ECON 251: Financial Theory
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ECON 251 - Lecture 20 - Dynamic Hedging
Chapter 1. Fundamentals of Hedging [00:00:00]
Professor John Geanakoplos: The subject of today’s lecture is hedging. So this is what hedge funds do. It’s what almost everyone on Wall Street does nowadays, at least to some extent, say half the people on Wall Street nowadays.
It was hardly done at all in the past. So first of all, just to mention what a hedge fund is, a hedge fund is a firm that manages money. And why is it different from any other firm that manages money? Well, the definition of hedge fund basically has three parts.
One is that hedge funds hedge. Now I’ll say what that is in a second. Secondly, hedge funds use borrowed money in addition to their investors’ money to buy assets. So they do what’s called leveraging, which is going to be an important subject for the last few lectures of the course. And thirdly, they charge their investors very high fees.
That’s basically the definition of a hedge fund, because they’re supposed to be so good at what they do, they can charge high fees and still get the investors. So what is hedging? Hedging is the idea that you want to cancel out some of your risks. So for instance, you might know that a company’s not going to default, whereas the rest of the market thinks it is going to default. So you know the company’s worth more than the rest of the market thinks, so you might be tempted to buy the company.
But if the interest rate were to change, say go way up and the company’s paying, has constant cash payments, if the interest rate goes way up and the company has constant cash payments, you could end up losing money anyway, because the present value’s going to go down, just because the interest rate has gone up.
So what a hedger would do is, the hedger would say, “Look, I’m relying on my expertise as an evaluator to realize there’s no default risk. I want to bet that there’s no default risk in this company. It’s not going to default. I don’t want to bet on which way interest rates are going to go, because I don’t know which way interest rates are going to go, so I want to hedge myself against that.” So to take– sorry, I’m just starting here. Hello. Sorry, hard to talk when everyone else is talking. So what does hedging a risk means?
It means no matter which way the risk factor goes, you’re going to still end up with the same amount of money. So the first person to do this and call himself a hedge fund was somebody named Jones in the 1940s and he was a stock picker who would try to find the best possible stock to buy. So before him, people would say, “Okay, Ford is a great auto company. We’re going to buy Ford.”
The trouble is that Ford may in fact turn out to be the best auto company, but because the whole economy collapsed, it may be that Ford collapsed with the rest of the economy, even though it did better than all the other auto companies. So what Jones said is, “I’m not going to just buy Ford. What I’m going to do is, I’m going to buy Ford and I’m going to sell General Motors, so that way I’m going to be betting not on whether Ford is better than General Motors, and in addition, whether the whole economy’s going to go up. I only want to bet on whether Ford is better than General Motors.”
So if the whole economy goes down, Ford and General Motors will go down together. So what you own in Ford, you won’t own something as valuable any more, but what you owe by having sold General Motors short also won’t be as valuable. So what you’ve bought and what you’ve sold will cancel each other out and you won’t have lost any money.
On the other hand, if Ford ends up doing better than General Motors, you’ll have gained more with Ford than you’ll have lost with General Motors. So whether the economy goes up or down, if the gap between Ford and General Motors stays the same, you won’t lose or win money, so you’re hedged against general changes in the economy.
You’re only going to make or lose money depending on the spread between Ford and General Motors. So you’ve canceled out the economy wide risk and you’re only holding one risk, which you really think that you’ve understood. So that’s what hedging is.
So I’m going to now show you that hedging is actually a slightly delicate thing. It’s quite complicated and we’re going to learn the technique of dynamic hedging. Now why is dynamic hedging so important?
Dynamic hedging is so important because there are so many things that can happen in the future, and as I said, for example, if you hold a mortgage, you may think the mortgage is more valuable, because you know people aren’t going to default, and the rest of the economy thinks they’re going to default. On the other hand, the value of the mortgage will change with the interest rate, so you want to protect yourself against the interest rate change.
But it’s not just the interest rate change. Over ten years or 30 years, the life of the mortgage, there are going to be a huge number of interest rate changes. There’ll be a whole path of interest rate changes, and the mortgage will be paying different cash flows along the whole path. In over 30 years, even if you think of only one change a day, there’s an exponentially growing set of changes, so there’s an exponential number of paths. You’ll never be able to equalize your cash flows, you would think, over every single path. There are just too many of them.
But dynamic hedging says you don’t have to equalize over every path from the very beginning. That is, you don’t have to hold a portfolio at the very beginning that equalizes your cash flows by the end of every path. You only have to equalize your cash flow one step at a time and then dynamically change your hedge, so it’s a much easier thing to hedge than it sounds at first glance. So I want to illustrate that, because you can’t know what that means yet until we see an example.
So let’s start with a very easy thing to understand, the World Series, where we started before. So you have some expertise, in fact, there are other people besides you who have the same expertise, these other bookies. You all know the Yankees have a 60 percent chance of winning every game against the Phillies, and let’s just assume for simplicity that the World Series is only going to take a couple of weeks. The interest rate is 0 over such a short period of time. You don’t care whether you get the money now or at the end of 2 weeks, just so long as you get it. You can borrow or lend at 0 interest over those short number of days.
And secondly, you know that the probability of winning is 60 percent. Now suppose that some naïve person comes to you and says, “I’m a Phillies fan. I want to bet at even odds that the Phillies are going to win for 100 dollars.” So you’re going to now get a payoff that’s -100 dollars at the end of game 7, assuming I counted right. So this is the start. This would be game 1 here, so it’s 1, 2, 3, 4, 5, 6, 7.
So by the end of game 7, if you played it to the end, you would have made 100 dollars if the Yankees won a majority of the games, and you’d have lost 100 dollars if the Yankees didn’t win a majority of the games. So how much is that worth to you?
Well obviously it’s worth a lot more than 20 dollars, because if they only played 1 game and the Yankees had a 60 percent chance of winning, the expected payoff for the Yankees, that bet for 1 game would be 20 dollars, because you know in 1 game a .6 chance of 100 and a .4 chance of - 100, that gives you an expectation of 20. 60 - 40 is 20. However, we’re playing a seven game series. The whole point of playing a long series is that the Yankees are more likely to win. The better team is more likely to win, and in fact, we know that by doing this backward induction calculation, we know that the expected winnings for the Yankees is 42 dollars.
However, the Yankees–you could lose 100 dollars. If the Phillies just got lucky, you could lose as well as win 100 dollars. You don’t want to face that risk, so what could you do? Well, the simplest thing to do is to go to another bookie and say, “I want to bet on the World Series,” with this other bookie. So if the bookie is willing to make a bet with you for the whole series, so the bookie has to be pretty sophisticated and do the same calculation, the bookie is going to let you bet however you want to at the right odds. So what would you do?
So here you can win 100. There are only two outcomes. You can win 100 or you can lose 100. Now the odds of winning the series, I forgot to say, we figured this out before. Oh dear, what were the odds? Hope I have another one of these World Series things. Okay, so what are the odds that the Yankees are going to win the World Series? We calculated this before. Do you remember what it was? Well, we know some probability P times 100 + some probability (1 - P), -100 gives you 42. So P times 100 + (1 - P) times - 100 = 42 approximately. That’s 42, right? So we know that therefore, 2P times 100 = 142, so P = 142 over 100 times 2, which equals about .71.
Okay, so this other bookie would have calculated the odds are 71 percent that the Yankees are going to win the World Series. We calculated this a few classes ago. The Yankees have a 71 percent chance of winning the World Series. So if you went to another bookie who could do this calculation, the other bookie would be willing to give you odds of 71 percent Yankees versus 39 percent [correction: 29 percent] Phillies, so you could just unload this whole bet and get 42 dollars for sure.
In fact, what would you bet with the other guy? What would you bet with the other bookie? What would you do if you could bet on the whole World Series? Somebody suggested that. He was sitting over there at the end of the last class. What would you do if you could bet on the whole World Series with the other bookie? What bet would you make?
You know that the bet is giving you 42 dollars on expectation. The trouble is, some of the time it’s giving you 100, some of the time it’s giving you -100. You don’t want to fix this risk. You want to end up with 42 dollars for sure. Now how can you do that? Well, if you did -68 [correction: -58] here, that would = 42, and if you did +142 here, that would also = 42. So could you figure out a way of betting with the other bookies 142 versus 58? You win 142 if the Phillies win, and you lose 58 if the Yankees win?
Professor John Geanakoplos: That’s not 42 by the way. Okay, would another bookie be willing to give you this bet? What’s the answer? The answer is yes. How do you know the other bookie would be willing to give you this bet? What is P times 42 + (1 - P) times 42? It’s 42. What’s P times 100 + (1 - P) times -100. It’s also 42. So therefore, what’s P times this + (1 - P) times that?
Professor John Geanakoplos: 0. So at the odds, P and 1 - P, this bet is perfectly fair. So the other bookie would say that’s the odds, 71 percent to 39  percent. That’s exactly what these odds are. So if P times this + (1 - P) times that is 42, and P times this + (1 - P) times that, that’s obviously 42, must be P times this + (1 - P) times that is 0. In other words, at the odds, P and 1 - P, this is a fair bet, so the other bookie would give you those odds.
So that’s what’s hedging is in its simplest form. We haven’t had to do anything dynamic. Some guy is willing to bet on the World Series with you, so you know that he’s done something wrong in his calculations. He’s got the wrong P, so you can take advantage of the situation. You know that P is higher than he does. On the other hand, there’s still God in the background and luck, which might make the Phillies win, and so you don’t want to subject yourself to that risk. So what do you do?
You take the bet, the advantageous bet, but you put together with the advantageous bet a hedging bet, where you’re betting on the wrong team, the team you think is going to lose. But you’re betting on the wrong team, but this time, at fair odds on the wrong team, and so you’re transforming your risky payoff, albeit with very favorable odds–P we just said was .71–you’re taking your risky payoff and turning it into a safe payoff of 42, no matter what.
That’s the essence of hedging. In order to do that, you might have to bet the opposite way that you think is going to turn out. So there’s principle number one. So let me pause and see if there are any questions for that, about that. The idea is that there’s something you know that you can take advantage of, but life is more complicated than knowing one thing, namely what P is.
Life also involves all kinds of other things that might happen. You want to insulate yourself from those other things and concentrate entirely on what you know about P and 1 - P and therefore assure yourself of a bet of 42, no matter what. And you’re taking advantage of the fact that other people will also be able to bet at P with you. This one outsider doesn’t understand that all the bookies are willing to take odds of P, he’s offering odds of 1 half, which is crazy. It’s a big gap, 71, 29 to P. So you can take 42 of his dollars for sure.
Okay, so now let’s go to–okay, this is hedging. It’s also an arbitrage. This is–hedging created a pure arbitrage. In this case, it’s done even better than in my example with Ford, because you’ve taken now a bet and transformed it with an expectation you’re going to win, into a bet where you can’t possibly lose.
Chapter 2. The Principle of Dynamic Hedging [00:15:38]
Okay, now suppose that these other brokers weren’t as clever as you. Suppose that the other brokers–okay, so that’s hedging. Now we’re going to do dynamic hedging.
Suppose the other brokers, they’re not as smart as you. They don’t know how to build these trees. They can’t do backward induction. They just know the odds are 60 percent that the Yankees are going to win any game against the Phillies. Now what would you do? Okay, you can’t go to another broker and say, “I want to bet on the whole 7 game series.” The guy’s going to say, “It’s just too complicated for me. I’m a simple man. I make a simple living. I just do 1 game bets. I’ll let you bet any game you want. You tell me what you want to bet, we’ll bet one game at a time, 60/40 odds. The whole series, it’s much more complicated for me.” So is there anything you can do now?
What can you do now? So now you have to do something more complicated, which is called dynamic hedging. So we started to talk about this last time, so what would you do? You only now have the opportunity of protecting yourself against bad luck by betting one game at a time with these other brokers. So what should you do? Well, let’s look at this picture here.
You know that this bet of (+100, -100) by doing backward induction is worth 42. So if you were a trader, a hedge fund manager, you would be marking your position at 42. You’re expecting now–you’ve just made a trade which you know on average is going to make you 42 dollars.
Of course, if you had bad luck, you could end up losing 100 dollars, which means you’re going to go out of business, you’re going to be fired, your name’s going to be in the newspaper, you’re going to be probably sued by your investors. Or you can–we haven’t been sued by any investors. Or you might make 100 dollars.
Now you don’t want to face that. You want to hedge yourself, protect yourself, so you’re going to make the money for sure.
Now some people don’t hedge. They say when you hedge, you’re betting against yourself, and something bad can happen. The hedge might go negative, right? It may be that the Yankees win, just as you think, but because you’ve hedged, you’re not going to get 100 dollars. You’re only going to get 42 dollars. So the essence of hedging is you give something up on the upside to get something on the downside. So people don’t like hedging because they’re giving something up.
Okay, so you’re giving something up. The point is, you’re getting something back on the downside, and you’re making it so you’re locking in your profit. So hedging is a good thing. So people who don’t hedge I think they’re just very simple minded. So there are a lot of people who still don’t hedge, but anyway, there’s a revolution in financial thought in the last 30 years, anticipated by Ford [correction: Jones], but it didn’t really get–people didn’t really catch on until the ’70s that you should hedge.
So how can you hedge this if you can only bet 1 time, 1 game at a time? That was the question we dealt with last time. What should you do? Well, you see the principles we’ve learned so far tell us what to do. You started here thinking on average you’re going to make 42 dollars. You should be marking your position correctly that it’s worth 42 dollars. Now if you win the first game, you’re at a big advantage. The Yankees are up a game and by backward induction we know that then your position will be worth 64 dollars. If you lose, your position will be worth 8 [correction: 9] dollars.
Now what is 60 percent of 64 plus 40 percent of 9? It’s 42. 60 percent of 64 is 38.4. So you’re at 42 and you could go with 60 percent probability to 60– I can’t even remember the number–64, or down to .9, 40 percent. Sorry, I’m getting senile, 9 with 40 percent probability.
So 60 percent times that is 38.4 + 40 percent of 9 is 3.6 and the two add up to 42. So that’s why–so to say the value’s worth 42, how did you get the 42? We were doing backward induction. We took 60 percent of that + 40 percent of that and we got 42. So if you’re marking yourself to market, after the first game of the series, you’ll have to admit to the world that you’re either now worth 64 dollars, or you’ve gotten crushed, and you’re down to 8 [correction: 9] dollars. That already might get you fired. Going from 42 to 8 [correction: 9], that’s a pretty drastic loss.
So even the first day, if you have to mark to market, you’re going to be revealing to the world that you’ve screwed up and now your position’s only 8 [correction: 9]. You’ve told all your investors, you’ve done this brilliant thing. They’re up 42 dollars. The next day, you’re going to have to tell them, “Well, I’m sorry, you’re only up 8 [correction: 9] dollars.” They’re going to be very upset. So what would you do? You know what to do. What would you do?
Professor John Geanakoplos: Hedge. Good answer. How would you hedge?
Professor John Geanakoplos: Right. So what do you say? This is 9, so you’d go (+33, -22). So that’s equal to 42 and that’s also equal to 42. Can you do this? Is it fair odds to do this? Well, yes, it has to be fair odds to do this, because 60 percent of 64 + 40 percent of 9 is indeed 42. 60 percent of 42 + 40 percent of 42 obviously is 42. So therefore the difference, 60 percent of this + 40 percent of that must be 0, so in other words, it’s fair odds.
And sure enough, this is obviously 3 to 2 odds, right? 60/40 means you’re betting at 3 to 2 odds. That’s 3 to 2 odds. So it is a fair bet, so the other bookie who’s willing to bet at 3 to 2 odds is going to give you this bet. So again, you think the Yankees are going to win, you make a bet with the naïve Philly fan, you’re betting the Yankees are going to win. You go to your broker friend and you bet against your Yankees. You’re going to lose 22 dollars if the Yankees win, and win 33 if the Phillies win.
You’re betting against yourself, but of course you’re betting against yourself to a smaller extent than you’re betting against the other guy, and therefore you still locked in 42 dollars.
So now we don’t have to go any further. It would only be confusing to go further. I usually go further, so it would just be complicated. If you can do something once, you can do it many times. So by making the right hedging bet, you can lock in a profit up here and up here. But that means you can lock it in going forward to the very end, because you can keep repeating that hedge over and over again.
Maybe I’d better go one step further. You don’t really believe me. So what could you do going from here to here? Well, the average of here and here is about 9. You’ve collected 33 dollars, don’t forget, from your hedging bet. So although this says 9, you’ve got the 33 in the background, so it’s 42. But it’s 9.
So what you want to do, this is 9, you just want to transform this and this into 9. Once you’ve transformed this and this into 9, which you can do because 60 percent of this + 40 percent of that is 9, so the other broker–you can do an offloading hedging bet with another broker at 60/40 odds to get 9 for sure here. Why is that? Because 60 percent of this + 40 percent of that is 9, so therefore there must be offsetting–a bet he’d take that gave you 9 for sure in both cases, because the offsetting bet would also be valued at 0 according to 60/40 odds.
So you can transform this and this into 9s but don’t forget, you’ve been paid the 33, so actually this is 42 and 42. So you can keep 42 and 42 going forever and by the end, you’ll have 42 forever. So that’s what dynamic hedging is. It’s as simple as that. Yes.
Chapter 3. How Does Hedging Generate Profit? [00:24:26]
Student: In order to make money with hedging, wouldn’t you have to have someone who would bet against you with different odds?
Professor John Geanakoplos: Yes. So how do you make money in hedging? You effectively were betting against the Phillies fan at 50 percent odds, and you’ve got your broker who’s willing to give you 60 percent odds, and you’re exploiting that gap. So let’s do another–excellent question you’re asking. I haven’t answered his question yet. I want to give another example to answer his question. What is really going on with the hedging?
What am I relying on? How many brokers do I have to deal with to hedge? What’s going on here? I’ve given an example of a World Series where it’s very simple to figure out what to do. So now I want to give an example of a bond.
It doesn’t matter what bond. I have to confess, I haven’t done this in advance. Here we have a 30-year bond, 8 percent 30-year bond. It’s an 8 percent 30-year bond. So we know what its payment is. Homeowners are all perfectly not paying attention, the bond is going to be worth some amount of money. So here are the interest rates, I forgot to say. So we have to assume what the risk is. So in fact, what does this mean? We know that the risk is 6–the interest rate starts at 6 percent and it has a 20 percent volatility. So what does that tell us?
It tells us that the interest rates could go up or down by this amount. Now, we’ve assumed that the odds that it can go up or down are 50/50. So we’ve assumed that everybody thinks the odds are 50/50 that the interest rate could go up or down. That’s an assumption, and so now I’m going to make–we’ve assumed that the odds are 50/50.
Let’s assume that everybody is willing to trade you 50/50 odds. So what does that mean? This is like your other broker. So the interest rate is here, r0. It can go up to r0 times up, or it can go down to r0 times some down. Up remember, is–up, by the way, this = r0 e to the volatility times e to the drift, and down = r0 e to the - volatility e to the drift. It doesn’t matter. r0can go up or it can go down. So U is bigger than 1, D is less than 1 probably. So we see examples of that.
Now I’m making an assumption here that the odds are 1 half and 1 half and that everybody understands and is willing to bet on these odds of 1 half and 1 half. So betting on an interest rate move is called–that kind of betting on interest rates means trading interest rate derivatives.
So there are a bunch of ways that you can trade in the interest rate market to get 1 dollar if the interest rate goes up to this number, in exchange for losing 1 dollar if it goes down to this number. And these things are called swaps and they’re called interest rate derivatives and they’re called a whole bunch of other things, which it’s going to take too long to explain. So let’s suppose that you can go out into the interest rate derivative market and simply trade a security that’ll pay 1 dollar if things go up or you can buy another security that will pay 1 dollar if things go down.
Now what is the price of this security that pays a dollar if the interest rates go up? Let’s say they can only go up or down to one of two values. So what would be the cost if everyone agrees the odds are 50 percent of getting up here? What is the price today going to be of getting 1 dollar if the interest rate goes up?
Student: X divided by the interest rate.
Professor John Geanakoplos: Excellent. Who said that? Perfect. I didn’t expect anyone to say that so fast, so right. So the price here, we could write is .5 divided by (1 + r0), and this price going down is going to be .5 divided by (1 + r0). How did he figure that out? Well, the odds are 50 percent of going up here and 50 percent of going down here, but don’t forget, you’re getting the money later and this is a 1 period interest rate, 1 + r0.
So therefore, what you’d be willing to pay here is 50 percent of the dollar that you’ll get, discounted by 1 + r0. What you paid here for that is 50 percent of the dollar you’d be paid, discounted by 1 + r0. So, exactly right.
But now these are like the World Series, game by game odds. They’re going to change depending on where you are in the tree, because if you go here in the tree, you’ve got a higher interest rate, so it’s going to be .5 divided by (1 + rU) or something, which is a bigger number. So the price of betting on 1 dollar–what you have to pay to get 1 dollar if interest rates go up is going to be more [correction: less] here than it is here, say, because you’re going to still have 50 percent chance of getting here, but you’re discounting by a higher interest rate.
So it’s slightly subtler, but not much. It’s like the broker in the World Series who game by game, day by day, there’s the interest rate derivative market, and they’re willing to trade you, they’re willing to let you bet on which way the interest rates are going to go. So what have we got now?
We’ve got this mortgage that’s sitting there. So let’s say that you know that the mortgage is owned by a very rational guy. The homeowner–he’s not the owner of the mortgage–the homeowners are very rational people. The market is very foolish. This is an exaggeration. The market might think the homeowner never does anything, so they think the bond is worth 120. You’ve studied prepayments much more than the market does. You realize that the homeowners aren’t going to just sit there, dumbly paying whatever they owe if the interest rates go down. They’re going to prepay, so in fact the mortgage is only worth 98.
So the market thinks it’s worth 100. The market understands the interest rate probabilities and is willing to give interest rate risk. They agree with you about that. You don’t know anything more about interest rates than the market does. What you do know is you know something about the prepayments that the market doesn’t know, and so you know the bond is really worth 98 instead of 120. So what should you do?
Let’s suppose that the market is actually treating–so what the market thinks, mortgage is = to a 30 year bond–well, it’s not actually a bond because it doesn’t have a principal at the end, so I’m going to have to do a different sheet here to get it together. The market thinks mortgage never prepays. Let’s suppose that what you can trade in the market is these interest rate derivatives.
So what should you do now? You can trade the mortgage, you can buy it, you can sell it short, you can buy or sell these interest rate derivatives. What would you do? Let’s say you can only buy or sell 1 unit of the mortgage. There’s only 1 mortgage that is trading, and since you know there’s a mistake being made, you’re going to use all of it. So what would you do? Do you think the mortgage is overvalued or undervalued?
Professor John Geanakoplos: It’s overvalued, so therefore what should you do?
Student: Sell it.
Professor John Geanakoplos: Sell all you’re allowed to sell. I’ve assumed a little artificially. All you can do is sell 1 unit of the mortgage short. So you’re going to sell it for 120 dollars. But what does it mean to sell it short? It means you have to make the payments that the mortgage is making. So you know those payments are only worth 98 dollars? So what does that mean? It means that you know you’re going to get a profit of 22 dollars. However, you could get completely crushed if things don’t turn out exactly the way you thought, so you want to lock in a constant profit for sure.
So you don’t want to get a huge profit more than 20 and no profit in some other cases, or even a negative profit and a huge profit. You want your 22 dollars for sure. So how would you do that? What would you do? Yes.
Professor John Geanakoplos: Okay, so let’s be more precise. It’s exactly right, but let’s just see what this really means. So I’m going to copy these numbers down which I can’t see yet. You have a mortgage that’s 98.
You know it’s 98 something, .8 and it could go in the up state to 92.6 or in the down state it could go to 99.11. Now on the other hand, the market thinks that this mortgage, this other instrument, you’re selling at the same time something that’s worth 120. You’re selling it for 120.
It doesn’t matter what the mistake is the market makes. That’s the reason why the market made its mistake. You don’t care why they made their mistake. You know they’re just willing to give you 120 for it. So you’re making a 22 dollar profit, so what should you do?
If you were marking yourself to market, what would you do? What would you mark yourself to market here? Well, you’ve gotten 122 dollars so you’ve marked yourself to market. Here you’d say you’re 21.16 dollars up. Here you’d have to say you’re not up so much money anymore, because you’ve just lost some of the value of your mortgage, you’ve lost 6 bucks. So you’d have to say you’re only up 15 dollars now. And here you’d have to say you made a little bit of money, so you’re up 22 dollars now. But you want to lock in this profit, 21.16 for sure.
So what should you do? If you’re marking yourself to market, you want to turn this money into 21.16, but more than 21.16. You want to turn this into 21.16 times (1 + r0). You want to turn this into 21.16 times (1 + r0). Because then, having locked in 21.16 times (1 + r0), that’s no risk and it’s got the same present value as this and so every period, you want to keep locking your money into 21.16.
That’s the gap between that and up there. Maybe 16 wasn’t the right number, by the way. What was this price up here that they were willing to pay? They were willing to pay 120.58. So that’s a little bit more. 21.7 or something. So it’s 21.7, that was the gap. So it’s 120.58, so the gap here is 21.7. So you want to turn this into 21.7 and this into 21.7. So 21.7 here, you want to ensure that that’s where you’re marking yourself to position.
You want to know by the end of the next year, you could turn this into 21.7 discounted up and then keep it discounting till the very end. So whether you take your money now, you take your money later, it’s always a sure thing with exactly the same present value. So how would you turn 21.7, knowing that you’re going to run this risk if you’re marking yourself to market, how would you do that?
Well, you know the value of this + this at this interest rate, in the mortgage derivative market, it’s worth 98.84. So you can always transform this into exactly 91.84. By doing a fair bet you can get this to go up to 91.84 and this to go down to 91.–oh, there’s also a payment here. Don’t forget the payment. So sorry, we’ve forgotten a payment.
The mortgage is much subtler because you’ve actually gotten 8 dollars here + 8, and this is + 8. So we forgot about this. So you got an 8 dollar coupon + what was left was 92.6. this is 99.1 + 8. So actually, the total value here is 100.6 and the total value here is 107.11. You discount this by this 6 percent interest and you get 98.84. So I left out a crucial step.
Remember, the cash flows here are the coupon + the value of the mortgage that’s left. So the value of the mortgage that’s left is 92.65,+ the 8 dollar coupon. So over here, you’re going to have a value of 100.6. Down here, you’re going to have a value of 107.11. So how can you–you want that to equal, locked in forever–so you’ve just sold this mortgage for 120, so you’ve got 21.7 dollars sitting here.
So now what can you do with this 21.7 dollars? You can cancel your cash flows so what you owe on the mortgage is always exactly the same thing, 98.84. So you want to make this 98.84, so you have to subtract something off from here, 100.6. You want this to be equal to–let’s do it this way. Let’s make this 98.–this is a simpler way of saying it, I think–98.84.
Let’s turn this thing, the present value of this and this is 98.84. Therefore you know there must be some fair trade in the derivatives market that turns this thing and this thing into 98.84 times (1 + r0), and this into the same thing, 98.84 times (1 + r0). How do you know you can do this? Because the present value of 98.84 times (1 + r0) and 98.84 times (1 + r0), 1 half of that + 1 half of that, divided by (1 + r0) is 98.84.
So therefore the present value of this thing has to be the same as the present value of that thing, and therefore there must be some trade in the derivatives market you can make, where you sell some of these interest rate derivatives and you buy some of these interest rate derivatives, which guarantees you 98.84 in both cases. So that means the present value of what you have to deliver in cash–the value of what you’d have to deliver in cash–is 98.84, still 21.7 dollars, in present value terms, less than what you’ve received.
What have you done with this 120 dollars? You’ve invested that at the same interest rate and so now you’ve got a profit. You owe 98.84, jacked up by the interest rate, but the 120 you’ve invested. You’ve got that 120 + the interest rate. You’ve always got more that you’re carrying on than the present value of what you owe, and you keep going to the end of the tree.
When it ends, you’re still going to owe 98.84, increased by the interest rate a bunch of times, but then the value of the money you have is going to be 120 increased by the same interest rate, and so you’re going to end up with money for sure. So then you can just reverse it all, and get the money 21.7 locked in for sure at the beginning and owe nothing later in the tree.
So to say it all one more time, you have to think of yourself as marking to market. If you’re marking to market correctly, you know what you owe is only worth 98 dollars and what you’ve sold it for is 120 dollars. That’s a profit of 21.7.
Chapter 4. Maintaining Profits from Dynamic Hedging [00:43:48]
But you want to make sure you can maintain the same profit forever, or to put it another way, if you’ve bought this bond–you want to make sure you can lock in your profit forever. So how would you do this? There are different ways of doing it. Let’s say you want this 21.7 dollars for sure at the beginning and you don’t want to ever have to make any payments. In the end you want everything to cancel out.
What would you do? How would you do this? This number tells you, you owe in present value terms 98.84 dollars. So you’ve gotten 120 dollars. You could take the 120 dollars, put 21.7 in your pocket, and now, with this 98 dollars, you can buy these cash flows. What can you buy?
You can invest money so that you–you can buy the right interest rate derivatives so you’re able to make all the payments of the future bonds. So this bond, the actual payments the bond is going to make, you can buy all those payments for 98.84 dollars. I guess that’s the simplest way of saying it. I’m going a little bit in circles.
What is the mortgage security down here if you know how the mortgage is going to pay? All this is, is a bunch of payments at different parts in the tree. It pays 8 dollars here it pays 8 dollars here. Somewhere low in the tree, it’s going to just prepay for sure and pay the whole remaining balance. Down here, it’s just paid off for sure, so it’s prepaid. Sorry, it’s prepaid at the top–at the bottom, it’s prepaid. The fact that all these payments together are worth 98.84 means that by taking 98.84 dollars and trading on the derivatives market, you can buy all those payments for exactly 98.84.
So therefore as you go farther into the tree, you always will have the payments to make. You’ve sold the bond short. What does that mean? Any time this bond makes a dividend payment, you have to make the payment. But you can always buy those payments for 98.84.
How do you know you can buy those payments for 98.84? Well, by backward induction. Here, the payments are 8 dollars here, + 92 dollars of remaining dollars. So there’s a way in the interest rate derivatives market of buying 8 + 92.6, that’s 100.6 dollars here, and buying 107.11 dollars here. So by spending this 98.84, you can get, buy this many interest derivatives, so you can get 100 dollars of payments here, and 107 dollars of payments here. Half of 100 + half of 107, discounted at 6 percent interest, gives you 98.84. So it’s possible on the market to buy 100 dollars up here and 107 dollars down here with this 98.84.
Now out of those 100 dollars up here you’ve bought, give 8 of them to the guy you sold the bond short to. That’s the coupon payment. Down here, give 8 of these dollars to the same guy you sold the coupon short to. Now that leaves you with 92 dollars up there and 99 dollars there.
But this 92 dollars is the present value of what’s going to happen next, and what’s going to happen next up here is that you’re going to owe 8 dollars to the guy here and you’re going to owe 8 dollars to the guy here, but what’s left will be worth 83 dollars and 97 dollars. So therefore with this 92 dollars, you in fact are able to buy this 92 + 105 here, because the present value of 92, almost 92, + 105 here, is exactly 92. That’s how we got this number. It’s 1 half times this number + the coupon, + 1 half times this number + the coupon, discounted at the interest rate corresponding to here. That is 92 dollars.
So therefore, by investing here in the interest rate market, you can buy that cash flow here, the 84 + 8, 92 dollars. Pay the guy the 8, you’ll still have 83 dollars left and with the cash you have over here, remember you bought exactly this much cash. You’ve got this much cash left. You can buy all the payments you need to make here, the 8-dollar coupon, plus you’ll still have 97 left. So all the way forward through the tree, you could always afford to buy all the cash payments of this bond.
So you can pay off the guy you sold it short to all the coupon payments that this bond is making, by having invested 98.84 at the beginning. And similarly, you got 120 by selling the bond short, so you’ve made 21.7 dollars and there’s never any more cash coming out of your pocket. You pocketed the 21.7 and you invested the 98.84 in the derivatives market over and over, changing your investment, and therefore reproducing all the cash flows that you have to make to the guy for having sold it short.
I didn’t say it very well. How followable was that? Not very followable. Sorry, I didn’t do a good job. Let’s try again. What is the essence of what’s going on? The essence of what’s going on is that gain-by-gain or month-by-month, you can find someone else who’ll always trade at fair odds with you. Fair odds means–in the interest market. They don’t know, these people trading the interest rate market, they don’t understand prepayments. It’s not like the bookie here. They don’t understand prepayments. They’re not willing to be with you on what prepayments are going to be. They’re just going to bet with you on what interest rate is going to turn up.
But you see, the cash flows from the bond depended on prepayments and depended on the interest rate. So you calculated it was worth 98.84 because you know what prepayments to put in the future. So if you’re absolutely confident on your prepayments, you’re going to know what this bond is going to do in the future. So remember the prepayments. What are these prepayments? It’s this stuff. So down here, where the 1s are, the guy’s prepaid for sure, so you’d have to deliver to the market the entire remaining balance at that point. But see, when you calculated this bond, you were anticipating what all the payments were going to be, whether they were going to be the whole remaining balance or just the coupon.
Taking the present value, what calculations were you doing? In the present value, you’re doing calculations that this guy can do. You’re just taking 1 half times this + 1 half times that discounted. The interest rate derivative guy is willing to do that. He doesn’t know what the prepayments are going to be, but you’re never betting on prepayments with him. With the interest rate derivatives guy, you’re just saying, “Let me pay you some money now. You give me money if things go up here, or I’ll pay you some money now and you give me money if things go down here, if interest rates go up here or interest rates go down here.”
So because the present value of the mortgage cash flows evaluated according to the interest rate probabilities and discounting is 98.84, that means that you can successively trade on the interest rate market and reproduce all the mortgage payments, assuming you’re right about what they’re going to be. And therefore with 98.84 dollars, you can replicate what the mortgage is going to pay. And therefore, you make 21. 7 for free, because you sold it for 120. You only need 98.84 of that to reproduce all the payments of the mortgage bond. And how do you know that you can do that?
Because it’s just one step repeated over and over again. At every step, there’s going to be a payment of say 8 dollars and a present value of what’s left. And there’ll be a payment down here of 8 dollars and a present value of what’s left. Or maybe this thing is just a prepayment, in which case it’s a single payment of 99.5. That’s the remaining balance after 1 period.
But when you figure out this number, it’s always taken by averaging this number discounted with this number discounted. So therefore, averaging at the odds the interest rate guy will give you. So therefore, using this money, we’re now doing the opposite of the World Series, using this money, you can buy this total payment, and you can buy this total payment, because that’s exactly what the interest rate guy will give you.
This is equal to 1 half of all that divided by 1 + r0, plus 1 half of all this divided by 1 + r0. That is this number, so therefore in the interest rate market, by using this cash, you can buy that and you can buy that. And that’s enough to make the payment A8 that the mortgage is paying, you keep your promise. And over here, if the mortgage prepaid, the remaining balance, that’s that number, 99.5, you can pay that too. And then the mortgage is done.
Here the mortgage is going to go on, but see, it’s going on at present value you’ve calculated at 92.6. You bought 92.6 + 8. You use the 8 to pay off the coupon. You still have 92.6 dollars in your pocket here. You’re going to use that to buy the future payments of the mortgage, some of which will be a coupon, and maybe this will be another coupon, plus the present value of what’s left. If it goes down here again, the guy might prepay, but you’re going to have enough money to make that prepayment.
So with this 98.84 dollars, you’re just doing fair odds over and over again. You’re buying the future cash flows of the mortgage and therefore with 98.84 dollars, you can keep all your promises, and yet you’ve gotten 120 at the beginning, so you’ve locked it in. That’s it. Any questions? I don’t know if that was too clear, so somebody ask a question. You’re great at asking questions. Is this followable?
Professor John Geanakoplos: Okay, that’s too good. Too great.
Chapter 5. Dynamic Hedging in the Bond Market [00:54:08]
Okay, let’s see if I can do another example. Let’s suppose that we’ve got this bond.
Let’s say you can’t trade in the interest rate derivatives market directly, but you can trade in the bond market. So let’s get more realistic now. Let’s suppose that there’s a bond market. Now here we had interest rates–so it was a 30-year bond, the starting interest rate was 6, and the volatility was 20. So we had that sheet that was bond market trading, so let’s just go to that. File, where’s open. So callable bond.
Here we are in the bond market. Let’s say there’s a 9 percent coupon bond, doesn’t matter what it is. We start at the interest rate of .06. Was that where we were starting? And the volatility was 20. So here we have the same exact–exactly the same interest rate process as before, so those are the interest rates, just as we had before.
Now let’s talk about a bond. So it’s a non-callable bond. I forgot already what I said. How long was this bond? It was a 30-year bond. Didn’t have to be 30 years, but anyway, it’s a 30-year, 9 percent bond, and its value was 140.93. So the bond is worth 140.93. If the interest rates go up, same interest rate process, it can go to 121.4 or to 159.4. So now let’s suppose you can’t trade in the interest rate market any more. What should you do now?
Student: Don’t you need another bond?
Professor John Geanakoplos: Well, you’ve got the interest rate. You’ve got the 1-month [correction: year] bond that you can trade at an interest rate of 6 percent and you’ve got this bond market. So anyone will buy and sell with you. Remember, this is the 30-year, 9 percent coupon bond.
They’re all willing to buy and sell this with you. And you’ve got this mortgage that everybody else thinks is the price is worth 120, but you’re sure it’s worth 98.84, because you know the prepayments. The homeowners are smart and they’re going to prepay and nobody else realizes they’re going to do it.
So what should you do now? You can’t trade interest rate derivatives any more. All you can do is trade in the bond market, so do you have any idea what you could do? What would you do? This is even slightly more realistic example. You’ve got a mortgage that you know is overvalued. You’ve got a bond that you can trade, and of course you can trade a very short bond, the 1-year bond, which is trading at 6 percent interest.
You can trade a 1 year treasury, and here’s the 30 year, 9 percent treasury, which happens to be worth 140, so what should you do? Well, you know that you could sell this mortgage short, get 120 dollars for it, and you think the value of the payments is only 98, so you’d love to do this.
The trouble is, the future values depend on a lot of uncertainty. It may be that the payments you might have to make are more than 98, so what are you going to do? How would you lock in your profit for sure? Can anybody figure out what to do, just intuitively? It seems pretty complicated to figure out what to do. So you sell this bond for 120. You put the 21.7 dollars in your pocket. You have 98.4 dollars now that you have to spend to somehow acquire payments that are going to just allow you to make the bond payments you’ve sold short, the mortgage payments you’ve sold short at every node, so you never have to give up any more money.
So what should you do? You want to acquire some assets with this 98.84 dollars that will put you in a position where you have 100.6 dollars up here and 107.1 dollars down there. So how can you do that?
Well, you have a combination of the short–you have this 30-year bond you could invest in. If you spent 98 dollars on the 30 year bond, that’s 70 percent of that, so that would give you 84 up here and 105 down there. So it wouldn’t match what you really need. On the other hand, you could put your 98 dollars in the 1-year bond, which would give you 98.84 times 1.06 and 98.84 times 1.06.
So you’ve got this choice, or you can put 100 dollars here and get 106 dollars there. The interest rate we said was 6 percent, so that means there’s a 1-year bond that pays 106 in these two cases. There’s the 30-year bond, where 140 dollars could be worth this in the two cases. So let’s say you can’t go trade on the derivatives market, it doesn’t exist. Or at least you don’t know how to trade on it. But what you do know how to trade is on the bond market, and–the long bond market and the short bond market. So let’s call this the long bond market, and the short bond market.
So what would you do? Any thoughts here? Let me repeat the problem. You find, like you typically do when you’re a trader, you know something other guys don’t do. That’s what you’re making your whole livelihood on. You understand that prepayments are going to be very fast. The people know how to prepay.
The rest of the market, hasn’t occurred to them yet that people can prepay. They’re grossly overpaying for the mortgage, paying 120 dollars. So you go and sell the mortgage short. Now that means you have to deliver what the mortgage is paying. You have to deliver it to other people, so that’s a lot to take on. But I say that you can sell this bond for 120, put 21.7 in your pocket, take the remaining 98.84 dollars and now buy the cash flows that you’re going to have to deliver in the future.
How do you know you can buy all the cash flows? There are 30 years of cash flows. There are coupon payments, there are prepayments. There are all kinds of complicated payments of the cash flows. How, with this 98.84 dollars, are you going to be able to buy them all exactly?
All you can do is trade, buy the long bond and buy the short bond, or maybe sell the long bond short or sell the short bond short. So how, by trading these two securities over and over again, can you replicate the cash flows of the mortgage? That way you will have hedged out your risk, and you’ll have the 21.7 locked in without ever having to worry about anything in the future, provided your prepayment calculations are correct.
If you’re wrong about prepayments, then you’re going to be in big trouble, so you’re betting on knowing what the prepayments are. So what should you do? Well, here this thing is going to–the cash flows at this point are going to be worth 100.6. The 8 that you have to deliver, 92.6 for the present value of the future cash flows.
Here they’re worth–let’s just do the real thing. Never mind that stuff–here they’re worth 107.11. Now you know that 1 half times 1 over (1 + r0) of 106 + 1 half over (1 + .06) times 107 is 98.84. So the average of these discounted is that number. You know that the average of these numbers–so this is the non-callable bond, so it’s not these numbers. It’s this number + 9 and this number + 9. It’s paid a coupon, so the average of these numbers, discounted by 6 percent, is that. So what does that tell you you should do?
I claim by combining this bond and this bond, you can produce 100.6 and 107.11 here. So we know that this top bond is going to pay you 130.4 up there and 168.4 down here. If you bought X units of the long bond, that’s what your payoffs would be. The short bond is going to pay you 106 in both cases. So if you bought X units of the short bond, that’s what your payoff would be.
You want the payoff to be 100.6 and 107.11. So there must be an XL and an XS that equals that. How do you know there has to be? Because it’s 2 equations and 2 unknowns and these are independent equations. In fact, you can tell what XLhas to be. This gap between 100.6 and 107, that gap is about 7, and this gap is about 38.
So 7 is like 1 fifth of 38, so XL is going to have to be 1 fifth. I know in advance that this is going to turn out to be about 7 over 38. Just by general principles, you know there’s a solution for XL and XS. So I know there’s a solution to that because it’s 2 linear equations and 2 unknowns and they’re not degenerate, so I know there has to be a solution to that.
And I also can tell you what XL has to be, because the gap, that’s the difference between the top number and the bottom number, is 7. That gap over there is 38. So obviously the middle gap, 106 and 106 is 0. So if I take XL to be 7 thirty-eighths of that number, the gap is going to go down from 38 to 7. Then I’ll just have to find the right XS to make that equal.
So that combination of XL and XS will produce this payoff here and this payoff there. So I know what combination of the long bond and the short bond to hold so I produce exactly these things. So I’m going to get a cash flow of 100.6 and 107.11 and I’ll be able to use that to make the coupon payment of 8 in both cases and on top of that, have enough money to continue buying future cash flows. Now what will the cost of the XL and X short be?
It will be XL–I’ll have to pay 100 dollars for the short bond, if I have to do XS, Xshort, and I’ll have to take 7 thirty-eighths of 140, that’s how much it’ll cost me to buy the long bond. So the right combination of short bond and long bond will give me the right payoffs. And it’s clear that there is an XS here and an XL here, such that I get the payoffs I want of 100 and 107. But the last thing to notice is the cost of this XL and this XS, which is going to be 140 times XL + 100 times XS. That cost will be exactly 98.84.
How do I know that–therefore I’ll just be able to do it. I’ll ask you that question in a second. Here’s what this hedging amounts to. The hedging amounts to again, you know because of your superior knowledge of prepayments that the mortgage cash flows, not the mortgage itself, the cash flows that are going to come from the mortgage are only worth 98.8. Someone’s handing you 120 dollars in exchange for making those mortgage cash flows. Okay, so what do you do for that?
You take the 98.84 dollars out of the 120. You take 21.7. That goes into your pocket. The 98.4 you have to use to buy the future cash flows. The first cash flows are going to be 8 dollars, but you’re going to need more money to buy the cash flows that come after. How much more money do you need? 92.6 here, 99.1 there. So how can you put yourself in a position to have 100.6 dollars here and 107.11 dollars here? Can you use this amount of money to buy this value at the next step?
We saw in the derivatives market, yes you can. But even if you can’t trade in the derivatives market, you don’t need to if you could trade a short bond and a long bond. You would just find how much of the long bond do I hold and how much of the short bond do I hold, so I can get these cash flows of 100.6 and 107.11?
The answer is, you have to solve that equation up there. What XL and what XS gives you that number? You know that there’s some XL and some XS that will solve that, because it’s 2 equations and 2 unknowns. I’m repeating myself. You even know what XL is without writing down the equations, because the gap has to be–one thing, there’s no risk, and the other thing, there’s a big risk. So the only risk in the payoff has to come from the long bond. So that gap of 38, if you take 7 thirty-eighths of that, you’re going to get the gap to go down to 7. So the XL’s going to turn out to be 7 thirty-eighths. XSyou have to solve by algebra. So you know how much that XL is and what the XS is.
That’s what you hold, that combination of XL and XS, you hold here and you get exactly the payoffs. And then the final step is to notice that you can exactly afford it with 98.84 dollars. How do you know that you can afford it? Because what you want is 100.6 and 107.11 and that costs 98.84. You’re buying two things together whose payoffs together are worth 98.84. So therefore the cost of the two things separately have to be worth 98.84. So it will turn out when you solve for XLand XS, you’ll exactly be able to afford it with this amount of cash.
Chapter 6. Conclusion [01:10:30]
So you can just do that going forward all the time, constantly re-hedging your portfolio. So that’s the essence of dynamic hedging.
It’s a very beautiful idea which I probably haven’t explained in the optimal way, but the point is that, to summarize the whole thing again, you know something about a bond, but you’re subject to more risk besides what you know. You hedge out the extra risk, still relying on yourself to be right about what you know.
You relied on your being right about the prepayments. You don’t know anything about the future interest rates. But there’s another guy like the trader in the World Series, the broker in the World Series, who’s willing to make bets on the interest rate with you. Or to say the same thing, who’s been calculating the values of the bonds, the short and the long bond, as if he was making bets on the interest rate. He’s calculated at the same 50/50 odds, discounting by the interest rate.
So because there are all these guys on the market who are willing to make these game by game bets, either directly–I’m almost done–they either make the game by game bets and the interest rate directly in the derivatives market, or what’s equivalent to that, they’ve used that calculus to figure out the value of these long and short bonds. So by trading through the long and short bonds, you’re effectively doing the same interest rate bet. So either way, by trading through the two bonds, or trading directly in the interest rate derivative market, you’re able to buy, by going game at a time. Year by year, you’re able to buy all the cash flows of your mortgage, the cash flows that you’re predicting it’s going to have.
And therefore, you can make the profit for sure. It’s predicting something, being confident in your prediction, then being able to buy what you’ve predicted the cash flows are for a smaller price than you can sell the security for. That’s how you locked in your profit. Okay.
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