[User] Please transcribe this text

[Assistant] I bet $100 with the original Philly fan. I bet $100 on the Yankees. If the Phillies win all four, first four games, I've got to give the guy $100. Okay, of course I made these side bets with all these bookies on the way, so what were the side bets?


[User] Please transcribe this text

[Assistant] Situations, you're up $42, but we're trying to understand why that is. That's your question. Right? So, I'm, you're gonna be able to answer the question.


[User] Please transcribe this text

[Assistant] Uh, although you don't think so yet, but, so what happened the first time? The first time the Phillies won, okay, that bet with the original Philly fan is looking worse. Now it looks like I had forty-four, forty-two dollars I expected to win. Now I only expect to win nine dollars, because, you know, the series is turning against me. But, the bookie paid me thirty-three dollars.


[User] Please transcribe this text

[Assistant] So I'm $33 already in my pocket. This is 9. Now let's say I got to the end, to the last game, where this was a 0. Okay, let's say this is the last game where I've lost.


[User] Please transcribe this text

[Assistant] Okay, well this is minus 32. Let's say it was minus, let's say this is the last game. I lose 32 dollars here.


[User] Please transcribe this text

[Assistant] The Phillies win the first two games, right? So you're saying, I'm down $32, that's the last node, how can I possibly be up 42? Well, so what's the answer?


[User] Please transcribe this text

[Assistant] Right, because I bet on the first game, and the bookie, the first bookie gave me 33 dollars. Then the second game, I'm gonna bet with the bookie, and I'm gonna bet minus 27 and 41. Okay, so although I now owe the Philly fan 33 dollars here,


[User] Please transcribe this text

[Assistant] I made $41 with the second bookie and $33 with the first bookie. That's $74.


[User] Please transcribe this text

[Assistant] Okay, now just, so it sounds like you have to do a lot of arithmetic. You don't have to do any arithmetic. I knew I was going to end up $42 up. How is that?


[User] Please transcribe this text

[Assistant] Because every bet I set, let's look at the number here, wherever we are in the tree, like here, at this node, after all the stuff that's happened before, the bet I'm gonna make with the bookie, that one game, is gonna be enough to turn this random thing into 6464.


[User] Please transcribe this text

[Assistant] Okay, so I always take what happens further down the tree and turn it in to what's happened at the beginning of the tree. But then I work it all the way back and it's 42. Okay, so that means


[User] Please transcribe this text

[Assistant] See, this thing was 64, 9, but I made a bet 22, 33 with the bookie to turn it into 42, 42. This thing was 64, I made a bet to turn this into 64, 64. Okay, but with the 22 here, that's also 42. So no matter how far I go in the tree, I'm still exactly 42 ahead. Okay, any other questions? Yes.


[User] Please transcribe this text

[Assistant] In this case, the mortgage, so now we're moving from the World Series to more real life. In this case, I know the cash flows from this mortgage are worth 98.8, but somebody else not understanding the homeowner and how rational she is, someone else is willing to pay me 120 for that mortgage. So I sell the mortgage for 120.


[User] Please transcribe this text

[Assistant] So that's money now I have to do something active with. What do I have to do? I owe this guy a lot of payments. Every time she, let's say it's you, makes your payment, her payment,


[User] Please transcribe this text

[Assistant] You know, in the market, I've sold the thing short. That means I owe, I've promised to deliver whatever that mortgage delivers. That's what it means to sell it short. So I've got to deliver the 8 here, say. Okay? So how much, I've got to deliver the 8 here, and here she's prepaying. I have to deliver the whole 107.


[User] Please transcribe this text

[Assistant] Right? So with 98.8 dollars, I can afford to buy 100.6 dollars in this state and 107.11 in this state. How do I know that? Because that's where the 98.8 came from. It said this total amount, 100.6,


[User] Please transcribe this text

[Assistant] Times the price, which is the odds of it happening discounted, that's the price, plus the 107.11 times its price, the odds it happens discounted, okay, is exactly 98.8. That's where I got 98.8.


[User] Please transcribe this text

[Assistant] So therefore with 98.8 dollars, remember the bookie's willing to go either way, with 98.8 dollars I can buy all these cash flows and all these cash flows. So what do I do?


[User] Please transcribe this text

[Assistant] With the 100.6, with the 107.1 down here, I paid 8 her coupon and her 99.11 prepayment. I pay it all to that guy I sold the bond short to and I've kept my promise. That's the end of the mortgage.


[User] Please transcribe this text

[Assistant] The mortgage only delivered 8, so I only owe him 8. Okay? But I've got another 92.6 left.


[User] Please transcribe this text

[Assistant] Why is that exactly what I need? Because from here on, I have to use that money to buy the future cash flows because she's going to continue to pay her 8 and maybe eventually will prepay. But by induction, by repeating this step over and over again,


[User] Please transcribe this text

[Assistant] I can always make the payment and have the present value left over to afford to buy the future payments plus the present value left over.


[User] Please transcribe this text

[Assistant] So by induction, I'm always making the coupon, like in this first step, I make the coupon payment, plus I've got enough money left over to match the present value. So that means if I go forward and forward and forward, I keep making the payment


[User] Please transcribe this text

[Assistant] Yes, I could. Okay, so I was hoping you wouldn't ask me that because it's, uhm, so what happens in the next period?


[User] Please transcribe this text

[Assistant] She didn't prepay. This was 92. The present value is 92.6. So we're here. 92.6. Now she didn't pray.


[User] Please transcribe this text

[Assistant] Let's look at what she's doing, whether she's prepaying or not. Okay, so this is why I did the second graph.


[User] Please transcribe this text

[Assistant] So if the interest rates go up the first time and then back down, she's still not going to prepay. Okay, so we can now go up and see.


[User] Please transcribe this text

[Assistant] And see what's happening here. It's 92.65. Go up to 83.95.


[User] Please transcribe this text

[Assistant] 97.17. Okay, so what does that mean? That means with an, okay, so with 98.8 dollars, she bought


[User] Please transcribe this text

[Assistant] He, I bought, I'm the guy selling the thing short, you're the smart, rational prepayer. So you paid $8 here, that means because I've sold the mortgage short and you've paid, you've made that $8 payment, I owe the guy


[User] Please transcribe this text

[Assistant] And I knew that she was going to prepay. I knew how smart she was. And so I kept my promise. You didn't think she'd prepay. I knew it. I kept my promise.


[User] Please transcribe this text

[Assistant] Okay, and then up here, I've got the $8 so I can make the payment.


[User] Please transcribe this text

[Assistant] Not by 1.6, by the interest rate here, which happens to be, you have to figure out what that interest rate was up there. Uhm, so the interest rate went up to, it started at 6%, it went up


[User] Please transcribe this text

[Assistant] How did this number, 92.6, get calculated? It was exactly, take the payment of 8, take the present value of what's left, multiply by half and discount it at the interest rate here, plus a half times, uh,


[User] Please transcribe this text

[Assistant] Per your eight dollar payment that you make to the bank, I have to make to the other guy because I sold the mortgage short. I've got the money to pay him. I've also got the money to pay him if things have gone down again, eight dollars.


[User] Please transcribe this text

[Assistant] And I've got more cash in my pocket, this and this in the two cases. With this money, I can continue to buy the future payments that you're going to make. So I always have enough money to make whatever payments it is that you, that I have to, to keep my promise.


[User] Please transcribe this text

[Assistant] Well don't forget, I'm locking in my $21 profit for sure, so no matter what happens, I'm gonna just be able to make all the payments no matter what.


[User] Please transcribe this text

[Assistant] He's saying that I was able to take it, this is what my hedge run does, literally. We think we know better what the prepayments are gonna be than the rest of the market. So we're willing to take on a commitment, someone will pay us $120, we think we're only gonna have to, you know, $98 are enough to buy all the, all the promises that we've made by selling the thing short, say. Now how do we, we could be wrong. We have been wrong. Last two years we made a lot of mistakes.


[User] Please transcribe this text

[Assistant] Okay, uh, the reason why the guy is willing to pay so much is because down here, he thinks he's gonna get a lot of money. The interest rates are going down, so you know, discounting at a lower rate.


[User] Please transcribe this text

[Assistant] So if the world really goes, interest rates going up, then our theory of prepayments never gets put to the test, and so we just make money, and this guy, you know, and so, I mean, so our theory never gets put to the test. If interest rates go down, then our theory's been really put to the test, and we have to see whether she's really smart enough to prepay or not.


[User] Please transcribe this text

[Assistant] So you're right. That's absolutely true. Yep. But in any case, we're going to assume our theory is correct and hedge in such a way so that if our theory is correct about prepayments, we will exactly be in a position to keep our promises without ever running a risk of loss. Yes.


[User] Please transcribe this text

[Assistant] Okay, because it couldn't have happened that there were four


[User] Please transcribe this text

[Assistant] The Yankees win four in a row, I lose.


[User] Please transcribe this text

[Assistant] Again, lost four bets with the other bookie, but I've actually won a bet with the other bookie, so it seems like somehow things are better for me. And so it seems unbelievable that I'm still at 42 in both cases. That's your question, right?


[User] Please transcribe this text

[Assistant] My bets are changing. So after I don't make the same last bet, okay, so the first three bets, there's a Yankee win, Yankee win, Yankee win.


[User] Please transcribe this text

[Assistant] Each of those nights I'm making the same bet in the same scenario. But when the Phillies lose and then win again, I've changed my bet. So I'm still going to end up with, uh, I'm still going to end up... So the Yankees lost the, uh, your scenario was the Yankees lost the fourth game, okay? So that outcome is different from what it would have been in the first


[User] Please transcribe this text

[Assistant] Okay, so that was, uh, but now the last bet I made is not the same bet as I would have made the night before. It's a different bet, so that it just ends up


[User] Please transcribe this text

[Assistant] With the different direction in the fourth bet, plus this different fifth bet, it adds back up to what would have happened had the Yankees won the fourth time.


[User] Please transcribe this text

[Assistant] I don't know if that helps, but that's what happens, and, and the, and the proof that it happens is just by induction. So really, you're, you're, you know, I gave a proof.


[User] Please transcribe this text

[Assistant] And you're saying the proof is just, it's amazingly good. Remember the proof is very simple. You always make a bet so that no matter what happens, your expectation is back to where it was the night before. And if you do that, you have to end up, that's the proof that you've always locked in 42 from the beginning.


[User] Please transcribe this text

[Assistant] And so, you know, you can ask a thousand questions like that, and it's always going to be a complicated answer, but there has to be an answer, because we proved it. There was one.


[User] Please transcribe this text

[Assistant] So with the mortgage example, you're hedging against the interest rate going, uh,


[User] Please transcribe this text

[Assistant] No. When the interest rate goes up, you actually pay less.


[User] Please transcribe this text

[Assistant] You've made a promise, remember the price, if the interest rate goes up, where's the mortgage, sorry.


[User] Please transcribe this text

[Assistant] The value of the non-call, the value of the call. So as the interest rate, uh, goes up, you owe less money.


[User] Please transcribe this text

[Assistant] Right, cause you're promising to deliver this stuff. When the interest rates up, the mortgage you've sold short is less and less valuable.


[User] Please transcribe this text

[Assistant] So, that's the good scenario for you. But if the interest rate goes down, that's the scenario where you could get into trouble.


[User] Please transcribe this text

[Assistant] So you're hedging, you're trying to protect against the interest rate going down. In order to do that, you have to give money up in case the interest rate goes up.


[User] Please transcribe this text

[Assistant] Any other questions about this? It's quite a ingenious thing, I think.


[User] Please transcribe this text

[Assistant] This is the high, so we're gonna have one more step of this, but this is the high watermark of the standard, uh, you know, the, the finance guys who made finance such an important subject and, uhm,


[User] Please transcribe this text

[Assistant] You know, the rational expectation school of finance, this is kind of the most clever thing that they did. So this was this kind of reasoning, you know, uh,


[User] Please transcribe this text

[Assistant] Bookies maybe knew it for a long time, but Black Schold in 1973 started this kind of thinking.


[User] Please transcribe this text

[Assistant] So, I certainly didn't invent this idea of dynamic hedging.


[User] Please transcribe this text

[Assistant] Let's take a step back. The crucial idea is marking to market. The crucial idea is that you want to hedge something through millions of scenarios and really the outcome you won't know until the end. What you should do is hedge at each


[User] Please transcribe this text

[Assistant] Okay, so you only have to do things a step at a time. You know, there's an exponentially growing number of paths, but there are only two possibilities from today till tomorrow. So it's very simple to hedge two things. Seems incredibly complicated to hedge two to the


[User] Please transcribe this text

[Assistant] 100 things, you know, 2 to the 10th is a very big number, 2 is a very small number. You can hedge 2 things, it may be very complicated to hedge 2 to the 10th things, but you only have to hedge 2 things on every day.


[User] Please transcribe this text

[Assistant] Okay, so once you realize that, hedging is actually quite a simple operation. You don't have to do it by trading in the interest rate, uh, derivatives market. You could trade in a simpler market. Suppose, for example, that you knew that, uh, there were bonds being traded, you know, let's say 30 year bond.


[User] Please transcribe this text

[Assistant] So let's just take a look at a 30 year bond here.


[User] Please transcribe this text

[Assistant] So this is exactly the same thing. This is the 30 year bond under the same circumstances. It's a 9% bond. 9% bond.


[User] Please transcribe this text

[Assistant] Okay, coupon bond is starting with the same 6% interest rate. It's exactly the same process as before.


[User] Please transcribe this text

[Assistant] Okay, so the interest rates, those are the same interest rates as before. Okay, so here's the bond. Now the bond,


[User] Please transcribe this text

[Assistant] The bond starts at $140, can go to $121 at $159. Okay, remember this thing that you're trying to hedge.


[User] Please transcribe this text

[Assistant] But you know, you could just buy the right percentage of bonds and be able to accomplish the same thing.


[User] Please transcribe this text

[Assistant] In fact, this bond is worth 38 dollars more here than here. So the gap in the bond is 38. The gap in the, uh, in here is something like, uh,


[User] Please transcribe this text

[Assistant] And something that paid the right constant amount in each case.


[User] Please transcribe this text

[Assistant] You would get the same payments here and here. So what is the right constant amount? One sixth of the bond, by the way, would cost you twenty


[User] Please transcribe this text

[Assistant] Okay, one-sixth of the bond. So if you took the other 98.8 dollars and put it to the one-year treasury, I claim it's obvious. So take one-sixth of the, so one-sixth of 30-year 9% bond. Okay, that's one-sixth of this thing. Okay, plus, okay, which costs you, cost you what? It costs 23 dollars. Price of that is 23 dollars. Now combine that with


[User] Please transcribe this text

[Assistant] So added that to the treasury's constant payments, I'm gonna get something over here and six and a half dollars more than that down here. But since I've paid the same amount, 98.8,


[User] Please transcribe this text

[Assistant] And the 30 year bond were calculated by discounting whatever their payments were here and here by the same prices, this and this.


[User] Please transcribe this text

[Assistant] Okay, so there's a second way, instead of trading in the, in the,


[User] Please transcribe this text

[Assistant] In the treas- in the derivatives market, I could just hold the, uh, a combination of two bonds. So, one year treasury and the, uhm, 30 year 9% bond, and I'd still be perfectly hedged. But then when I got to here, I'd want to change my mix. I'd want to hold a different one year bond.


[User] Please transcribe this text

[Assistant] I have to buy a new one year bond in a different amount of the 30 year...


[User] Please transcribe this text

[Assistant] Uh, non-callable bond, 9% bond. But I could always reproduce these same two payments. So I don't have to hedge by holding


[User] Please transcribe this text

[Assistant] Okay, and I can tell at a glance how much of the longer term bond to hold because I just have to match this gap in the prices. Since the one year bond is always going to pay the same thing up and down. Okay, did that go too fast?


[User] Please transcribe this text

[Assistant] Okay, so I want to find, I want to hold one-sixth of the 30-year bond, which cost me 140.


[User] Please transcribe this text

[Assistant] Okay, so I know, alright, now I want to also hold


[User] Please transcribe this text

[Assistant] Uh, a one-year treasury, so that the payment, one-sixth of 121 at the top,


[User] Please transcribe this text

[Assistant] Plus whatever the one year treasury pays me is equal to a hundred point six and such that one sixth times this value at the bottom. Oh, sorry. It's one sixth.


[User] Please transcribe this text

[Assistant] This coupon is paying, actually it wasn't 121.6, it's not 9% bond, I forgot it's coupon. Very bad of me, won't change the numbers but, uh, so it's, it's paying, I didn't write it down.


[User] Please transcribe this text

[Assistant] So this thing is not paying 121, it's also paying the coupon. This is the present value of what's left. So it's paying 9 plus that. So it's 130, and here it's 168. The gap is still 38. Okay, so I want 1 sixth of a hundred, uh, 130, so 1 sixth of 130 cost, uhm, 20,


[User] Please transcribe this text

[Assistant] Twenty is still the same number, twenty-two, bad arithmetic to begin with, so still twenty-two. No, sorry, one-sixth.


[User] Please transcribe this text

[Assistant] Sorry, I didn't change. Okay, so the payment here is 130, and the payment here is 168. So I want, so 1 sixth of 130, that's the up payment, and this is 1 sixth of 168.


[User] Please transcribe this text

[Assistant] Okay, I want that plus whatever the one year treasury gives me to equal


[User] Please transcribe this text

[Assistant] 100.6 up here and 107.11 down here. That way I would have replicated things and hedged myself perfectly, right?


[User] Please transcribe this text

[Assistant] But I don't have to figure out what that one year treasury is. I think it's obvious. All I have to do is say...


[User] Please transcribe this text

[Assistant] Subtract 23 from 98.8. So I just have to buy 75.8 dollars worth.


[User] Please transcribe this text

[Assistant] Times this, this is the price, 1 half times 1 over 1.06.


[User] Please transcribe this text

[Assistant] Alright, so that bond, which cost 140.53, how did I get 140.53 there? I took the price of a dollar in the upstate, which is 50% likely discounted, multiplying by its payoff, 130, plus a half discounted, multiplied by the payoff of 168. This is 168 because it's a 9% bond.


[User] Please transcribe this text

[Assistant] So, it's one-sixth times one-sixth. So, if I get a sixth of the bond, it'll cost me 140.43 divided by six, which is 23. Okay, so.


[User] Please transcribe this text

[Assistant] What about the one year treasury? That's gonna give me some payment, you know, an additional payment here and here, namely 1.06 times whatever I put in.


[User] Please transcribe this text

[Assistant] Okay, so you notice that the gap between here and here, since I multiplied by one-sixth, is exactly the six-and-a-half dollar gap.


[User] Please transcribe this text

[Assistant] So, if I spend, the total I have to spend is 98.8 over here.


[User] Please transcribe this text

[Assistant] If I spend 23 dollars on the treasury and the remaining amount, 23 dollars on the 30 year 9% bond, the remaining amount of money, which is 75.8 on the treasury, I will have spent this 98.8 dollars. But then I have to get the same payoff.


[User] Please transcribe this text

[Assistant] Why is that? Because they're all being priced at the same prices. So it's just the distributive law of arithmetic.


[User] Please transcribe this text

[Assistant] Okay, plus the treasury payoffs equal 100.6 and 107.11 then it must be that the sum of the money I spent on the


[User] Please transcribe this text

[Assistant] Okay, so to say it again, just in a big picture look, there are two things that can happen. If you hold two bonds, a one year treasury and a 30 year thing, you're obviously going to be able to match these two things by holding the right proportion of them.


[User] Please transcribe this text

[Assistant] Okay, that's just obvious. And then less obvious is how quickly I'm figuring out


[User] Please transcribe this text

[Assistant] How much of those two bonds to hold, and the way I'm figuring it out so quickly is because the one year treasury is paying the same thing up and down, so I have to get all the variation out of the 30 year bond, so I have to hold the right proportion of the 30 year bond to get the same variation.


[User] Please transcribe this text

[Assistant] Okay, so in the very big picture, if you're trying to hedge some instrument with another instrument, you wanna, uh, and something, you know, a portfolio of stuff, much of which is certain, and only one thing that's risky, the amount of the risky thing you have to hold has to produce as much variation as the thing you're trying to hedge.


[User] Please transcribe this text

[Assistant] So the key is being able to tell at a glance how much variation each thing has. Then you can tell in what proportion to hold them.


[User] Please transcribe this text

[Assistant] Okay, and so the very last thing I wanna tell you today, which is unfortunately gonna take me a few minutes, is how traders


[User] Please transcribe this text

[Assistant] And guess, without doing almost any calculations, what the right hedge is. So just like we did the doubling rule, I want to do the variation rule of thumb. Okay, and so, this is, uh,


[User] Please transcribe this text

[Assistant] So I ask myself, suppose I have this 30 year bond. 30 year non-callable bond, that's this one. Okay, 9% bond. How could I guess kind of what the variation in price would be?


[User] Please transcribe this text

[Assistant] Without having to do, you know, I had to do this very complicated backward induction calculation.


[User] Please transcribe this text

[Assistant] Remember, once the interest rates go up here, it's a random walk. So on average, they're pretty much, not pretty much, they're exactly what they were here. You know, they start here, they go up and down, but on average, they're still here. And over here, they've gone down.


[User] Please transcribe this text

[Assistant] Okay, so the average is lower going forward, in fact it's exactly equal to that.


[User] Please transcribe this text

[Assistant] Uncertainty, you would know that the present value with no uncertainty is gonna be the coupon divided by 1 plus the interest rate plus the coupon over 1 plus the interest rate squared.


[User] Please transcribe this text

[Assistant] Plus in the last year, the coupon over 1 plus the interest rate to the t plus 100 times 1 plus the interest rate to the t. That's the value of the bond.


[User] Please transcribe this text

[Assistant] Okay, so although it's making things even more complicated, you would think. But,


[User] Please transcribe this text

[Assistant] This turns out to be not so complicated. Okay, because this is a number, so let's put the minus on the outside. I'll get rid of this.


[User] Please transcribe this text

[Assistant] This is a number which you can sort of get in your head.


[User] Please transcribe this text

[Assistant] So let's try writing it, let's pull out one of the 1 plus r's. Minus 1 over 1 plus r times c over 1 plus r plus


[User] Please transcribe this text

[Assistant] And the answer is yes, you can interpret that. Because what is that?


[User] Please transcribe this text

[Assistant] That's saying, these are all the payments you're getting over the life of the bond.


[User] Please transcribe this text

[Assistant] Okay, it's cc and the last one is c plus 100. This number that keeps coming in front of them though, is the year...


[User] Please transcribe this text

[Assistant] In which you're getting the payment. This is in the first year. I can put a 1 there. This is a payment in the second year. There's a 2 here. This is a payment in the teeth year. There's a T here.


[User] Please transcribe this text

[Assistant] Okay, so it's like waiting the year in which the payments come by


[User] Please transcribe this text

[Assistant] This thing in here is the average life, because I've multiplied the year in which the payments come by the present value of the cash flows there, and then I've made those multipliers add up to one by dividing by the PV. This is the, this in parenthesis, all these terms are the things multiplying the year in which the thing came. So if I add up all these


[User] Please transcribe this text

[Assistant] Coefficient here, I'm just getting the present values. So now the coefficients add up to one. So I'm just taking an average of the year in which the thing came.


[User] Please transcribe this text

[Assistant] And that's what the average life is. But that's equal to the derivative of the present value divided by the present value.


[User] Please transcribe this text

[Assistant] Okay, so it's the percentage change in the present value is the average year in which the payments come. So let's just think about it. Suppose you have a ten-year bond. It pays C, C, C, C, C, and then a hundred at the very end. What do you think the average life is? Well, if there were no discounting,


[User] Please transcribe this text

[Assistant] The average life wouldn't be five because there's such a huge payment in the end. The average life would be, you know, eight or nine.


[User] Please transcribe this text

[Assistant] Okay, but there is discounting, so this thing at the end won't count as much. The average life is probably 7. If it's a 30 year bond, the average life


[User] Please transcribe this text

[Assistant] You know, you're gonna get C C C C and then all the way at the end you're gonna get C plus 100.


[User] Please transcribe this text

[Assistant] Well, if the payments were all equal and no discounting, it would be 15.


[User] Please transcribe this text

[Assistant] But there's a huge amount of discounting, so that last payment is really pretty irrelevant.


[User] Please transcribe this text

[Assistant] So probably, it's not clear which way it goes, probably the average life will be less than 15. Okay, so, so you can see just by looking at the bond, and I'm ending with this thought, by looking at the bond itself, you can figure out what the average life is.


[User] Please transcribe this text

[Assistant] Okay, so a trader, just by a little experience in common sense, can figure out the average life. I mean, here all the payments, the way they're coming, they're coming, you know, evenly across the whole history of the bond with something at the end, so the average life is probably a little bit over the average.


[User] Please transcribe this text

[Assistant] Weigh off, you push the final payment too far to the end, then it's becoming negligible, it's almost irrelevant, and so the average life by discounting probably be before the average. So a 10 year bond probably has an average life of 7, a 30 year bond maybe has an average life of 12 or 13. Once you know the average life, you just have to hold bonds in proportion


[User] Please transcribe this text

[Assistant] To their average life. The dollar amount you put in each bond to hedge it is just proportional to its average life.


[User] Please transcribe this text

[Assistant] So that's how a trader, without having to do any arithmetic, can kind of guess how much of a 30 year bond will it take to hedge a 10 year bond, and how much of a, so which has the,


[User] Please transcribe this text

[Assistant] So which has a higher average life? The longer the bond, the higher the average life. The more sensitive it is to, uh,


[User] Please transcribe this text

[Assistant] Uh, to interest rate changes, so the less of it you need to hedge your position.


[User] Please transcribe this text

[Assistant] So if you hold a 30-year bond, you don't have to hold that much of it, a sixth of it, in order to hedge this mortgage. The mortgage has a very short average life because it's prepaying so quickly. So it's not going to last very long. So it's an amazing connection between average life


[User] Please transcribe this text

[Assistant] And sensitivity to interest rate. Average life is something you can guess.


[User] Please transcribe this text

[Assistant] Okay, so in the problem set, we'll see if you can do it, uhm, I ask you a couple problems to do with, uh, average life.


[User] Please transcribe this text

[Assistant] In this video, we'll talk about sampling. We'll get into the details of it and how it works across multiple different iterations.


[User] Please transcribe this text

[Assistant] So before we talk about how to train this neural network, let's talk about sampling or what we do with the neural network after it's trained at everyone's time.


[User] Please transcribe this text

[Assistant] So what happens is you have that noise sample. You put it through your trained neural network that has understood what a sprite kind of looks like. And what it does is it predicts noise. It predicts noise as opposed to the sprite. And then we subtract that predicted noise from the noise sample to get something a little bit more sprite-like.


[User] Please transcribe this text

[Assistant] Now realistically, that is just a prediction of noise, and it doesn't fully remove all the noise, so you need multiple steps to get high quality samples. That's after 500 iterations, we're able to get something that looks very sprite-like. So now let's step through this algorithmically.


[User] Please transcribe this text

[Assistant] And then you want to step through time, and actually you're stepping backwards through time, all the way from the last iteration, 500, where it's completely noisy, all the way down to one. And just think of your ink drop, you're actually going backwards in time, it's fully diffused initially, and then you're going back all the way up to when it was first dropped into the water.


[User] Please transcribe this text

[Assistant] Here's where you actually pass that original noise, that sample, back into your neural network, and you get some predicted noise. And this predicted noise is noise that the trained neural network wants to subtract from the original noise to get something that looks more sprite-like.


[User] Please transcribe this text

[Assistant] And finally, there's a sampling algorithm called DDPM, which stands for denoising diffusion probabilistic models, a paper written by Jonathan Ho, AJ Jane, and one of my good friends, Peter Rebiel.


[User] Please transcribe this text

[Assistant] And this sampling algorithm essentially is able to get a few numbers for scale. That's not super important, but what is important is that this is where you are actually subtracting out that predicted noise from the original noise sample. And again, you're adding that little extra noise back in, which we'll return to in a moment.


[User] Please transcribe this text

[Assistant] Alright, let's jump to the notebook. So you'll see here some setup code. Uhm, I think all that's really important here is that you're importing, uh, PyTorch and a lot of, uh, utilities from PyTorch. We also import some helper functions here that we had written, uh, for the neural network. So I'm just gonna hit shift enter to run that cell so that we import everything.


[User] Please transcribe this text

[Assistant] And now here's setting up the neural network, which we're going to use for sampling. Uh, we'll go into the details of this later. So I'm just going to run this.


[User] Please transcribe this text

[Assistant] And no need to really follow everything that's going on there just yet.


[User] Please transcribe this text

[Assistant] Here we're setting up some hyperparameters and that includes those time steps that you've seen there. So that's the 500 time steps. Beta 1 and beta 2 are just some hyperparameters for DDPM.


[User] Please transcribe this text

[Assistant] And this is just a noise schedule defined in the DDPM paper. And all a noise schedule is, is it determines what level of noise to apply to the image, uh, at a certain time step.


[User] Please transcribe this text

[Assistant] So this part is just constructing some of the parameters for the DDPM algorithm that you remember the scaling factors. Those scaling values, S1, S2, S3, that's being computed here in the noise schedule and it's called a schedule because it is dependent on the time step.


[User] Please transcribe this text

[Assistant] Remember you're looking through 500 time steps because you're going through those 500 iterations that you see here of slowly removing noise.


[User] Please transcribe this text

[Assistant] So I'm just going to run that here. So just depending on that time step that we're on next, I'm just going to instantiate the model, that unit, uh, which we will come back to.


[User] Please transcribe this text

[Assistant] And then here is the sampling algorithm, the denoise add noise that you had seen previously, which is really the main important part is that it is removing the predicted noise, which is what the model thinks is not a sprite from the original noise.


[User] Please transcribe this text

[Assistant] And then here is what we had just stepped through that sampling algorithm. And specifically here is running the model to get the predicted noise and then running the denoise.


[User] Please transcribe this text

[Assistant] Now let's visualize what sampling looks like over time.


[User] Please transcribe this text

[Assistant] This may take a few minutes depending on what kind of hardware you're running on, and we're going to speed this up in the video. In the next video, you'll also see a more efficient sampling technique as well. Alright, let's see it in action.


[User] Please transcribe this text

[Assistant] Wow, look at those sprites. So you should definitely pause and try these yourself.


[User] Please transcribe this text

[Assistant] Alright, so there's one more extra detail. So right now you have your neural network that's predicting noise from your original noise sample. You subtract it out. Great. And you get something a little bit more sprite-like.


[User] Please transcribe this text

[Assistant] But the thing is, this neural network expects this noisy sample, this normally distributed noisy sample as input.


[User] Please transcribe this text

[Assistant] So actually what you have to do after each step and before each next step is to add in additional noise that's scaled based on what time step you're at.


[User] Please transcribe this text

[Assistant] This actually helps stabilize the neural network so it doesn't collapse to something that's close to the average of the data set, meaning it doesn't look like this thing on the left. When we don't add that noise back in, the neural network just produces these average-looking blobs of sprites versus when we go add it back in, it actually is able to produce these beautiful images of sprites.


[User] Please transcribe this text

[Assistant] So here the algorithm is where this happens. So we actually sample a random noise again at each time step based on the time step. And then down here,


[User] Please transcribe this text

[Assistant] We actually add it back in with that scaling factor S3. So now let's take a look at the notebook.


[User] Please transcribe this text

[Assistant] So now in this function, denoise, add noise, we are talking about the add noise part. And what that is, is this Z that you randomly sample, that's that extra noise, and you scale it by some factor, and then you actually do add it back in.


[User] Please transcribe this text

[Assistant] Alright, so let's pick up where we left off then. So for the incorrect way where we don't add the noise back in, we actually just set Z to zero and we pass that in. It only subtracts the predicted noise from the original noise and it doesn't add any extra noise back in. And let's run this with shift, enter, shift. And again, this will take a couple minutes. Alright, so let's take a look at what this does instead.


[User] Please transcribe this text

[Assistant] The Transpose Convolution is a key part of the UNAT architecture. How do you take a 2x2 input and blow it up into a 4x4 dimensional output? The Transpose Convolution lets you do that. Let's dig into the details.


[User] Please transcribe this text

[Assistant] You're familiar with the normal convolution in which a typical layer of a neural network may input a 6x6x3 image?


[User] Please transcribe this text

[Assistant] Convolve that with a set of say 3x3x3 filters and if you have 5 of these then you end up with an output that is 4x4x5. A transpose convolution looks a bit different. You might input a 2x2 set of activations.


[User] Please transcribe this text

[Assistant] Convolve that with a 3x3 filter and end up with an output that is 4x4, that's bigger than the original input.


[User] Please transcribe this text

[Assistant] Let's step through a more detailed example of how this works.


[User] Please transcribe this text

[Assistant] In this example, we're going to take a 2x2 input, like that shown on the left,


[User] Please transcribe this text

[Assistant] And we want to end up with a 4 by 4 output.


[User] Please transcribe this text

[Assistant] But to go from 2x2 to 4x4, I'm going to choose to use a filter that is


[User] Please transcribe this text

[Assistant] Three by three, so the filter is F by F. And I'm going to choose


[User] Please transcribe this text

[Assistant] Instead of placing the filter on the input, you would instead place the filter on the output.


[User] Please transcribe this text

[Assistant] Let me illustrate that by mechanically stepping through the steps of the transpose convolution calculation.


[User] Please transcribe this text

[Assistant] So let's start with this upper left entry of the input, which is a tube.


[User] Please transcribe this text

[Assistant] We are going to take this number 2 and multiply it by every single value in the filter. And we're going to take the output, which is 3 by 3, and kind of paste it in this position.


[User] Please transcribe this text

[Assistant] Now, the padding area isn't going to contain any values, so what we're going to end up doing is ignore this padding region.


[User] Please transcribe this text

[Assistant] And just throw in four values in the red highlighted area. And so specifically, the upper left entry is zero times two. So that's zero.


[User] Please transcribe this text

[Assistant] The second entry is 1 times 2, so that is 2. Down here is 2 times 2, so that's 4.


[User] Please transcribe this text

[Assistant] And then over here is 1 times 2.


[User] Please transcribe this text

[Assistant] So that's equal to 2. Next, let's look at the second entry of the input, which is a 1.


[User] Please transcribe this text

[Assistant] I'm going to search the green pen for this. So once again, we're going to take a one and multiply by one every single element of the filter. Because we're using a stride of two, we're now going to shift the box in which we copy the numbers over.


[User] Please transcribe this text

[Assistant] Again, we'll ignore the area which is in the padding and so the only area that we need to copy the numbers over is this green shaded area. You may notice that there is some overlap between the places where we copied the red colored version of the filter and the green version.


[User] Please transcribe this text

[Assistant] And we cannot simply copy the green value over the red one.


[User] Please transcribe this text

[Assistant] So where the red and the green, uh, boxes overlap, you add the values, you add the two values together.


[User] Please transcribe this text

[Assistant] So where there's already a 2 from the first weighted filter, you add to it this first value from the green region, which is also 2. So you end up with


[User] Please transcribe this text

[Assistant] 2 plus 2. The next entry, 0 times 1 is 0, then you have 1,


[User] Please transcribe this text

[Assistant] 2 plus 0 times 1, so 2 plus 0, followed by 2, followed by 1.


[User] Please transcribe this text

[Assistant] Okay, we shifted two squares over from the red box here to the green box here because it's using a stride of two.


[User] Please transcribe this text

[Assistant] Next, let's look at the lower left entry of the input, which is a 3. Uh, so we'll take the filter, multiply every element by 3, and, and we've gone down by one step here. So we're gonna go down by two steps here. So we will be filling in our numbers.


[User] Please transcribe this text

[Assistant] In this three by three square, and so you find that the numbers you're copying over are two times three, which is six, one times three, which is three, zero times three, which is zero, and so on. Three, six, three.


[User] Please transcribe this text

[Assistant] And then lastly, let's go into the last input element, which is 2.


[User] Please transcribe this text

[Assistant] So we will multiply every element of the filter by two and add them to this block. And so you end up with adding one times two, which is plus two and so on for the rest of the elements.


[User] Please transcribe this text

[Assistant] So the final step is to take all the numbers in these 4x4 matrix of values in these 16 values and add them up. So you end up with 0 here, 2 plus 2 is 4, uh, 0, 1, uh, 4 plus 6 is 10,


[User] Please transcribe this text

[Assistant] I know this looks like a little bit of a mess because I'm writing on top of things, uh, that's 0, 3 plus 4 is 7.


[User] Please transcribe this text

[Assistant] And so that's your 4x4 output.


[User] Please transcribe this text

[Assistant] In case you're wondering, why do we have it to do it this way?


[User] Please transcribe this text

[Assistant] I think there are multiple possible ways


[User] Please transcribe this text

[Assistant] Small input and turn it into a bigger output, but the transpose convolution happens to be one that is effective. And when you learn all the parameters of the filter here, this turns out to give good results when you put this in the context of the unit, which is the learning algorithm we'll use now.


[User] Please transcribe this text

[Assistant] So let's go back and keep on building the unit. In this video, we step through step by step how the transpose convolution lets you take a small input, say two by two, and blow it up into a larger output.


[User] Please transcribe this text

[Assistant] This is the key building block of the UNet architecture, which you get to implement and play with in this week's programming exercise as well.


[User] Please transcribe this text

[Assistant] Now that you understand the transpose convolution, let's take this building block you now have and see how it fits into the overall unit architecture that you can use for semantic segmentation. Let's go on to the next video.


[User] Please transcribe this text

[Assistant] The subject of today's lecture is hedging. So this is the


[User] Please transcribe this text

[Assistant] It's 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 do it nowadays. Uh, it was hardly done at all in the past. So,


[User] Please transcribe this text

[Assistant] 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.


[User] Please transcribe this text

[Assistant] Okay, now, uh, I'll say what that is in a second.


[User] Please transcribe this text

[Assistant] 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.


[User] Please transcribe this text

[Assistant] 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?


[User] Please transcribe this text

[Assistant] Hedging is the idea that you want to cancel out some of your risks. So for instance,


[User] Please transcribe this text

[Assistant] You might know that a company's not going to default.


[User] Please transcribe this text

[Assistant] Whereas the rest, whereas the rest of the market thinks it is going to default.


[User] Please transcribe this text

[Assistant] So you know the company's worth more than the rest of the market thinks. So you might be tempted to buy the con- the company. But if the interest rate were to change,


[User] Please transcribe this text

[Assistant] Say you go way up and the company's paying, has constant, uh, cash payments. If the interest rate goes way up and the company has constant cash payments, you could end up losing money anyway because, you know, the present value's gonna go down just because the interest rate has gone up.


[User] Please transcribe this text

[Assistant] So what a hedger would do is the hedger would say, look, I'm relying on my expertise as a evaluator to realize there's no default risk. I want to bet.


[User] Please transcribe this text

[Assistant] 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, to take a, sorry, I'm just starting here, a lot of, hello.


[User] Please transcribe this text

[Assistant] Sorry, it's hard to talk when everyone else is talking. So, if, uh, if you, uhm, so what does hedging a risk mean?


[User] Please transcribe this text

[Assistant] Means, it means no matter which way the risk factor goes, you're gonna still end up with the same amount of money. So the first person to do this,


[User] Please transcribe this text

[Assistant] Uh, and call himself a hedge fund with somebody named Jones in the...


[User] Please transcribe this text

[Assistant] 1940s, and he was a stock picker.


[User] Please transcribe this text

[Assistant] Uh, 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


[User] Please transcribe this text

[Assistant] 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,


[User] Please transcribe this text

[Assistant] General changes in the economy, you're only gonna make or lose money depending on the spread between Ford and General Motors. So you've cancelled out the economy-wide risk,


[User] Please transcribe this text

[Assistant] And you're only holding one risk which you really think that you've understood. So that's what, uh, hedging is.


[User] Please transcribe this text

[Assistant] And we're going to learn the, the, the technique of dynamic hedging.


[User] Please transcribe this text

[Assistant] Now why is dynamic hedging so important? Dynamic hedging is so important because


[User] Please transcribe this text

[Assistant] There's 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,


[User] Please transcribe this text

[Assistant] You know, over 10 years or 30 years, the life of the mortgage, there are going to be a huge number of interest rate changes. There will be a whole path of interest rate changes.


[User] Please transcribe this text

[Assistant] And, and the mortgage will obtain different cash flows along the whole path. In over 30 years, you know, even if you only think of 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's just too many of them.


[User] Please transcribe this text

[Assistant] 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 on every path. You only have to equalize your cash flow one step at a time and then dynamically change your hedge.


[User] Please transcribe this text

[Assistant] Let's start with a very easy thing to understand, the World Series, where we started with before.


[User] Please transcribe this text

[Assistant] So you have some expertise, in fact there are other people besides you who have the same expertise, these other


[User] Please transcribe this text

[Assistant] Bookies, you all know the Yankees have a 60% 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 weeks.


[User] Please transcribe this text

[Assistant] The interest rate is zero over such a short period of time. You don't care whether you get the money now or at the end of two weeks. It's just so long as you get it. And secondly, uhm, and you can borrow or lend, you know, zero interest over those short number of days. And secondly, uhm,


[User] Please transcribe this text

[Assistant] Now, suppose that some naive person comes to you and says,


[User] Please transcribe this text

[Assistant] You know, I'm a Philly fan, I wanna bet at even odds that the Phillies are gonna win. For a hundred dollars.


[User] Please transcribe this text

[Assistant] So this is, uh, the start. So this would be game one here. So it's one.


[User] Please transcribe this text

[Assistant] So how much is that worth to you? Well, obviously it's worth a lot more.


[User] Please transcribe this text

[Assistant] Then twenty dollars, because if they only played one game and the Yankees had a sixty percent chance of winning, the expected payoff for the Yankees would be, that bet for one game would be twenty dollars.


[User] Please transcribe this text

[Assistant] Right? Because, you know, in one game, a .6 chance of 100 and a .4 chance of minus 100, that gives you an expectation of 20. 60 minus 40 is 20.


[User] Please transcribe this text

[Assistant] 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, uh, we know that the expected winning for the Yankees is $42.


[User] Please transcribe this text

[Assistant] However, the Yankees, you know, you could lose $100. If the Phillies just got lucky, you could lose as well as win $100. You don't want to face that risk. So what could you do? Well, the simplest thing to do


[User] Please transcribe this text

[Assistant] Is to go to another bookie and say, I want a bet on the World Series with this other bookie. And if that, and so, if the bookie is willing to make a bet with you for a hunt, you know, for the whole series.


[User] Please transcribe this text

[Assistant] Okay, so the bookie has to be pretty sophisticated and do the same calculation. The bookie is going to give you, you know, let you bet however you want to at


[User] Please transcribe this text

[Assistant] I forgot to say that the series, we figured this out before. Oh dear, what were the odds?


[User] Please transcribe this text

[Assistant] Well, we know some probability p times 100 plus some probability 1 minus p times minus 100 gives you 42. So p times 100 plus 1 minus p times minus 100 equals 42. Approximately, that's 42, right? So we know that, uh, therefore 2p times 100. Okay, 2p times 100.


[User] Please transcribe this text

[Assistant] Okay, so you know, so this other book you would have calculated, the odds are 71%.


[User] Please transcribe this text

[Assistant] That the Yankees are going to win the World Series. And we calculated this a few classes ago. The Yankees have a 71% chance of winning the World Series.


[User] Please transcribe this text

[Assistant] Okay, so if you went to another bookie who could do this calculation the other bookie would be willing to


[User] Please transcribe this text

[Assistant] To give you odds of 71% Yankees versus 39%, uhm, Phillies. So you could just unload this whole bet and get $42 for sure. In fact, what would you bet with the other guy?


[User] Please transcribe this text

[Assistant] What would you do if you could bet on the whole world series with the other...


[User] Please transcribe this text

[Assistant] Bookie, what bet would you make? You know that the bet is giving you $42 on expectation. The trouble is, some of the time it's giving you $100, some of the time it's giving you minus $100. You don't want to fix this risk. You want to end up with $42 for sure. Now how can you do that?


[User] Please transcribe this text

[Assistant] Well, if you did minus 68 here, that would equal 42. Okay, and if you did plus


[User] Please transcribe this text

[Assistant] Uh, 142 here, that would also equal 42.


[User] Please transcribe this text

[Assistant] And you lose 68 if the, uh, Yankees win.


[User] Please transcribe this text

[Assistant] What is p times 50, what is p times 42 plus 1 minus p times 42? It's 42. What's p times 100 plus 1 minus p times minus 100 is also 42. So therefore, what's p times this plus 1 minus p times that?


[User] Please transcribe this text

[Assistant] Zero. Okay, so at the odds p and 1 minus p, this bet is perfectly fair.


[User] Please transcribe this text

[Assistant] So the other book you would say, that's the odds, 71% to 39%, that's exactly what these odds are.


[User] Please transcribe this text

[Assistant] So if, if p times this plus 1 minus p times that is 42, and p times this plus 1 minus p times that, that's obviously 42, must be p times this plus 1 minus p times that is 0. In other words, at the odds, p and 1 minus p, this is a fair bet. So the other bookie ought to be able to, would give you those odds. Okay, so that's what hedging is.


[User] Please transcribe this text

[Assistant] In its simplest form, we haven't had to do anything dynamic. The guy, some guy's willing to bet on the World Series with you,


[User] Please transcribe this text

[Assistant] So you know that he's done something wrong in his calculations. He's got the wrong P. So you want, okay, 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.


[User] Please transcribe this text

[Assistant] You know, which might make the Phillies win. And so you don't want to subject yourself to that risk. So what do you do? You do


[User] Please transcribe this text

[Assistant] You take the bat, the advantageous bat, but you put together with the advantageous bat,


[User] Please transcribe this text

[Assistant] A hedging bet, where you're betting on the wrong team. The team you think is gonna lose. Okay, 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, you know, P we just said was .71.


[User] Please transcribe this text

[Assistant] 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.


[User] Please transcribe this text

[Assistant] Uh, at P with you. Okay, and this one outsider doesn't understand that all the bookies are willing to take odds of P. He's offering odds of a half, which are crazy. It's a big gap, 71, 29 to P. So you can take 42 of his dollars for sure.


[User] Please transcribe this text

[Assistant] Okay, so this is hedging. It's also an arbitrage. This is a hedging created a pure arbitrage.


[User] Please transcribe this text

[Assistant] Because now that you've, in this case it's done even better than, uh, in my example with Ford because you've taken now a bet and transformed it where an expectation you're gonna win into a bet where you can't possibly lose.


[User] Please transcribe this text

[Assistant] Okay, now suppose that these other brokers weren't as clever as you.


[User] Please transcribe this text

[Assistant] Suppose that the other broker, okay, so that's hedging. Now we're gonna 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% that the Yankees are gonna win any game against the Phillies. Now what would you do?


[User] Please transcribe this text

[Assistant] Okay, you can't go to another broker and say, I wanna bet on the whole seven game series. The guy's gonna say, it's just too complicated for me, I'm a simple man, I make a simple living, I just do one game bet.


[User] Please transcribe this text

[Assistant] Tell me, you know, I'll let you bet any game you want, you can, you know, you tell me what you want to bet, we'll bet one game at a time, 60-40 odds. The whole series, you know, it's much more complicated for me. So what could you, is there anything you can do now? What can you do now?


[User] Please transcribe this text

[Assistant] So now you have to do something more complicated, which is called dynamic hedging.


[User] Please transcribe this text

[Assistant] 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?


[User] Please transcribe this text

[Assistant] Well, let's look at this picture here.


[User] Please transcribe this text

[Assistant] You know that this bet of plus 100 minus 100 by doing backward induction is worth 42. So if you were a trader, a hedge fund manager, you would be marking your position


[User] Please transcribe this text

[Assistant] You're expecting now, you've just made a trade which you know on average is going to make you $42. Of course, if you had bad luck, you could end up losing $100.


[User] Please transcribe this text

[Assistant] You know, which means you're gonna go out of business, you're gonna be fired, your name's gonna be in the newspaper, you're gonna be probably sued by your investors. Okay, or, or you can, uhm,


[User] Please transcribe this text

[Assistant] Now you don't wanna face that, you wanna hedge yourself, protect yourself, you're gonna make the money for sure. Now some people don't hedge, they say, you know, when you hedge, you're betting against yourself, and you know, something bad can happen. The hedge might go negative, right? I mean, it may be that the Yankees win, just as you think.


[User] Please transcribe this text

[Assistant] 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.


[User] Please transcribe this text

[Assistant] Very simple minded. So there are a lot of people who still don't hedge. But anyway, there's a revolution in, in financial thought in the last 30 years. Anticipated by Ford, but didn't really get, people didn't really catch on until the 70s.


[User] Please transcribe this text

[Assistant] Uh, that you should hedge. So how can you hedge this if you can only bet one time?


[User] Please transcribe this text

[Assistant] One game at a time. That was the question we dealt with last time. What should you do?


[User] Please transcribe this text

[Assistant] You should be marking your position, you know, correctly, that it's worth $42. Now if you win the first game, you know, 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. Okay, if you lose, your position will be worth $8. Now, what is


[User] Please transcribe this text

[Assistant] Okay, so you're at 42, and you could go with 60% probability to 60, well I can't remember the number, 64, or down to .9, 40%, sorry, I'm getting senile, 9, with 40% probability, so,


[User] Please transcribe this text

[Assistant] And the two add up to 42. Okay, so that's why, okay, so to say that the value is worth 42, how did you get the 42? We were doing backward induction. We took 60% of that plus


[User] Please transcribe this text

[Assistant] 40% of that, we got 42. So if you're marking yourself to market after the first game of the series, you'll have to say to the world, admit to the world that you're either now worth $64, or you've gotten crushed and you're down to $8. That already might get you fired. You know, going from 42 to 8, that's a pretty drastic loss.


[User] Please transcribe this text

[Assistant] So, even the first day, you'd be, if you have to market to market, you're gonna be revealing to the world that you've screwed up, and now your position's only 8. You've told all your investors, you've done this brilliant thing, they're up 42 dollars. The next day, you're gonna have to tell them, well, I'm sorry, you only have 8 dollars. They're gonna be very upset. So, what would you do?


[User] Please transcribe this text

[Assistant] So here, this is minus 9, no, this is 9, right, so you go plus 33 minus 22.


[User] Please transcribe this text

[Assistant] Okay, so that's equal to 42, and that's also equal to 42.


[User] Please transcribe this text

[Assistant] So how, can you do this at a fair, is it fair odds to do this?


[User] Please transcribe this text

[Assistant] Well yes, it has to be fair odds to do this, because 60% of 64 plus 40% of 9 is indeed 42. 60% of 42 plus 40% of 42 obviously is 42. So therefore the difference, 60% of this plus 40% of that must be zero. So in other words, it's fair odds. And sure enough,


[User] Please transcribe this text

[Assistant] 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.


[User] Please transcribe this text

[Assistant] So again, you, you think the Yankees are gonna win, you go to the other, and you make a bet with the naive Philly fan, you're betting the Yankees are gonna win, you go to your broker friend, and you bet against your Yankees.


[User] Please transcribe this text

[Assistant] You know, you're, you're putting down 22, you're gonna 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 at the, against the other guy and therefore you still locked in 42 dollars.


[User] Please transcribe this text

[Assistant] Okay, so now, we don't have to go any further. You would only be confusing to go further, and I usually go further and then, so it would just be complicated. If you can do something once, you can do it many times.


[User] Please transcribe this text

[Assistant] So what could you do going from here to here? Well the average of here and here is 8.


[User] Please transcribe this text

[Assistant] Okay, but of course you've co- is nine, about nine. You've collected thirty-three dollars, don't forget, from your...


[User] Please transcribe this text

[Assistant] You're hedging bet, so although this says 9, you've got the 33 in the background, so it's 42. Okay, but it's 9. So what you want to do, this is 9, you just want to transform this and this until 9.


[User] Please transcribe this text

[Assistant] Once you've transformed this and this into 9, okay, which you can do, right, because 60% of this plus 40% of that is 9.


[User] Please transcribe this text

[Assistant] So the other broker is gonna, you can do an offloading hedging bet with another broker at 60-40 odds.


[User] Please transcribe this text

[Assistant] To get 9 for sure here. Why is that? Because 60% of this plus 40% of that is 9. Okay, 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 in 0 according to 60-40 odds.


[User] Please transcribe this text

[Assistant] 44, 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. Okay, yeaah.


[User] Please transcribe this text

[Assistant] Yeah, so how do you make money in, uh, hedging? You effectively were betting


[User] Please transcribe this text

[Assistant] Against the Philly fan at 50% odds and you've got your broker who's willing to give you 60% odds and you're exploiting that gap. Okay, so let's, uh, let's do a


[User] Please transcribe this text

[Assistant] Another, so okay, so it's an excellent question you're asking, so I haven't answered his question yet. I want to give another example to answer his question. So, what, what is really going on with the hedging? I mean, who, what am I relying on? I mean, how many brokers do I have to deal with to hedge? What's going on here? So I've given an example of a world series where it's very simple, uh, to figure out what to do. So now I want to give an example of a


[User] Please transcribe this text

[Assistant] Done this in advance. Okay, so what is the...


[User] Please transcribe this text

[Assistant] Okay, so here we have a year, we have a 30 year bond, 8% 30 year bond, okay, and it's an 8% 30 year bond.


[User] Please transcribe this text

[Assistant] Okay, so we know what it's payment is, and now if the guy, if the, if the homeowner, homeowners are all perfectly, uh, perfectly, you know, uh, not paying attention, the bond is gonna be worth some amount of money.


[User] Please transcribe this text

[Assistant] Okay, so here are the interest rates, oh, I forgot to say. So we have to assume what the risk is. So we know, so in fact, what does this mean? We know that the risk is, uh, we know that the, that the, uh, risk is, uh,


[User] Please transcribe this text

[Assistant] The, the, the interest rate starts at 6% and it has a 20% volatility. Okay, so what does that tell us?


[User] Please transcribe this text

[Assistant] It tells us that the interest rates could go up or down by this amount.


[User] Please transcribe this text

[Assistant] Now, we've assumed that the odds that it can go up or down are 50-50.


[User] Please transcribe this text

[Assistant] Okay, 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.


[User] Please transcribe this text

[Assistant] And so now, I'm, I'm going to make, okay, we've assumed that the odds are 50-50. Let's assume that everybody, uh, is willing to trade you 50-50 odds. Okay, so, what is the, what does that mean?


[User] Please transcribe this text

[Assistant] This is like your other broker. Okay, so the interest rate is here, r0. It can go up to r0 times up.


[User] Please transcribe this text

[Assistant] Or it can go down to r0 times some down.


[User] Please transcribe this text

[Assistant] Okay, and up, remember, is, you know, up, by the way. This equals r, 0, e.


[User] Please transcribe this text

[Assistant] To the volatility times e to the drift and down.


[User] Please transcribe this text

[Assistant] Up, or it can go down. So u is bigger than 1, d is less than 1, probably. And so we see examples of that.


[User] Please transcribe this text

[Assistant] Now I'm making an assumption here that the odds are a half and a half and that everybody


[User] Please transcribe this text

[Assistant] Alright, so there are a bunch of ways that you can trade in the interest rate market to get a dollar if the interest rate goes up to this number in exchange for losing a dollar if it goes down to this number.


[User] Please transcribe this text

[Assistant] Okay, and these things are called swaps and they're called interest rate derivatives and they're called a whole bunch of other things which is going to take too long to explain. Okay, but so let's suppose that you can go out into the interest rate derivative market and simply trade a security that will pay a dollar if things go up.


[User] Please transcribe this text

[Assistant] You know, or you can buy another security that will pay a 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.


[User] Please transcribe this text

[Assistant] So what would somebody, what would be the cost if everyone agrees the odds are 50% of getting out the peer? What is the price today going to be of getting a dollar if the interest rate goes up?


[User] Please transcribe this text

[Assistant] Okay, I didn't expect anyone to say that so fast, so right. So the price here we could write is .5 divided by 1 plus r zero. And this price


[User] Please transcribe this text

[Assistant] Going down is going to be .5 divided by 1 plus r0. Okay, now how did he figure that out?


[User] Please transcribe this text

[Assistant] Well the odds are 50% of going up here and 50% of going down here, but don't forget you're getting the money later. And this is a one period interest rate, 1 plus r is 0. So therefore, uh, what you'd be willing to pay here is 50% of the dollar that you'll get discounted by 1 plus r is 0. What you'd pay here for that is 50%


[User] Please transcribe this text

[Assistant] Are like the World Series game by game odds.


[User] Please transcribe this text

[Assistant] They're gonna 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 gonna be .5 divided by 1 plus ru or something, which is a bigger number. Okay, so the price


[User] Please transcribe this text

[Assistant] Of betting on a dollar, you know, what you have to pay to get a dollar for interest rates to go up is gonna be, uhm, more here.


[User] Please transcribe this text

[Assistant] Then it is here, say, because you're gonna still have 50% chance of getting here, but you're discounting by a higher interest rate. So, okay, so this is 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 gonna go.


[User] Please transcribe this text

[Assistant] Okay, so what have we got now? We've got this mortgage that's sitting there.


[User] Please transcribe this text

[Assistant] Is owned by a very rational guy.


[User] Please transcribe this text

[Assistant] You know the mortgage is owned by a very, you know, the homeowner is not the owner of the mortgage. The homeowners are very rational people. The market is very foolish.


[User] Please transcribe this text

[Assistant] This is an exaggeration. The market might think, oh, the homeowner, you know, he's got this, the homeowner never does anything, the market thinks, so they think the bond is worth 120. But you're much,


[User] Please transcribe this text

[Assistant] You, you know, you studied prepayments much more than the market does. You realize that the homeowners aren't gonna just sit there, you know, dumbly paying whatever they owe. If the interest rates go down, they're gonna prepay. So in fact, the mortgage is only worth 98. Okay, so the market thinks it's worth 100.


[User] Please transcribe this text

[Assistant] 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 pre-payments that the market doesn't know. And so you know the bond is really worth 98 instead of 120. Okay, so what, what should you do? Now let's suppose that the market is actually treating


[User] Please transcribe this text

[Assistant] Okay, it's not actually a bond, okay, because it doesn't have a principle at the end. So the market, so I'm going to have to do a different sheet here to get it together. The market thinks mortgage never prepays.


[User] Please transcribe this text

[Assistant] Now, let's suppose that what you can trade in the market is you can trade these interest rate derivatives. So what should you, what should you do?


[User] Please transcribe this text

[Assistant] What should you do now? You can trade the mortgage, you can buy it, you can sell it short, you can, uh, buy or sell these interest rate derivatives. What would you do?


[User] Please transcribe this text

[Assistant] Since you know there's a mistake being made, you're gonna, you're gonna use all of it. So what would you do?


[User] Please transcribe this text

[Assistant] All you're allowed to sell, I've assumed, you know, a little artificially, all you can do is sell one unit of the mortgage short. So you're gonna sell it for $120.


[User] Please transcribe this text

[Assistant] What does it mean to sell it short? It means you have to make the payments that the mortgage is making.


[User] Please transcribe this text

[Assistant] Okay, however, you could get completely crushed if things, uh, don't turn out exactly the way you thought. So you want to lock in a constant profit for sure.


[User] Please transcribe this text

[Assistant] 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?


[User] Please transcribe this text

[Assistant] Okay, so let's be more precise. Uh, that's exactly right, so let's just see what that really means.


[User] Please transcribe this text

[Assistant] Okay, so I'm going to copy these numbers down, which I can't see yet.


[User] Please transcribe this text

[Assistant] You have a mortgage that's 98, you know it's worth 98 something, .8.


[User] Please transcribe this text

[Assistant] Okay, now, on the other hand, the market thinks that this mortgage


[User] Please transcribe this text

[Assistant] You're selling at the same time something that's worth 120.


[User] Please transcribe this text

[Assistant] You're selling it for 120, okay? It doesn't even matter what, what, 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?


[User] Please transcribe this text

[Assistant] If you were marking yourself to market, what would you do? What would you mark yourself to market here?


[User] Please transcribe this text

[Assistant] Well, you've gotten $122, so you've marked yourself to market. Here you'd say you're $21.16 up. Here you'd have to say, well, you know, you, uh,


[User] Please transcribe this text

[Assistant] You're not up so much money anymore because you've just lost some of the value of your mortgage. You've lost, you know, six bucks. You have to say you're only up fifteen dollars now. And here you'd have to say, well you made a little bit of money, so you're up, you know, twenty-two dollars now.


[User] Please transcribe this text

[Assistant] This profit, 21.16 for sure.


[User] Please transcribe this text

[Assistant] So what should you do? If you're marking yourself to market, you want to turn this money into


[User] Please transcribe this text

[Assistant] Okay, because then, having locked in 21.16 times 1 plus 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.


[User] Please transcribe this text

[Assistant] You know, you wanna ensure that that's where you're marking yourself the position. You wanna know by the end of the next year, you could turn this into 21.7, you know, discount it up.


[User] Please transcribe this text

[Assistant] Okay, and then keep a discount until all the very end. So whether you take your money now or you take your money later, it's always a sure thing with exactly the same present value.


[User] Please transcribe this text

[Assistant] Okay, 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?


[User] Please transcribe this text

[Assistant] Well you know the value of this plus this at this interest rate in the mortgage derivative market, it's worth 98.84.


[User] Please transcribe this text

[Assistant] Okay, 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 91.84.


[User] Please transcribe this text

[Assistant] Oh, there's also a payment here. Don't forget the payment. So sorry, we've forgotten the payment. So this is, the mortgage is much subtler because you've actually gotten, uhm, eight dollars here, plus eight, and this is plus eight.


[User] Please transcribe this text

[Assistant] Okay, so we forgot about this. The present, so you got an $8 coupon plus what was left was 92.6. This is 99.1 plus 8. So actually the total value here is 100.6 and the total value here is 107.11. Okay, and you discount this by this, you know, 6% interest and you get 98.84.


[User] Please transcribe this text

[Assistant] Okay, so I left out a crucial step. So remember, the cash flows here are the coupon plus the value of the mortgage that's left. The value of the mortgage that's left is 92.65.


[User] Please transcribe this text

[Assistant] Okay, plus the 8 dollar coupon. So over here you're gonna have a value of 100.6. Down here you're gonna have a value of 107.11.


[User] Please transcribe this text

[Assistant] Okay, so how can you, you want that to equal, locked in forever, you want that to, so you know that you've, you've just sold this mortgage for, uhm, a hundred, uh, what do you call it, a hundred twenties. You've got 21.7 dollars sitting here.


[User] Please transcribe this text

[Assistant] Okay, so now what can you do with this 21.7 dollars? You can...


[User] Please transcribe this text

[Assistant] Cancel your cash flows, so what you owe on the mortgage is always exactly the same thing, 98.84. If you want to make this 98.84, so you have to subtract off something from here, 100.6, you have to subtract, so you want this to be equal to, so let's do it this way, okay, let's make this 98.84.


[User] Please transcribe this text

[Assistant] Okay, so let's turn this thing and this thing. The present value of this and this is 98.84.


[User] Please transcribe this text

[Assistant] Therefore you know there must be some fair trade in the derivatives market that turns this thing and this thing into 98.04 times 1 plus r0 and this into the same thing, 98.84 times 1 plus r0. How do you know you can do that?


[User] Please transcribe this text

[Assistant] Because the present value of 98.84 times 1 plus r0 and 98.84 times 1 plus r0, a half of that


[User] Please transcribe this text

[Assistant] What's the half of that? Divided by 1 plus r zero is 98.84.


[User] Please transcribe this text

[Assistant] 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, uh, you sell some of these


[User] Please transcribe this text

[Assistant] That means the present value of what you have to deliver in cash, the present, the value of what you'd have to deliver in cash is 98.84. Okay, still 21.7 dollars in present value terms less than what you've received.


[User] Please transcribe this text

[Assistant] Okay, so what have you done with this $120? You've invested that at the same interest rate, and so now you've got a profit. You owe 98.84


[User] Please transcribe this text

[Assistant] 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, uhm,


[User] Please transcribe this text

[Assistant] You have to think of yourself as marking to market. And 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.


[User] Please transcribe this text

[Assistant] But you want to make sure you can maintain the same profit forever. Okay, or to put it another way, if you bought this bond and got those, you want to make sure you can lock in your profit forever. Okay, so how would you do this? Well, uhm, there are different ways of doing it. I was saying, suppose you, you, uh,


[User] Please transcribe this text

[Assistant] So let's say you want this $21.7 for sure at the beginning, and you don't want to ever have to make any payments.


[User] Please transcribe this text

[Assistant] On, you know, in the end you want everything to cancel out. What would you do?


[User] Please transcribe this text

[Assistant] How would you do this? Well, this number tells you you owe in present value terms 98.84 dollars.


[User] Please transcribe this text

[Assistant] Okay, so, uh, you've gotten 120 dollars. So with this, you could take the 120 dollars, put 21.7 in your pocket, and now, with this 98 dollars, you can buy, uh, these cash flows. What can you buy? You can invest


[User] Please transcribe this text

[Assistant] You can invest money so that you, uh,


[User] Please transcribe this text

[Assistant] So this bond, the actual payments the bond is gonna make, you can buy all those payments for 98.84 dollars. Okay, I guess that's the simplest way of saying it.


[User] Please transcribe this text

[Assistant] 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 of the tree, right?


[User] Please transcribe this text

[Assistant] 8 dollars here, it pays 8 dollars here, somewhere low in the tree, it's gonna just prepay for sure and pay the whole remaining balance, you know, down here, it's not paying any, it's just paid off


[User] Please transcribe this text

[Assistant] For sure. Okay, so it's prepaid. So what is the, uh,


[User] Please transcribe this text

[Assistant] The fact that all these payments together are worth 98.84


[User] Please transcribe this text

[Assistant] Means that by taking 98.84 dollars and trading on the derivatives market, you can buy all those payments for exactly 98.84.


[User] Please transcribe this text

[Assistant] So therefore as you go futu- farther into the tree, you always will have the payments to make. You've sold the bond short. What does that mean? Anytime this bond makes a, a payment, 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, because, you know, by, by backward induction.


[User] Please transcribe this text

[Assistant] The payments are $8 here plus $92 of remaining payments.


[User] Please transcribe this text

[Assistant] So there's a way in the interest rate derivatives market of buying 8 plus 92.6, that's 100.6 dollars here, and buying 107.1.


[User] Please transcribe this text

[Assistant] By this many, uh, interest rate derivatives, you can get $100 of payments here and $107 of payments here. You know, half of 100 plus half of 107.


[User] Please transcribe this text

[Assistant] Discounted at 6% interest gives you 98.84. Okay, so it's possible on the market to buy $100 up here and $107 down here which is 98.84. Now out of those $100 up here you've bought, give 8 of them


[User] Please transcribe this text

[Assistant] To the guy you sold the bond short to. That's the coupon payment. Down here, give eight of these dollars to the same guy you sold the coupon short to. Now that leaves you with 92 dollars.


[User] Please transcribe this text

[Assistant] 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


[User] Please transcribe this text

[Assistant] Up here is that you're gonna owe $8 to the guy here, and you're gonna owe $8 to the guy here, but what's left will be worth $83 or $97. So therefore with this $92, you in fact are able to buy this


[User] Please transcribe this text

[Assistant] 92 plus 105 here because, you know, the present value of 92, that's 91, you know, almost 92 plus 105 here is exactly 92. That's how we got that number. It's a half


[User] Please transcribe this text

[Assistant] Times this number plus the coupon plus a half times this number plus the coupon discounted at the interest rate corresponding to here. That is 92 dollars.


[User] Please transcribe this text

[Assistant] So therefore, by investing here in the interest rate market, you can buy that cash flow here, the 84 plus 8, 92 dollars. Okay, 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.


[User] Please transcribe this text

[Assistant] You can buy all the payments you need to make here. The $8 coupon, plus you'll still have $97 left.


[User] Please transcribe this text

[Assistant] 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 98.4 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.


[User] Please transcribe this text

[Assistant] of what's going on. The essence of what's going on


[User] Please transcribe this text

[Assistant] You can find someone else who will always trade at fair odds with you. Okay, fair odds means in the interest rate market.


[User] Please transcribe this text

[Assistant] They don't know these people trading the interest rate market. They don't understand prepayments. Okay? It's not like the bookie here. They don't understand prepayments. They're not willing to bet with you on what prepayments are gonna be. They're just gonna bet with you on what interest rate is gonna turn up.


[User] Please transcribe this text

[Assistant] But you see, the cash flows from the bond depended on prepayments and depended on the interest rate.


[User] Please transcribe this text

[Assistant] 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 prepayment,


[User] Please transcribe this text

[Assistant] Okay, you're going to know what this bond is going to do in the future. So remember what the prepayment, you know, and so what is, what are those prepayments? You know, this stuff.


[User] Please transcribe this text

[Assistant] So down here, where the ones are, the guy's prepaid for sure. So you'd have to deliver to the market the entire remaining balance at that point.


[User] Please transcribe this text

[Assistant] Taking the present value, what calculations were you doing? In the present value you're doing calculations


[User] Please transcribe this text

[Assistant] The interest rate derivative guy is willing to do that. He doesn't know what the prepayments are gonna 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.


[User] Please transcribe this text

[Assistant] 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.


[User] Please transcribe this text

[Assistant] And therefore, with 98.84 dollars, you can replicate what the bond, 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.


[User] Please transcribe this text

[Assistant] And how do you know that you can do that? Because it's just one step repeated over and over again. At every step,


[User] Please transcribe this text

[Assistant] There's going to be a payment of say $8 and a present value of what's left.


[User] Please transcribe this text

[Assistant] And there'll be a payment down here of 8 dollars in the present value of what's left. Or maybe this thing is just a pre-payment. In which case, it's a single payment of 90...


[User] Please transcribe this text

[Assistant] You know, 99.5, that's the remaining balance after one period.


[User] Please transcribe this text

[Assistant] So, but when you figure out this number, it's always taken by averaging this number discounted with this number discounted. Okay? So, therefore,


[User] Please transcribe this text

[Assistant] 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 cash pa- 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 a half of all that


[User] Please transcribe this text

[Assistant] 92.6 plus 8, use the 8 to pay off the coupon, you still have 92.6 dollars in your pocket here. You're gonna use that to buy the future payment of the mortgage, some of which will be a coupon, and maybe this will be another coupon, okay, plus the present value of what's left.


[User] Please transcribe this text

[Assistant] 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. Okay, so that's the, that's it.


[User] Please transcribe this text

[Assistant] Any questions? I don't know if that was too clear. Will somebody ask any questions? You are great at asking questions. Is there? Is this followable or? Okay, well that's, that's too good. Uhm, too great.


[User] Please transcribe this text

[Assistant] Okay, so let's see if I can do another example.


[User] Please transcribe this text

[Assistant] In the interest rate derivative market directly.


[User] Please transcribe this text

[Assistant] But you can trade in the bond market.


[User] Please transcribe this text

[Assistant] So let's get more realistic now. Let's suppose that there's a bond market.


[User] Please transcribe this text

[Assistant] Here we had, okay, so it was a 30 year bond, 8%, the starting interest rate was 6 and the volatility was 20.


[User] Please transcribe this text

[Assistant] So we had that, that, uh, sheet that was bond market trading. So let's just go to that. File was open.


[User] Please transcribe this text

[Assistant] Okay, so here we are in the bond market. So let's say there's a 9% coupon bond, doesn't matter what it is. Uh, we start at the interest rate at .06. Was that where we were starting? And the volatility was 20.


[User] Please transcribe this text

[Assistant] Okay, so here we have the same exact, uhm,


[User] Please transcribe this text

[Assistant] Got exactly the same interest rate process as before. So those are the interest rates just as we had before.


[User] Please transcribe this text

[Assistant] Okay, and now let's talk about a bond. So it's a non-callable bond. I forgot already now what I said. How long was this bond? It was a 30 year bond.


[User] Please transcribe this text

[Assistant] Didn't have to be 30 years, but it was a 30 year 9% bond.


[User] Please transcribe this text

[Assistant] So the bond is worth 140.93. Interest rates go up.


[User] Please transcribe this text

[Assistant] So now, let's suppose you can't trade in the interest rate market anymore.


[User] Please transcribe this text

[Assistant] Well, you've got the interest rate. You've got the one month bond that you can trade at an interest rate of, uh, 6%. And you've got this bond market. So anyone will buy and sell with you. This, this is the, remember this is the 30 year


[User] Please transcribe this text

[Assistant] 30 year, 9% coupon bond. They're all willing to buy and sell this with you.


[User] Please transcribe this text

[Assistant] And you've got this mortgage that everybody else thinks that the price is worth $120, but you're sure it's worth $98.84 because you know the prepayment, the homeowners are smart and they're going to prepay.


[User] Please transcribe this text

[Assistant] And no one else realizes they're gonna do it. So what should you do now?


[User] Please transcribe this text

[Assistant] You can't trade interest rate derivatives anymore. All you can do is trade in the bond market. Do you have any idea what you could, uh, do?


[User] Please transcribe this text

[Assistant] 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.


[User] Please transcribe this text

[Assistant] And of course you can trade a very short bond, the one, one year bond, which is trading at 6% interest. You can trade a one year treasury, and here's the 30 year 9% treasury. Okay, which happens to be worth 140. So what did you do?


[User] Please transcribe this text

[Assistant] And you think the value of the payment is only 98, so you'd love to do this. The trouble is, you know, the future values depend on a lot of uncertainty. Okay, so uhm,


[User] Please transcribe this text

[Assistant] Can anybody figure out what to do just intuitively? It seems like pretty complicated to figure out what to do.


[User] Please transcribe this text

[Assistant] 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 sold short, the mortgage payments you sold short, at every node. So you never have to give up any more money. So what should you do?


[User] Please transcribe this text

[Assistant] You want to acquire some assets with this 98.84 dollars that will put you in a position where you have a hundred point six dollars up here and a hundred point seven one


[User] Please transcribe this text

[Assistant] One dollars down there. So how can you do that?


[User] Please transcribe this text

[Assistant] Well, you have a combination of the short, you can put, you have this


[User] Please transcribe this text

[Assistant] 30 year bond, you could invest in. If you, if you spent $98 on the 30 year bond, you know, that's...


[User] Please transcribe this text

[Assistant] So that would give you 84 up here and 105 down there. Okay, so it wouldn't match what you really need. On the other hand, you could also put


[User] Please transcribe this text

[Assistant] You're 98 dollars in a one-year bond, which would give you 98.84 times


[User] Please transcribe this text

[Assistant] You know, 1.06 and 98.04 times 1.06. So you've got this choice.


[User] Please transcribe this text

[Assistant] Or you can put, you know, you can put $100 here and get $106 there. Right, that's what the one year...