. You go to A and then to D, you go to B and then to C. Okay, let's bring the tree up further. How many ways to go there? Well, there's gonna be one way to get to the top, three ways to get to position. Two, three ways to get there. One way to get there, go one. One. It'll be 1, 4, 6, 4, 1. What am I doing here? Well, what I'm doing is I am going from a model that looks like this. There's only one way, there are two ways to get here. One way to get here, one way to get here, go on like this, I will eventually end up with something that looks like a normal distribution, which we've already done in our lecture on portfolio theory. So if you have a binomial model with tiny little intervals, we'll end up with something that looks exactly like a regular stock price. We along the stock price to change every millisecond by tiny little amounts every period. And we end up with that. And that brings us to the final formula today, the Black Shaws formula. That is basically a continuous time version of the OMI model we just saw. What does this formula look like? We are first gonna assume the stock prices are log normally distributed. What does that mean? That means a log of the stock prices are normally distributed. Why do I need that? Because stock prices only go to up to zero. You can't have a negative stock price, which means you can't fit it into a normal distribution because normal distributions go from minus infinity to plus infinity. So how do you get that? Take the log log of anything less than one is negative. So you can go from minus to plus infinity. And that's the final point here. The Blackshaws formula. And you look at this and you say, oh my god, what the hell is that? Right? Well it turns out that black and Scholtz couldn't solve this formula by themselves, right? They were sitting in their office, they were wrestling with this, but they're economists. They weren't really, uh, physics people, right? So it turns out apparently the story is that Robert Merton, who was also at Harvard at that time, walked into the office and he said, I've seen this formula before and I have a buddy who was, you know, in college with me, he's a physics guy. I think I saw him work on this formula. So they call up the guy and the guy said, yeah, yeah, that's a fame man CAC formula, very famous stochastic differential equation physics. Here's the solution. So those guys took the solution and they said you applied it straight to stop prices. So this is based on a physics formula called the Finman CAC formula. Alright? Now, of course they published the paper after a lot of difficulty because no, the, none of the referees could understand the paper either, and then nobody could understand the paper once it was published. So eventually people said, okay, how do we get a simpler way to think about it? They went back to the black binomial model and proved it became the Black Shoals model. And then people understood how to get to this formula. So let me show you quickly what the formula is, right? Value of an option is just the maximum of the price of the ticket minus the price you've agreed to buy. Remember, I've agreed to buy the ticket for 21. So if the ticket price is 22, what is this? What am I doing? I'm saying 22 minus 21, the option price is one. But if the ticket price the 18, it's maximum of 18 minus 21, which is minus three or zero, the option cannot be less than zero. So I get zero, right? That's all I'm doing. That's exactly what this formula is. I have S and I have K. So S is the exercise, is the value, the asset, the price of the ticket. K is the exercise price, the price at which I've agreed to buy it, but K times either the power minus RT is the present value of K. So I've said I'm getting an asset worth s I'm giving up the present value of K and then I'm multiplying it something by ND two and ND one. Now what's ND two? Well, you know something, if the option is exercised, we know that the price of the option, the price of the stock or the price of the asset must be bigger than 21. Otherwise the option would not be exercised. So ND two is just a probability that the final day asset price is bigger than K, otherwise the option is not gonna be exercised. What about ND one? Well, you know, one thing else, if you don't know anything about the value of the asset, the ticket price could be anywhere between zero and a million or infinity, but you know it's been exercised. I know the option allows me to buy a ticket for 21. That means the moment's been exercised. The boundary, the lower boundary changes from zero to 21. That means the expected value goes up, right? Your previously, your expected value was anywhere between zero and infinity. Now it's between 21 and infinity. So it towards the right hand side. So what you have in the first one is the expected value of the ticket, given that the ticket is more is gonna be exercised. And uh, same thing over there, right? And how do you get those numbers? Normal distribution, basically a plug in the number of the normal distribution and the number on the other side will give you ND one and ND two. So essentially traders use this all the time. This is one of the, the most common formula in finance. And what do they, why do traders use it so much? Do they all understand this formula? No, they program it into the financial calculators. So literally all you need to do plug in that formula into your financial calculator. You need five things, sorry, we need five things and you're done. Right? So you are call a trader and you say, I wanna price an option on this. You'll just type in the numbers num, whatever it gives him on the other side. He'll quote you that number. He'll add a little fudge figure to make up for his profits. But essentially that's how it works. So they don't understand this either, right? But you do. Alright, and that brings that to the end of today. Thank you Professor Rao. Um, we'll be taking questions from the audience and we do have a roving mic. Um, it does beg me to ask you the question, how do we forward price your next next lecture and put a value on that<laugh>? Good question. It depends on the risk free rate, which fortunately is quite high nowadays. So <laugh>, it'll be <laugh>, but then zero times anything will still be zero. So the next lecture guys is also free. Um, thank you very much for very interesting if somewhat taxing lecture <laugh>, it, um, definitely exercised the, the brain power. My question is perhaps slightly off mm-Hmm <affirmative>, um, the direct subject of the lecture. But if, um, what, when you are talking about options mm-hmm <affirmative> and things of that ilk, what is the actual value of that in terms of the contribution to society, if you like? In other words, if I, if I make a product and sell it to you and you pay for it and so forth, we can understand what's going on there. If I'm standing sort of watching that happen and effectively taking bets on what's going to happen, how does that contribute to the, um, the benefit of humanity? That's my question. That's a great question, right? So that's only looking at one of the reasons why options are are important. That is basically you're saying it's a casino, right? Some people are taking bets, it doesn't really contribute anything. But the second reason why an option is important is for hedging your risk. So for example, think of an airline company which is hedging the price of its fuel oil. It might use an option for that and that allows it to stabilize its earnings so as to prevent itself from going bankrupt, for example. Same thing with a farmer. So the idea is that many of us cannot take risks which are too big. Like for example, would you insure, um, your pen? Probably not. It's still so small. You can buy a new pen if you lose it. But if you don't insure your house, if something happens, we can't afford to replace our house. So that means you want insurance on your house. The insurance company has a lot of other houses so they can take the risk, but you can't. So it allows you to transfer risk from someone who is not able to bear it, to someone who is able to bear it. And that's what the real effect of an option is. Financial returns and equity prices are not normally distributed. Uh, they usually have fat tails. That means there's a free lunch. How do I earn it?<laugh>? Well, let me put it this way. Um, there's a very famous example of a company long-term capital management, which was indeed managed by three people, two people, uh, one person, uh, two people whose names you already know. One was John Merriweather, who was head of trading at Sal Brothers, but the others were indeed Myron Shu and Robert Merton. And they used their own formula to lay bets on these transactions all over the world. And in 19 94, 19 95, they were earning 40% per year just on these tiny little arbitrage earning free lunches everywhere. And of course the people would ask them, what are you doing? And I was like, sorry, we won the Nobel Prize. Don't just just give us the money, we won't tell you anything about what we're going to do. And then in 1997 after they got the Nobel, um, Russia defaulted in orange corporate bonds, and essentially what happened was all their bets, it was like 200 bets. They had people panicked and everyone rushed for the safe exit. So that means they watch safe assets with these guys had shorted and they, you know, dumped ba unsafe assets equivalent to the others. But these guys had gone long on them. They had bought the unsafe assets and sold the safe assets hoping to make money free lunches, but essentially fact tailed argument. They ended up losing close to a hundred million dollars a day for five days. Eventually the feds walked into the entire thing and shut them down and netted off everybody's staging against each other. So the originators of the Black Shoals model, the Black Shoals Myrtle model, they themselves were caught by the fat tail because they assumed a distribution which didn't work exactly as you said, right? So a moral of the story is essentially, you know, if you trust your models really well and you don't account for human behavior, you can be in trouble. And in fact, that's one lecture we are gonna cover in June when we talk about market efficiency. Thank you for the lecture who's very interesting. I wanted to ask you, it's more of a simple question. To what extent has the understanding of options pricing theory affected your successful investments? Like has it affected your win rate by a lot? Uh, how valuable really is this for like an investor? Okay, well in my particular case, I can tell you that the easiest way I've invested is basically index funds, index ETFs, which is as we again go back to the lecture on, um, market efficiency in the, in in the future. A preview of that one, uh, that lecture says there is no way to consistently beat the market. And so I don't try to beat the market, I just basically invest everything in the market. So as long as I beat inflation, that's all I care about, right? So, but um, I don't use options, definitely I can tell you that, right? So it's best vanilla ETFs is pretty much the way to go Before, before exchanges. Mm-hmm, <affirmative>, how were people trading for what was probably not called forward contracts? So what was a mechanism of trading before exchanges and this Nobel Prize winning idea? That's actually a very, very good question. Um, it's, it turns out, by the way, do you guys have a wild idea when the earliest options traded ever? I mean, what was the recorded date for the earliest options ever traded? Wild guess anybody? 3000 BC <laugh>, right? So in Sumaria, in Babylon, they have actually examples of people writing contracts on grain harvest and so on, which were contingent on the harvest being good or bad. Those were options. So how did those people price it before the Black Shaws formula came along? The answer is they didn't use the formula, but they did. They derived what are called boundary conditions. The price of an option cannot be more than this, it cannot be less than this. There's a narrow bound where you do not require complicated mathematics. For example, a simple example, the price of a call option, the price of right to buy something cannot be worth more than the something, right? So we know the upper bound must be the ticket price. The price of a ticket is 20 pounds. You cannot have an option the right to buy the ticket more than 20 pounds. It'd be, you can create a free lunch for that. So that's an example of a boundary condition. There are lots of these where you could see prices trading in those conditions. You couldn't narrow down an exact price, but to get to the exact price you need a lot of assumptions. And those formulate needed no assumptions at all. Thank you. I'll take one question gentlemen here. Yes. Um, I wonder if, um, is a bad idea for force, uh, energy supplier to, um, have long and short options about energy prices in order to avoid? Well, the key question at the end of the day is if you do not allow energy providers to, you know, hedge their risks in some way or the other, as did happen a couple of years ago, a lot of energy suppliers went bankrupt, right? So you, I can have a contract with my, it happened to me three times honestly. So I, I had a contract with my energy supplier to supply me at a fixed price and I was very happy with that fixed price. And then they said, we can't supply this to you. And they went bankrupt. So I couldn't enforce it. So I was transferred to another one which went bankrupt, transferred to third one, and eventually I've got one, but it's a variable rate pricing and I'm, you know, so it's kind of annoying. But yes. So in a question like this, you would want to encourage them maybe to hedge their risk, but it's up to them because again, no free lunch, if they hedge their risk, it costs them money because they have to pay for the option. If they buy a forward or the future, one side will win, the other side will lose. But you don't guarantee that you'll always be on the winning side, right? So yeah, it's, again, no free lunch is always the way to think about it, right? I can buy insurance, but the value, the premiums I pay is equal to eventually gonna be equal to the value of the house. I'm insured. Thank you. We've got some more questions, but I know we're short of time. But once again, professor out, thank you for taking us through options, pricing, making the complicated, relatively simple to understand in, in the space of an hour. Thank you again, Chris, around. Thank you.

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