Hey there. Good morning. Good Afternoon, potentially. Good evening everyone, and welcome to this Felix Life session, um, on Equity delta one derivatives. Let's say it's gonna be the fundamentals of it, right? So my name is Thomas Krause, and I have the honor to take you through, uh, today's session. And before we start with the content, let me just remind you on what's on the agenda today. We're gonna start with a quick refresher on The fair value of an equity Forward contract. That's Just because it's the most important building block of the equity derivative space. And of course, um, leading into all the other products that we Are gonna talk about. We will then discuss the Mechanics of equity in next futures more Specifically, and then how they can be Used to hedge equity portfolios. And then towards the end of the session, we're also going to have a High level introduction to equity swaps, although due to time constraints, this probably won't go into, Um, too much Detail, but that's it. Um, Let's get started with today's content. And as I said, we're gonna start with a Quick refresher On the fair value of an equity forward. IE where is this forward price come from? Where should The Fair forward price, um, rather really be Now, um, as any Other forward contract, for example, on FX or on commodities or interest rates. The Starting point for a fair value, uh, calculation of the forward price is something called the no arbitrage principle or the law of one price, or whichever name you want to, uh, slap onto this. The idea is the following. If there's one, well, no two or more ways To achieve The same risk exposure or basically the same, um, outcome. Then in efficient markets, of course, the price for those two ways should be identical because if these prices were not the same, then what, um, a rational human being would do Is obviously to buy the cheaper way, Sell the more expensive way. And as The outcome of those two ways is identical, there would be no market risk at the end of the, as a, in this combination of these two trades. But because we're selling them more expensive and buying cheaper away, there would be a certain amount of risk free profit. And that, of course, would be arbitraged away by market participants relatively quickly. Now, of course, we might know that this no arbitrage principle is based on a couple of assumptions that might not always hold in practice. So that's why I call it the fair forward price, not the forward price, simply because the actual forward market price might move away from, uh, those fair forward prices. It wouldn't do so under a normal circumstance, at least in a Too large fashion, because At some point there would be at least other arbitrage activity setting in. Anyway, with that in mind, let's break down how this concept then leads to the formula that you see here on this slide, right? And of course, you know, with the, the, um, assumption we're making here is that we have an individual, and that obviously for an developing and intuitive understanding of this formula is that we have some, um, you know, investor here that is looking to, um, lock in the price of the purchase of a share, and they want to have those shares 12 months from now. So they don't need immediate access or ownership of the shares. They don't want necessarily to own them. Uh, on a t plus one or t plus two basis, wherever you own the world, that will be the spot settlement in equity markets. Um, they are looking to lock in the price today because obviously the expectation is here that prices will go up in the future, but they're looking to actually take ownership of this share, uh, 12 months, um, from now. Whatever the reason is, it could, for example, be, and this is a very often quoted reason, that at that point in time where they want to make the investment, the cash isn't available. That's, of course not necessarily, um, the case in, in, in, in practice, but it's a nice way of introducing this example. So we have an investor that's looking to buy some shares, doesn't have the money available right now, has it in 12 months, therefore it's looking for a 12 months forward contract on this specific, um, stock. Now, of course, the easiest thing to do would be to call, call a market maker in this particular, uh, stock and ask for a 12 months forward price. I, that's the definition of a forward rate. You agree on the price today. Um, but the settlement of this transaction will happen at a predetermined point in the future, and that would be in our case here, let's say 12 month in the future. So 12 months forward price, that's alternative. What now there's another way in achieving the same outcome, and that is having the shares in 12 months time at a price, you know, already right here, right now. However, that's not a forward transaction. That's actually a series of, um, spot transactions, right? And so first transaction that the client or the investor could make here is buy the shares at spot, right? And, uh, let's say the spot price is a hundred dollars. Now, of course, that is rather straightforward, uh, assumption here that because the spot price is the really the only available price we have, assuming there is no forward price for a second here, that we know exactly where it is right now, we know we could buy this share here at a hundred dollars per share in the spot market. However, when I introduced this example, we made some, like the reasoning of, uh, forward trade was that this investor didn't have the money available. That has not changed. So what needs to happen, um, from the investor side is because if they enter into the spot transaction, then in one or two days from now, they will have to pay a hundred dollars. They need those a hundred dollars, right? So in a second step, uh, there would be, um, the borrowing, um, borrow a hundred dollars, uh, um, at the 12 months interest rate. And let's put this at, uh, 5%. And what this end results in is that we now have borrowed a hundred dollars or the investor has borrowed a hundred dollars. These a hundred dollars can be spent on buying the shares on the spot basis, and then in 12 months, this borrowed money will have to be repaid. And we're assuming for simplicity at this point that the stock doesn't include, um, or doesn't pay dividends. So basically what would happen over the next 12 months is that in 12 months, these a hundred dollars will have to be repaid. We're simplifying here a little bit and saying let's ignore day count conventions as well. So if we borrowed a hundred dollars for a 12 month period at an interest rate of 5%, then obviously this amount of money we would have to repay or the investor have to repay, uh, would've grown to $105. Basically what this means is they will have the shares in 12 months time at a price, basically, what, 12 months forward price of $805. So that then indicates if we're thinking about it now, what I said before, we have those two different ways. A, is buy the shares on a forward basis or basically replicating this forward transaction through, uh, transactions in the cash market. In both cases, the investor ends up owning this share in 12 months time, um, but obviously, um, excuse me, in 12 months time at prices that are known today. And that basically means we have two alternative ways to achieve exactly the same. Hence the law of one price says both prices should be identical, and that means the theoretical forward price should be 105 because let's assume for a second that the forward price that we got quoted from, uh, a market maker was not $105, but it was in fact $110. Then not only would we buy the shares in the, uh, second alternative, we laid out here, IE buy the share spot, then borrow the money, and then effectively pay 105 uh, dollars for them in 12 months time. We would actually do an arbitrage trade, right? We would buy as many shares as we can and the spot price of a hundred and then, uh, borrow the money so that we are effectively paying 105 as described, and then we would sell at the same time, those shares on a 12 months forward basis at 110, and that gives us a $5 arbitrage. If we would all do this in size, then of course the spot price will go up and the forward price will go down until this transaction, this arbitrage opportunity is got that, as I said, under some sort of like idealized assumptions, but you get the, uh, general idea. So what this means is that, um, you know, we can obviously calculate the fair forward price by taking the spot price and adjusting it by what we call the cost of carry and take this literal, those are the costs that an investor faces by taking a position and then carrying it over time. And the most obvious part is the one we have already, um, discussed, and that is the borrow cost. If you enter into a position and you don't have the cash, then you will normally have to borrow the money, and that generates interest, um, that you need to pay. However, right? Um, the, we have excluded dividends so far in our, uh, examples. In reality though, many stocks out there do pay, um, dividends on a regular basis. And so we're thinking about a 12 months period. I'm just drawing the timeline here. Now, um, you know, this is the transaction date. This is basically the spot date of this today, and then this is 12 months forward. Uh, the settlement date there is in many cases, our stocks will have not just one, but maybe several dividend payments during this time. And of course, there's then a difference between the 12 months forward buyer that will buy the shares on a forward basis because they won't receive the dividends. Whereas as a spot buyer, you will receive the dividends that are occurring at these different points in time. And so what needs to happen to make, again, both, um, contracts sort of economically, um, identical, is that there must be a dividend adjustment on the forward price when there are dividend payments. So if we go back to our example here, we buy the shares at the spot price a hundred dollars, we borrow the money at 5%, and then obviously what happens over the next 12 months, because we're owning this stock already, we will receive all dividend payments that are happening over the next 12 months period. And that is then something that is an advantage for us. This is kind of like the opposite of the borrowing costs and therefore dividends have to be subtracted. And that is then how we get to this formula that you see here, saying that the fair value of an equity forward contract is nothing else but the spot price plus spot price times whatever the interest rate is times stays over basis. That's the day count fraction, really just, um, breaking down the per annum interest rate here to the relevant period of time we're looking at. And then we're subtracting dividends. Now, this is where sometimes you see, uh, a slightly different formula because you know many, um, books and, and, and individuals prefer the use of dividend yields, uh, simply because then what we're doing is we're, um, describing the yield not as actual dollar payments, but as a percent of the current share price, and then we're having an interest rate expressed in percentage terms, and we have the dividends expressed in percentage terms. And so the formula can sort of, you know, be rewritten in a somewhat elegant, uh, way, however, you know, it, it economically or mathematically, it shouldn't really make a difference. If you calculate the dividends appropriately or you calculate the dividend yield appropriately, then both should lead to the exact same outcome. What I would say though is, uh, is that it's really important that if you go for dividend absolute amounts or the dividend yields to be careful which dividend payments should actually be included in the calculation. Because even if you use a dividend yield that doesn't assume that we're gonna have dividends, you know, small amounts of dividend paid on a daily basis or anything like that. There are discrete points in time where dividends are paid, right? So the question is, which dividend payments should be included in the, um, forward calculation? And if you are familiar with the dividend timeline, then you know that there's a difference between the point in time where dividend is announced when, uh, you know, the stock goes X div and then when the dividend is paid. And the general rule is that, um, all those dividend payments that have an x diff date, um, that falls into the forward period. So all x diff dates between this point in time and this point in time, they need to be considered, even if the dividend that has an x diff date here is maybe paid there, IE after the forward has settled, that should be considered in the, um, um, you know, in in, in the forward, uh, calculation. The next thing then, if you're using absolute dividend amounts rather than dividend yields, uh, what you need to be careful of is obviously that those dividend payments happen at different points in time, right? So we cannot just sum up, let's say this is 0.5, this is 0.5, et cetera. So every, um, you know, four times there's dividend payment of 50 cents. That doesn't mean that over the next 12 months period, we get a $2 amount of dividends. Yes, that would be the simple sum of the dividends, but of course, the dividend that we receive in let's say three months will be reinvested for nine months and return interest on interest, et cetera, et cetera. So we need to be, um, a little bit more accurate here and say, okay, what's the future value actually of all those dividends, IE if we compounded from the time they are paid to the end of the four debt, that would be strictly speaking the correct amount of dividends, um, to, with, um, to, to subtract, right? And so, um, basically this then gives us the often, uh, looked at formula here that calculates the fair value affords by adjusting the spot price for cost of carry. And those two components in case of equity are clearly borrow costs and dividends. So, concrete example here, and you recognize the numbers, right? We have a spot price of the asset being a hundred dollars, the 12 months interest rate was 5%. And now what we've done in, you know, in addition to the previous slide is also give you, um, a dividend payment. And here we're giving an absolute number so no yield, um, but as I said, the yield would be the same if we use 6% here as a, as a dividend yield, the fair forward price wouldn't change. But anyway, 12 months future value of dividends being $6. And then what we, uh, have as a result is a 12 months forward price of $99. How's that generated? A hundred as borrowed right? Now. We then go and buy the spot, um, share or the share in the spot market. We have to pay 5% interest. So that brings us to 105 in 12 months time. However, over the next 12 months, we're also receiving $6 in total in dividends. That means the effective price we're paying for the share to have at in 12 months, we'll be 99. And that's an important yet, you know, or simple but important, uh, calculation here because what it shows nicely is a couple of things, right? First of all, the fact, um, that, uh, forward price is higher or lower than spot is not driven by expectations with regards to where the price is gonna go, right? So here the shares look, you know, as if they are cheaper, um, if we buy them 12 months out rather than spot market because you can pay a hundred right now or you get them for 99, uh, with a 12 month settlement. But the reason as to why, um, you get them cheaper is not that the market believes the share price is gonna fall, it's because the dividend outweighs the borrow cost, right? The dividend yield is higher than the interest rate, if you wish, or the dividend amount that you receive is exceeding the interest you pay, uh, for funding, hence the forward, uh, purchase is cheaper. Um, the other thing, um, that we can start to see here now is obviously that the 12 months forward price is not only driven by the spot price of the asset. Now of course, if the spot price of, of the asset would not be a hundred, but it goes up to 110, it's easy to see that the forward price will follow, right? Simply because we won't get more dividends because that dividend of $6 per stock most likely is gonna remain unchanged, not goes up because the stock price has increased. Um, but, um, you know, what's, what will increase certainly is the amount of foreign costs, because if the stock is now trading at 110, then it's not only a hundred dollars we need to borrow, but it's $110 we need to borrow. And then 5% on that is obviously a larger, uh, amount of money. So there is clearly a link between the spot price and the forward price. But what's also true is that there's a link between the 12 months forward price and, um, and, uh, interest rates and, uh, dividends, right? So let's just kind of make a couple of, um, scenarios here to just show the impact. So let's say, you know, interest rates are not, um, 5%, but they are at 7%, right? So interest rates have increased, that seems rather unlikely, uh, from today's perspective, a 7% interest rate. But nonetheless, what would happen to the forward price? Well, the calculation itself remains the same. A hundred is borrowed, but now it's 7%, right? That means 107 needs to be repaid. We get $6 dividends. So the fair forward price is 101. So if there was now this unexpected overnight increase in interest rates of 5% to 7%, the uh, forward price would change quite significantly. Now, as I said, unrealistic scenario maybe, but it's, it's showing quite clearly that the forward price is sort of having some sort of exposure to the level of interest rates. Now, the other thing, um, to consider is the exposure to dividends. Right now, one thing that I think is known about dividends is that companies generally, or first of all, a dividend is not mandatory, right? So it's kind of discretionary payment by, uh, the, the issuing company. And, um, that, you know, is, is that's a legal situation. Now, practically, companies obviously try to keep their dividend payments rather stable, not to surprise investors, at least not in a negative way. And they also try to manage expectations there so that any potential change in dividend is usually somewhat announced, um, to a certain time in, in advance. So dividends are probably best described as relatively sticky in comparison to other, um, market prices. But, um, you know, let's just assume what would happen though. Um, if a company, for some dramatic reason, because there's some sort of a cash crunch going on now has to decide, um, that they cannot longer pay dividends. So yesterday this company we're looking at looked relatively or looked relatively healthy. Uh, we didn't have any reason to expect that the dividend payments here of $6 that we anticipated over the next 12 months, uh, are not gonna materialize. And now overnight there's been a, you know, a new piece of information that that was made available. There's, um, you know, maybe an accounting, um, irregularity or something and, and, and there's a lot less, uh, cash available, or there's been an accident and there's, uh, probably, um, you know, compensation to be paid, whatever it is. Now, the company now, uh, decides that they will no longer be able to pay dividends over the next 12 months, then this will go from six to zero. And what's that gonna do to the forward price? It's gonna bring the forward price back to what we calculated on the first slide, and that is gonna be 105. Again, a very extreme example, not necessarily, um, too often observed in reality, but I just wanted to showcase that there is in fact an, uh, exposure of the forward price to dividends and or to expected changes in dividends and also to ex uh, expect changes or changes in interest rates. And that's something just to be aware of. And that should conclude, um, our sort of recap on, on forward prices. And now what we want to do is just sort of look at the, um, futures market. And more specifically, we're gonna have a look at equity index futures. And the first question that needs to be addressed, and I guess, um, although probably very briefly, is what's the difference between forwards and futures? Now, this is something you probably would've heard about, and that is forwards are basically, um, contracts that are traded in the OTC market. That means bilateral agreements between the two involved counterparties to forward buy and the forward seller. And that means there is, uh, much greater flexibility because, you know, counterparties can almost freely negotiate the terms of, um, you know, the actual transaction of course within the legal framework, right? But that is clearly, um, an advantage and the most often quoted advantage of OTC products in general and forwards in particular. Now, the downside or the, the thing to be aware of then obviously, um, if you choose a forward contract, is just the fact that there's no central market price for this, uh, particular product. So if you agree on a specific forward with a counterparty, you wouldn't see this particular contract or the market value of this contract being displayed on an exchange or anything, um, where you can feed the data, you will need to have some sort of valuation model in place to calculate and do a mark to market valuation of those contracts. But as we've seen on the previous slide slide, the mass here is not particularly challenging. So that I don't think is a particular concern, um, when it comes to, um, forwards in general and networking in, in particular anyway. So let's go and have a look at futures contracts. How do they differ? Well, they're basically just exchange traded forward contracts, right? And exchange trade means that there are certain terms of the contract that are standardized because that's, you know, simply, uh, simply a requirement for anything to be traded on exchange. There need to be some fungibility of, of, of contracts. And that means we have usually standardization with regards to what exactly is the underlying we can trade in the futures market. What is the size of one contract? Uh, when is the forward, what the, the, the, the, the, uh, settlement date? Basically, when does the contract expire? Is it cash settlement? Is it physical settlement? These things are usually set in stone by the, or determined by the exchange, the issues to the contracts. Now, of course, they do this with, you know, the interest of the potential users in mind. So they don't want to create a, uh, a product that then is not gonna be used and has to be pulled from the offerings. They will obviously do some research as to what is, what people most likely, um, find beneficial in terms of contract terms. But once they have been determined, and it doesn't mean we cannot change them at all, but then we will have at least a limited number of contracts that are available, uh, to trade. And that's then obviously, uh, one of the, uh, most often referred to considerations of using futures, right? That this standardization means that the terms are, you know, sort of limited in terms of flexibility, right? So that you cannot agree on any expiry date on any sort of contract size, et cetera, et cetera. There's sim certain number of contracts that are available, and if you are looking for something else, then, uh, you know, this will not be a futures product. And that is of course, you know, something that one needs to be aware of. However, there's a lot of advantages of using futures as well. And here, one of the most important points, uh, is clearly the liquidity that is often, often not always, but often, uh, superior in, uh, the futures market. Um, and, uh, here, if you kind of look at equity index futures, for example, and we're gonna see, uh, some numbers later on that show, how much is, is, is sort of like in, in terms of, uh, risk is held in those contracts. These are enormous numbers. And you know, I think that's a direct consequence of the fact that we're using standardized products because, you know, instead of, I'd say, you know, uh, splitting the, the liquidity, the interest in trading a particular underlying up over an almost unlimited number of forward contracts, all the trading interest gets channeled into, you know, a relative small number of futures contracts. And that then generally increases the amount of buying and selling interest per contract quite significantly.

And that will then, uh, positively impact liquidity with all its advantages, right? IE you can buy and sell a larger number of contracts relatively quickly without distorting the market price. Uh, too much you can, you will expect to see for a higher liquidity product, relatively tight bit of a spread so that the spread you have to cross as a market taker isn't, uh, as wide, um, as it would be, for example, then in, in some forward markets. And, uh, and of course other, um, advantages as well. And, and in addition to that liquidity, what's also a, you know, worth mentioning as an advantage is that there's a central market price, right? At any minute of any day, uh, we know where the current price of the s and p 500 futures contract is, and if I bought it at a specific price and I now see the prices right there, it's relatively easy for me to calculate what the, um, performance of my, uh, position looks like. I don't need any valuation model. I can just sort of, uh, do simple math as we're gonna see in a minute, right? And then one last point, and I think, um, you know, this is something that we need to understand, to understand futures behavior as well, is that, um, you know, exchanges do provide some additional information. So from a transparency point of view, there's an argument maybe to be made for you using futures. And the information I'm talking about in this context are trading volume and open interest. And as I said, this is going to, uh, come back. So we're gonna see, um, the evolve or the open interest and, and volume of, of a contract later on. So I think it's important we understand the difference, and I think volume isn't really something that we need to explain a great deal that basically measures the trading activity of a particular contract on any given trading day. And so the way we're usually doing is we're starting the day with a volume of zero, and then we're counting the number of contracts that have changed hands during the day. Uh, and then obviously, um, that is an at the end of day or at the end of the trading day. The volume that we've gone up to is the daily volume for this particular, uh, contract. Open interest is somewhat more abstract. And I think, you know, when it's the first time you're looking at this, it's, it's, it's really good to think about it. It, uh, by the way of an example and how numbers are due. And so that's what we're gonna do at the bottom. But first let's start with, um, a quick definition because basically what open interest generally describes, and you see this here on the slide, is that it's a number of outstanding future contracts that are held by market participants, right? Um, and that number is given to us, um, usually on an end of day, um, basis. So you don't necessarily have real time updates, um, on this. And the idea is here that we're only including positions that have not been closed or settled yet. And what this, the relevance of this then is, is that this open interest is basically indicating how much risk is held by the market using those contracts. Now, let's go, as I said through an example, to see really how volume and open interest differ, and then how to interpret those numbers. And, um, the example that we wanna look at here is outlined at the, um, bottom of the slide. And, you know, the first thing we do is we are assuming that a new futures contract has been launched today. Uh, yesterday, this contract simply wasn't available. This was brought to life this morning, and therefore the volume is zero, and the open interest is zero because nobody could have changed that, uh, could have traded this contract already. Hence, volume and or interest that will boast be zero. And now let's say the first ever trade in this contract is trader B, selling a hundred contracts via the exchange to, um, trader a IE Trader A buys a hundred contracts. And this is an opening trade, right, uh, for both traders because they didn't have any open position, there's nothing they can close. So per definition, both trades are an opening trade trader. A opens a long position of 100. Trader B opens a short position of 100. So what does it mean? A hundred contracts have changed hands, that means volume is a hundred and now a hundred, uh, contracts are held by market participants, a hundred long, a hundred short. We just kind of decide here to, uh, take 100 rather than 200. Um, that would obviously overinflate those numbers a little bit, but you know, that's not really of great importance. So a hundred volume, a hundred open interest, that's the situation after the first ever trade in this particular contract. Now, uh, there's a second trade happening, um, when it's irrelevant, it's a second trade in this contract. And here what happens is Trader B buys back, um, 50 of their short position. So that is a partial close, right? So they sold a hundred, uh, contracts in trade one, and now they are probably looking to, you know, take some profits or to, you know, um, stop their position out partially, whatever the reason is. It's not really, uh, important. But Trader B is closing 50 of their shorts. Who is the counterpart in that trade? Of course, the exchange, but the exchange only matches orders. And therefore there's another, um, trader involved here, and that's trader C who is taking a short position of 50 contracts. And that is an opening trade because Trader C didn't have any position before. Now, what does the impact on volume and open interest volume goes up by 50 because an additional 50 contracts have changed hands open interest on the other hand remains unchanged. Why is that? Well, because what happened is really the trader C has replaced or partially replaced Trader B. Trader C at the moment is short 50 contracts. Trader B is short 50 contracts, trader A is long, a hundred contracts. So the open interest is net altogether a hundred still until the third trade happens. And this is when Trader C buys back their shorts. Um, and that's a closing, um, transaction. And this time the counterpart is, uh, trader A, which is also partially closing, uh, their loan. So Trader A sells 50 of the contracts they bought in step one and tells it to Trader C, who just closes that position all together, impact on volume and open interest. While another 50 contracts have changed, hands volume has gone up from 150 to 200, but the open interest has actually gone down simply because Trader A has closed part of their long position is now long 50, uh, contracts, and Trader B is still short 50 contracts. And that's the open interest of this, uh, contract. Why this becomes relevant, we're gonna see, uh, later on, but hopefully you can now see, okay, this open interest shows us how much risk is held. So who's long, who's short by how much, uh, well not who, but obviously, you know, um, how many long and how many shorts are they in the market? And the volume just shows us how much, uh, you know, how many contracts are, are changing hands, um, on a particular day. And that was of course an an interesting indicator as well, simply because it allows us to gauge, if I wanted to sell 10,000 contracts here in this new, uh, futures contract, uh, and it would trade on a volume of 200 a day, uh, then I can probably get some information, uh, that it will probably take me quite a long time to, uh, should sell, uh, 10,000 contracts in, in, in total. And, you know, might put a question mark on whether or not this is a good idea. Anyway. So, um, that's the volume versus open interest, and that's information that you get, uh, when you're trading futures. And yes, there are some sort of trading statistics from the OTC market from various, uh, sources, but they are not quite as granular as the stuff that the exchange provides us. So there's a argument there to be made, but let's go and have a look at a specific futures contract example that then allows us to really look at, okay, what are the standardizations? Um, and uh, also, you know, look at what is the actual size of, uh, some particular markets, an example of p and l calculation coming up, et cetera, et cetera. So what we're gonna look at is probably the most often looked at, um, you know, futures contract, um, in the world that see CME As and P 500 futures, more specifically the mini, uh, contract. And what that means is that we have a, uh, contract size here of 50, uh, dollars per contract point, I think, uh, or per index point. And I think the original, um, you know, s and p 500 futures contract had 250 or 500, you know, um, that just gives me now the exact numbers. But, you know, um, that's why it's a little bit, um, or nicknamed mini, um, simply because it's a smaller contract size. Anyway, what does it mean? And I think the product code here is not really, uh, something worth, uh, remembering because that might differ across, um, market data providers. But what does the contract size actually, what does that represent? Now, as we said, you know, when you exchange or when you trade exchange traded contracts, there is a certain element of standardization. And one thing that is standardized is the size of the contract. IE what do you trade? How much risk do you take by trading one particular futures contract? And the way this is done in equity index futures is that we assign a value, uh, to every index point, and that is $50 in this specific case. And then we go over to the right hand side of this slide, and that is an example from, um, you know, uh, November, 2023. Um, so the s and ps uh, nowadays trades obviously a little higher, but you know, at that point it was 4565.75, that was the, uh, index level. And, uh, if we now have the contract size being, uh, $50 per index point, then basically this times $50, um, then gives us, uh, the contract size or the contract value that you see here. That's 228,287 and 50 cents. So basically buying one futures contract at that last price here effectively means, or you get an economic exposure that's similar to buying or investing, $228,287 and 50 cents in the 500 companies of the s and p 500, uh, in the different, um, relevant weightings, right? But by just one transaction, rather than having to trade 500 different names and different ratings, et cetera. So arguably a much more efficient way of, uh, going low, um, the index. Now, then there's other things that are standardized on the, on the left hand side, there's a tick, for example, that's nothing else but the smallest possible price change. So basically futures prices can go up or down by a quarter of, in the index point. Nothing, um, smaller than that. So in theory, a futures price of 4,565 point set, well, you know, 0.80, that's technically impossible only, um, you know, potentially happening at settlement. But before that, it can't happen because future move in quarter of point increments. Um, the other thing is then there's a tick value that's $12 50, that's just a quarter of the, um, value per index point that we have been talking about in the, um, context of the contract size. So 50 divided by four is 1250, and then there's, um, you know, uh, this contract months. And what it says there is that there's contracts listed for 21 consecutive quarters in the cycle March, June, September, December. Now that this 21, that changes every now and then. So, um, take that with a pitch of sold, but basically what it tells us is that there's not just one futures contract. So right now we're trading June, future 2025, there's a September contract, there's a December contract for 2025, that's March 26, and so on and so forth. So there's several futures contract outstanding. However important to note that just because these futures are technically available for trading, they are listed and they exist in the system and databases, et cetera, doesn't mean they are necessarily actively traded. And you know how that works in equity index world. So we're gonna see later on. Then there's also another standardization because not only at which months of the year do contracts expire, you need to be a little bit more precise exactly which day do these contracts expire. And that is for equity index futures, uh, the third Friday of the month. So the June future, um, is, you know, that we're trading right now is the futures contract that expires on the third Friday in June. I don't know which date that is, but you know, let to look it up if you wanted to. And the last thing that's standardized is there's cash settlement simply because it's a bit difficult and complex to deliver one piece of index, right? Yes, of course you could deliver certain number of shares of 500 different companies and you know, the relevant way, but arguably that is, um, quite cumbersome. So cash settlement seems to be the much easier way of doing this. So then we have talked about the right hand side to some degree already, but what I wanted to highlight is the open interest of that contract in November, 2023 was almost 2.2 million, uh, contracts, which meant 2.2 million contracts long short across the market. And that if you think about one contract is 228,000, that brings us north of, uh, $2 trillion worth of risk in equity markets held via those contracts. And that gives you a good indicator that this is actually quite an important, uh, contract out there. And now, uh, just to sort of reiterate my point, um, that it's quite simple to calculate the, uh, p and l um, of a position held in those contracts. Let's have a look at this example here in the bottom. Uh, right. And there we should assume that we bought a hundred of these, um, December contracts at the last price, which was this one here, and then we are able to sell it 10 index points higher. We are ignoring transaction cost, brokerage fees, et cetera, et cetera. So basically, uh, we have made 10 index points and ignoring transaction costs. The question is what's our profit? And that is relatively straightforward, right? We said we made 10 index points profit, every index point is worth $50. That's what it says, then the contract size. And we do not just have one contract, we have a hundred contracts. So the p and l ignoring transaction costs will then be 50,000, uh, dollars. Simple as that. No valuation model required, just purchase price minus sales price, uh, times 10 times a hundred, uh, times 50. And then, uh, sorry, not times 10 because that was the different times, 50 times a hundred. And that gives you, uh, the p and l that the trader should see after having closed their position, or that's the mark to market of that position at that particular point in time. Now, um, of course, you know, you can use futures for taking, um, speculative long or short positions, but also a great deal of future trading activity probably comes from the hedging side, right? Um, and the example that we wanna look at here is that there's, um, uh, investor that has long positions in stocks, and this is a portfolio that contains mostly, if not exclusively, um, shares of the s and p 500 index. And let's say the investor generally is happy with the portfolio composition and wants to maintain the stock exposure in the long run, but feels there's a bit of uncertainty, maybe correction potential over the next couple of weeks, right? Um, and, uh, is therefore looking to patch the market risk that they are currently having in their portfolios. And of course, one way to do this would simply be to sell all the shares that they're currently holding, uh, and turn everything into cash and invest in some sort of t be, for example. And arguably there wouldn't be, uh, any risk, especially not on the equity market left, however, that, again, maybe just a little bit of a too cumbersome, uh, transaction depending on obviously the size and composition of this portfolio, how many shares are there, and how many, um, you know, shares are held by different companies, et cetera, et cetera. So if you just want to eliminate the market risk, then a hedge comes to mind that is using, for example, the index futures contracts, uh, to take an offsetting, um, position. So the idea here is we're taking a long, or we're having a long position in, in the cash, uh, market, and we can hedge that with a short position in future. So when we did that, it was the, um, you know, index level was 4565.75. If now the investor expected a correction of let's say 10%, um, and they have a hundred million dollars, um, portfolio as we've been given here as the information, uh, on the slide, then that would probably mean they would lose $10 million on their cash portfolio if they don't do anything. And the correction happens as they, um, fear. Now, how can we get around this? Well, we basically create an offsetting, uh, risk position. And that could be done, for example, by selling short equity index futures. IE sell them at the current high index level. And then if we were correct where the investor was correct and the correction happens and the index drops 10%, then we can buy our futures back at a lower price, right? And that means we have made, um, positive p and l in our futures position and that we can use to offset losses in our cash position. So let's just go through, um, some, some numbers here. 'cause the first question obviously is, if you were to do this, how many futures contracts do you have to sell? Well, there's one piece of information we have already talked about that becomes incredibly important here, and that is the contract size. IE earlier we said if the index point has value of 50 and the futures level is 4,565, then that means that one contract is like selling or buying, um, stocks in the s and p 500 index, uh, at a notional of in total 228,287, right? And so the calculation of the, um, hatch ratio, IE how many futures I should sell to be hatched could be done relatively straightforwardly. And that is, that is we have a hundred million, um, money at risk and each contract eliminates 228,000 of that. So divide a hundred million by 228,000, that gives us 438.04, and that's the number of futures contracts that we should sell. Now, of course, we're not gonna sell 438, let's go for 400, uh, over the 38.4, we're gonna, uh, sell 438. And now let's just sort of, uh, make, um, make the, uh, calculation here. Let's assume a 10% correction, right? And that means we have minus 10 million on cash, right? On the cash position, and we see, what is it, 456, uh, index points decline, right? And so we now can basically say each of those 456 gives us $50, right? That's the, uh, index mm, or the, the, the value of the index point of the futures contracts. And so we're making 456 index points profit. Each index point gives us $50, and we have to multiply that with 438 because that's the number of contracts we are short. So let me just type that in here and just see if the numbers make sense. Um, and that means we are having a positive p and l here of 9,986,400. It's not quite the same, right? It's not a hundred percent. There's obviously, um, two main drivers. A I wasn't quite using exactly 10%, right? Because 10% of 4565.75 is not 460, uh, 56 index points exactly. And the other one is obviously that we have only in a comma sold 438 contracts rather than the suggested 438.04. But you get the point, right? So we have turned a loss of 10 million if we hadn't done anything into a loss of, you know, what is it? 13,600 here. That's obviously a much smaller, uh, amount of money here. And if we would've gone for 439, we would've gained potentially, uh, some money. So that's a general simple approach here to just kind of cut, um, your exposure temporarily. And if you then have the correction, you know, um, if the correction happened, then of course you can buy back those hedges. Or if you were wrong and the correction doesn't come, you have to buy them back as well. Um, but you know, that's a general idea of hedging. Now, the one critical assumption that this approach basically makes is that our cash portfolio will move exactly in line with the index. And that may not be, um, something that is a realistic assumption then that depends solely on the nature of our portfolio. So let's say we're an investor that is very, very, um, aggressive and has a very, very strong growth, uh, strategy there. So we have bought, um, you know, shares of sectors of companies that display a somewhat more aggressive behavior than the overall equity market. IE we're talking now about high beta stocks here, or high beta sectors, or higher beta sectors, right? So beta remember gives us a sort of like, you know, idea about how much more volatile than the index a particular stock or, um, a, you know, sector or a particular portfolio because we can calculate, um, a portfolio beta here as well is in, in comparison to the index. And then you have a beta of one that means we're moving in the same direction and at the same speed in the market. Beta of two would then mean we're moving in the same direction, but twice as fast and 0.5, same direction, but only, um, you know, half the speed, et cetera, et cetera. And that of course is something that we can calculate using historical data. And while this is no guarantee that the future portfolio behavior will be exactly like the portfolio behavior in the past, at least it's a good indication, right? So let's extend our example. Now that we have our investor here, they have still the s and p 500 portfolio or focused portfolio, a hundred million is the current market value. Uh, the front months contract trades at this level, as you know, discussed. But now we've been given the information that the portfolio beta is 1.2. Now, what does it mean if the beta health in the future and we see the correction of one, uh, of 10% here, what this means is that our portfolio should lose 12%. So it goes in the same direction, it goes down in value, but 1.2 times as fast, uh, from 10% loss will translate to 12% losses on our cash portfolio. That's 20 times more, uh, sorry, 20% more volatile, uh, as it says here, um, on this slide. And that isn't something we can obviously consider when we're putting our hedge together because the index or the, the, the futures that we are, um, using here, uh, to hedge, they have a beta of one, right? Because they are basically reflecting the index level of ignoring a little bit of interest rate and, and dividend stuff there. Now, um, so what this means is if we're using now a 1.2, um, or a one a beta of one instrument, uh, to hatch a beta, um, of 1.2 portfolio, uh, then of course we need to adjust for that. And the idea here is that we will simply have to multiply the hedge ratio we've calculated on the slide before, um, well two slides ago, um, by 1.2, which is the portfolio beta. So in other words, we're selling more contracts, um, and we're selling 1.2 times the number of contracts on, uh, non-beta adjusted, uh, hedge. And the reasoning for doing though is so is that yes, we have a hundred million cash portfolio and if the index goes down 10%, we're losing 12, or we expect to lose 12%, which means 12 million, and we're using a beta one instrument to hatch. And that means we're now going to, um, have to hatch at a 1.2 times larger size so that we're getting paid 10%, but on 1.2 times the notional. Hence, we are selling 525.65 contracts. Do the rounding as you wish. Okay? So that's the idea. And then if beta holds in the future, then again, our hat should be, um, you know, taking away or eliminate most of the losses by far, um, for us. And that's then obviously, you know, one of the main considerations that one needs to be aware of when one thinks about using futures for hedging, and that is that there is often a difference between the constituents of a particular portfolio and then the constituents that are making the index on which the future contracts then is designed of. And then also, although B two is something that we can calculate and they're well familiar with, question is how stable will this B two be, uh, going forward? And then there's other things, as we have already discussed, there's an interest rate and dividend exposure here because remember, the forward price and future prices are nothing else than forward prices just for exchange trader products, um, are sort of dependent on interest rate and dividend changes to some degree. I'm not saying it's the most important exposure here, but one should at least be aware of that. And then there's something that's called the role risk. And I wanna, um, you know, bring this together with the open interest and volume, uh, stuff that we've talked about earlier. And what is the future's role, right? This is something you hear quite a lot in, there's particular days of the month where, or, you know, every three months where we pay a little bit more attention maybe, uh, to trading activity. And those are, and usually when the contracts come to expire. So what we're doing here is we're looking at two futures contracts to September 23 and the December 23, if we would use this, uh, you know, data from 24 or now March and June, uh, contract for 2025. Uh, you know, while the numbers might be, um, slightly different, you know, the, the, the overall, um, pattern will look more or less, uh, the same. So let's have a look what we, what we have done. We looked at this time period from 1st of May 23, uh, and that was basically when, you know, the, um, September contract, um, was already listed as we talked about before. But as we can see here, by just looking at the open interest and also by looking at the volume, um, literally nothing was happening, right? So this contract wasn't really traded, this changed, um, in June, 2023, relatively sudden, right? At some point in early June, suddenly the open interest, um, climbs quite significantly and the, uh, trading volume goes up, um, quite significantly at around the same point in time. Now, why was that happening? It was a point of time when the June, 2023 contract was getting close to expiry. That's when suddenly the activity, um, shifted to the September contract. Then it stayed relatively unchanged. The open interest over, um, you know, there's some degree of variance, but overall it was a relative unchanged open interest until sort of, um, you know, September when the open interest, uh, crashed. And that same is more or less true for volume, although volume is a little bit more noisy simply because there's less active days. You know, maybe some bank holidays, some, you know, long weekends, whatever. Um, but overall, you know, there's some sort of like relative stable volume going through the market as well. So now question is what is that, what's the conclusion? And I think what we can conclude from here, because the same time the September contract open interest goes down, now the open interest in December contract goes up and the conclusion is that the market tends to focus trading activity on the next, um, contract that is, that is or expiring next. So that would be right now the June, uh, 2025 contract, and that's what we call the front month, uh, contract, IE the contract that expires next. Um, and, uh, and when then the expiry date of this contract nears, uh, then um, investors will have to adjust their positions because most of us take futures positions because we either wanna be, uh, building a long position or a short position or whatever it is that we do, but we don't necessarily wanna lose this position just because the contract expires. And so assuming we are long, um, these futures here now and the expired date of our contract, uh, approaches, but we still wanna be long, um, the s and p 500 futures contract. So we don't really wanna have this thing being cash settled in a couple of days. So what we're going to do is we're gonna sell our long position in the September contract and simultaneously open a new long position in the December contract. We're doing this as closing trades and as a result what we do is we're pushing open interest in September down and we're lifting the open interest in December, and that's what's called a future role. We're rolling our risk exposure from the, um, contract that expires next to the now, um, following, uh, contract. And that is then when the open interest gets pushed to the next contract. And also when the volume in daily trading generally sort of gets pushed to the next contract. And so despite the fact that there's these 21 or whatever, um, quarterly contracts available, uh, what's really trading most actively is the front months, maybe the second one as well, and a little bit of open interest in third and fourth. And after that it gets pretty, pretty, um, illiquid. Okay? But, uh, you know, just a word of caution there. That's only true for equity index contract as the general sort of, um, thought there some others which have same, um, um, same, uh, behaviors. But, you know, if you're thinking about, for example, sulfur, um, futures, et cetera, they might display a very different, um, behavior for, um, you know, for, for, for many, many reasons. So, um, always check before you, um, extrapolate this behavior to others. And with that, now, as I said, we just want to have a quick look at equity swaps and, um, you know, basically like any other, swap the agreement to exchange two streams of cash flows. And in many, many of those equity swaps, um, those cash flows of one leg are linked to the return of either a single stock of a custom equity basket or an equity index. And then the cash flows of the other, uh, lag are linked to interest rates. And so let's just dissect this, uh, and, and, and focus on, um, this sort of like, um, boxes scenarios here, and to get an understanding as to how conceptually equity swaps work. So let's say, um, you know, we're, uh, an investor here and we're thinking about, um, a particular stock and we want to build up a long position or, you know, long exposure, um, to this stock, then of course we can either buy the stock or we can maybe buy a futures contract on this stock. But one thing we could also do, we could enter into an equity swap as the so-called equity receiver. And being in that position means that we are going to get paid from our counterpart, any equity upside that will materialize between us entering into the trade IE the trade date, and then the, uh, maturity date of this equity swap. This is a 12 month swap. Then in 12 months, let's say the stock has gone up 10%, we get 10% off this, uh, oh no, we get, we get 10% paid on the, uh, trade notional as the equity upside. If, however, the equity or the underlying equity has gone down over this 12 months period, um, then we will have to pay the equity downside, right? So if the stock has declined in price, for example, by 10%, then we need to pay 10% of the notional to our counterpart, which is called the equity pay. They pay the upside, they receive the downside. Now, that's just the price return. Um, but most equity swaps are actually structured in what we call total return swaps. And that means not only do we receive the equity upside as the equity receiver, but we are also receiving any dividends that the underlying stock or, you know, uh, basket of stocks here, uh, generates. So economically speaking, the equity swap is really like being low. The stock, you get the price upside, you pay the price downside, and you get any dividend, and there's no sort of dividend expectations baked into the forward price or anything like that. It's really just if there is a dividend, you get it. If there's none, you won't get. And if the price goes up 10%, you get it. If the price goes down 10%, you have to pay that. So it's really like being long. The stock, um, the only, no, not the only. But one significant difference though is you never are the physically, uh, physical owner of the stock. And that means you, for example, don't have voting rights, right? You cannot vote on the annual general meetings. Um, for example. Now what we can see here, so what we have explained this, we have explained this, we have explained this, and now we need to sort of focus on that. What's all that about? Why is there a reference rate? What's interest rates have to do with this? Well, let's think for a second about the equity pay. That's probably gonna be a market maker, right? So the client calls the market maker and says, I want to get into this total return swap on this particular equities, you know, single stock, whatever. Uh, and I want to be the equity receiver. IE you pay me the upside and you pay me the dividends. Now, how can the equity payer now simply hedge, uh, there is, because now what they have to pay is any upside that materializes and they have to pay dividends. And the idea is the equity payer could, for example, go out and just buy the shares, right? So they go buy the shares, and that gives them obviously ownership of the shares. That means they get the dividends from the, um, issuer here, right? So issuer pays dividends, um, to the equity payer, and then the equity payer forwards. This dividends may be adjusted by withholding tax or whatever, um, to the equity, uh, receivers. So that gives them the dividend. And because they bought the shares at the price when this, uh, trade was created, and subsequently the share price goes up, they can sell the shares at higher prices, and that gives them then the equity upside that they will have to pay into the contract. And then the only problem really is that what happens if the stock price goes down? Well, you know, in assuming the counterpart will, um, you know, deliver on their obligations, nothing happens because the equity downside. So any potential losses will be paid by the equity receiver. So if the stock price has declined by 10%, yes, we're selling the shares 10% less, but we get those 10% from our client, the equity, uh, receiver because they pay the equity downside. So that would be a simple, um, hedge to put in place. The only you know, point to mention there is though, that to buy the stocks in the first place, what do we need? You, right? Um, cash, right? So this cash will then have to be borrowed and borrowing cash costs, interest rates. So we usually get this, um, interest rate paid by our equity receiver client simply because the reason why we're doing this trade as a market maker is because we want to facilitate that trade, um, for our client, and then they pay us our funding costs and of course, the spread, um, because there's capital charges and there's, you know, overhead and all this other things. And we also want to have, um, you know, a little bit of, bit of a spread there, uh, because we're facilitating a market liquidity, tech risk, et cetera, et cetera. So that is the overall structure of those equity swaps. You can see here beta says OTC, there's obviously a lot of, uh, things that we need to negotiate. What's the notional? What's a 10, what's the underlying, uh, what's the reference rate? Are we using UIOR? Are we using Ester or you know, SR whatever. Uh, and then how often do we make payments? Is this just at the end? Is it every month, whatever, and so on and so forth. And we could spend a lot more time on those equity swaps. But the general structure is that, and I guess the fundamental difference because now obviously you realize, okay, you can buy stocks in the spot market. You can, uh, enter into future contract or forward contracts. You can, uh, do swaps. And they obviously all have sort of, you know, similar exposures, but also, uh, these exposures, uh, differ, uh, in, in, in various ways. And so you start to see that there's actually a reason why all those different products exist because all of them fulfill slightly different purposes. That's all I wanted to share with you today. Thank you so much for your participation here. I'm still having a couple of minutes, so if you want to use the opportunity to ask any sort of final questions, use the, uh, q and A function right now, I'm happy to stay and answer them a little bit, uh, longer. If not, have a fantastic rest of your Friday, a great weekend ahead, and I hope to see you again soon of one of our, uh, on one of our sessions. Take good care. Thank you so much. Bye for now.