Good morning, good afternoon, possibly good evening for some of you, and a very, very warm welcome to today's Felix Live session, which is going to be on equity delta one derivatives fundamentals. My name is Thomas Carlsen.

I have the honor to take you through this session today, and of course, always one of the most important questions that you want answered right at the beginning, what exactly are we going to cover today? So, it's a fundamental session. What that means is we're going to start with a little bit of a refresher.

We're going to take a look at the fair value of an equity forward, because this provides a starting point for a lot of things that we're here to talk about today.

From there, we will then discuss some of the mechanics of equity index futures, and then see a practical hedging example, how we can use those contracts to hedge portfolios. And then towards the end of the session, we're going to give a brief high-level introduction into equity swaps. So that's at least the structure that I have in mind. Now, before we start, couple of important reminders.

As always, you can access today's course materials as PDF and several ways of doing so. The most convenient way, although it might not be possible for all of you, is just to go into the Zoom chat right now. I just shared the link to the relevant website there of this Felix Live session, where you can download the document in PDF format.

Points, let's get started with the content.

And as I said, we're going to start with a recap on the fair value of equity forwards. And this is really down to the question, where do forward prices generally come from? And then obviously, when this is the first time you're approaching this, you do sort of tend to think that's sort of somehow linked to where market participants expect prices in the future to be.

But as you can see here on the slide, that's actually for most forward contracts almost completely irrelevant what the future expectations are. It's all just based on this no arbitrage principle.

And the idea here is, and this is not just true for equity forwards, but literally for any other forward contract on any other asset class as well, is that the forward price is basically the spot price of the asset adjusted by the cost of carry, and that we should take quite literally. That is just the cost of taking a position right now instead of at a specific point in the future and then carrying this position over time. Now, of course, interesting to think briefly about what are the components of the cost of carry, and this is where we're now switching into the equity world, simply because the cost of carry components do differ across asset classes. One thing, though, they all have in common is that it all starts with funding costs, right? So basically what we can see here for the fair value of an equity forward, we're starting with the spot price, so that's the price at which we could buy the asset in the spot market. Then we're adding the product from spot price and an interest rate times days over basis. So here a disclaimer, right, where this formula only technically holds perfectly well for forward periods that are not exceeding 12 months. If you calculate a two-year forward, for example, then the maths would have to change a little bit. But most forwards do have probably a forward date, at least in many asset classes, that's the case of below 12 months period. So we're going to stick to this approach here, and that's basically nothing else than what we said, the borrowing costs, right. Now why do borrowing costs need to be considered in case of an equity forward, right? You think about stocks, where are interest rates suddenly coming in? And the idea is the following, right.

Let's imagine you are working as a market maker.

You're trading a particular instrument here.

Let's say it's a stock, and you now have a client that is approaching you and has sort of a specific requirement.

Let's assume for simplicity, forward markets don't exist at that point.

So the client now is asking you, "Well, can you somehow give me a price that we agreed upon today, but I pay you 12 months from now because I want to lock in the price right now, but I don't have the money to buy the asset immediately." And then, of course, you start thinking, how can I do this? And the way to do this without exposing the firm you work for to market risk, but you also want to deliver this requirement or you want to serve your client's needs, you will have to start by buying, at least conceptually, by buying the underlying stock at the only point in time where you know with absolute certainty where the price is, and that is right here, right now. So you go to the spot market and you will just buy the underlying spot, right.

But if you think about the timeline, right, then you have this is the trading transaction day, and in 12 months the client is going to pay you the money. So you bought the stocks today, meaning in T+2 or T+1, wherever you are, you will have cash leaving the building because you need to pay for those stocks and then you get the stocks coming in. Fair enough.

And the client is going to pay you 12 months later. So what do you have to do? You have to bridge this 12 months funding gap, right, because you're paying T+1, client is paying you T+12 months plus one day, and that means you have to sort of bridge that gap.

So you will, in a second step, then obviously borrow the spot priceborrow the cash for 12 months in this particular example, if we're just staying with the 12-month example. Now, of course, technically speaking, then you have to multiply it with days of a basis.

That's your day count convention.

But if we just assume for simplicity that the spot price here was 100 and we borrowed cash at a rate of 5% for 12 months, and we just assume it's a whole year, then this would basically bring us to $5. And then, of course, if we have nothing else involved here, then of course, we would borrow the money, $100 at spot. We would then buy the shares with it, and in 12 months, we will have to pay $105 to the lender that gave us 100 in the first place, and that's the minimum amount that the client should pay for the shares.

We're adding bid-offer spread around that, and that gives us then a fair approximate, a good starting point for our negotiations.

That is only true, though, if we completely ignore dividends, right? So far, we have just looked at what does it cost us to take this position and carry it over the next 12 months, and that's basically usage of cash and that's basically financing costs, right? However, there might be an economic benefit in holding onto these stocks over the next 12 months, and that's the case whenever there is a dividend payment, because the dividend is going to get paid to whoever's holding the stock, whoever's recorded as a shareholder on the record date.

And if there was a dividend payment at any point in time between the transaction date and the 12-month period, and then we're going to receive this dividend.

So now we have to apply the same logic than before. In the previous example, where we just looked at financing costs, we said, well, we're going to charge these financing costs onto our client because basically they are the reason why we're incurring these costs, right? The client wanted us to give them a forward price.

As a result, we just buy the stock spot, at least conceptually as I said, we don't necessarily have to do this, but from a conceptual point of view, it's easiest to imagine that. We go out and buy the stock with borrowed money, and that borrowed money incurs costs, and that's what the client has to pay, basically for our resource usage, right? But that also means that if there's any benefit that holding the asset gives us, then we should pass that back on to the client, because we're not holding the stock because we want to hold the stock. We're holding the stock because we're holding it on behalf of our client, and if the client then pays for the costs that this creates for us, we should pass on the benefits. And so that makes perfect sense from a fairness point of view, but also from an economic point of view because if you think about it, who eventually or finally holds the market risk in this transaction if we just ignore for one second now that there's counterparty exposure? So we just kind of eliminate that concern here from the equation and we just say, who is having the market risk in a situation where we bought the stock at the spot price, we borrowed the money all the way through for the next 12 months, and we have agreed upon a forward price.

Do we have market risk? No, because even if the stock rallies to 10,000, we're not losing money, we're not profiting here in any way through this transaction because we bought at 100, we're going to sell at 105, and we have to take this $105 and pay back the amount of money we borrowed.

And that is also true when the stock now has crashed to zero, i.e., we are having a stock that is worthless, but we have an agreement that says somebody's going to pay us 105 at the maturity of the contract. So once again, market risk-wise, we are completely unaffected when we have done the hedge as described, right? So the client, however, is locked into a contractual agreement to buy those shares at the price of 105, if that was our example forward price, no matter what. If it's at 10,000, great for him, or her, or they, them.

And if it's at zero, that's obviously not as great.

So they are holding the risk. And conceptually, I think it's fair to say that whoever holds the risk, or the market risk on a particular investment should receive all the returns of that investment. That's only sensible.

So dividends should be passed on to the forward buyer. And that usually happens in form of sort of an adjustment on the forward price, and that's now where we go, and I'm tidying this up a little bit because there's a lot of drawing already on there.

So forward equity price is spot price plus funding costs.

If we just sort of keep this as funding, and then we're subtracting dividends.

And that's easy said, right? And then you say, well, of course we need to deduct all dividend payments, but there's obviously always sort of when you look at these things on a formula level, yeah, okay, I understand it conceptually, spot price, I can see, then I know how to calculate interest rates and if I have a dividend amount given, then I can do the math here. That's not the challenge.

The challenge is to really think about the question, first of all, which interest rate should we use, right? Now, of course, in today's world, it's very tempting to say, well, given that there's no term deposit benchmark fixing anymore, LIBOR is gone, right? Instead of using 12 months LIBOR, which we might have done previously, we're now maybe using 12 months SOFR swap rates as a sort of indicator for SOFR over the 12-month period, and then we might just add a spread or not.

This is sort of already raising certain question marks. Also, SOFR, let's not forget, is a repo rate, which means you only can borrow money at SOFR, strictly speaking, if you pass a US Treasury over to your counterpart in return as collateral. And that in itself needs to be financed.

So I'm not sure if SOFR is, per se, the right rate to use here. And also, every financial counterparty out there will most likely face slightly different funding costs, right? Banks might be able to borrow at SOFR flat, butOther market counterparties might not be, or they have to take a significant haircut on their collateral, things of that nature. So finding the right interest rate is indeed not trivial and will probably there will be a different answer for different market participants.

So that's the first thing that I just wanted to create awareness for. The second point is dividends. Right? We're just saying there, okay, we deduct dividends.

And then very often people say, "Well, okay, cool, so we just take the dividend yield." And yeah, that's fine. But how exactly do you calculate the dividend yield? The dividend yield can be conceptually thought of the sort of dividend that a certain asset is paying, like a stock here in this case, over a certain time period. But the problem is, in reality, dividends don't necessarily accrue like interest does. So every day that goes by, you get a certain amount of interest. But they're specific points in time where dividends are paid, right? And that's causing the practical challenge in using dividend yields here, which sort of create the illusion that this is a running income. Now, that is not saying we cannot use dividend yields, by all means, but we need to be quite careful in how we're calculating the dividend yield.

But regardless of which way you want to use, we need to conceptually understand how we actually need to treat dividends. And the first question is, which dividend payments should we actually include? Right? And here's a little bit of information given to us at the bottom of the slide. Right? And specifically, we're going to include all dividend payments in the calculation.

So for calculating this amount here, where the ex-div date falls between transaction date and forward date.

So that means here, today is the starting point.

In 12 months, the whole contract ends.

If there's an ex-div date here, then we're going to include the dividend in our forward pricing because what happens on this day? On this day, the stock starts trading in a way that we don't, or not assuming, but basically expressing the fact that whoever buys the stock on this day will no longer receive dividends.

That means the price of the share is adjusted for the ex-div event on this particular day, even if the dividend is paid after the forward contract is over. So that's important. Not the payment date matters, but the ex-div date matters because that really decides who's going to get the dividend payment rather than the actual payment date because the dividend will be paid to everybody that is on record on a specific date. Right? So that's the first thing.

And then we need to also remember that there might just not be one dividend payment perfectly timed at the end of the forward period, but there might be multiple payments throughout. Right? Some companies, for example, pay dividends on a quarterly basis.

So we have a payment in three months, in six months, in nine months, in 12 months, or-- Oh, yeah, obviously that was double.

Or it might just be in one month, in four months, in seven, and in 11.

That's just dependent on whenever we're buying the stock here and when the dividend payments are.

Now, conceptually, though, what we cannot do is just simply say, okay, if we get, let's say, $1.50 here, and we get another $1.50 there, and another $1.50 there, and another $1.50 there, we cannot just say, "Okay, this is a dividend of $6." Right? That's too simple, because we're going to get one and a half dollars after three months. And technically speaking, we're going to pay it to our client here or give it back to them, basically, at the 12-month point. Right? So what we're going to do as rational market participants with money that we're going to get at this three-month point is we're going to invest it for the next nine months and generate interest. Right? So instead of just taking the dividend payments and summing them up, technically speaking, what we should do is we should calculate the future values of each dividend payment at the point in time when the forward is settled, and then use that as the actual dividend. Hence, I'm saying here the future value of all included dividends must be used to calculate this amount that we're discounting.

So this is just, as you can see, commonly known principles, but once you look at them, it was a little bit more detail and you want to be a little bit more precise about things, and actually, there's quite a few things that you have to think about. Okay. Here, just the numbers at work, where we see our example here. Spot price of the asset was 100, 12 months interest rate was given was 5%, as on the previous slide. Then we said the 12 months future value of dividends should be $6, also in line with what we said on the previous slide. And then we said, okay, that results in the 12 months fair forward price of 99, because this is spot, this is the financing cost. We're adding $5 to it.

We're subtracting $6 then from 105, and we get to 99. So there you can see, as I said, the forward price is not based on anyone's expectation.

It's just the result of cost of carry.

Now, that's, and I think it's important to mention, the fair market price of the forward. That doesn't mean that's exactly where the forward is going to trade.

Right? We're making a couple of simplifying assumptions here, and they might not hold in reality. So the real forward price can, of course, be slightly different. However, if the difference is too big, then potential arbitrage where trades will become profitable, and then obviously market participants will act to bring this price back in line.

Interestingly enough, though, what we can see hereIs that, with that in mind, the price of a forward contract on stocks now has a certain sensitivity to interest rates.

Right? And this is, I think, an important feature to at least have heard about. I'm not saying that trading equity forwards like the ones here is creating you a crazy high interest rate duration risk on your portfolios, but you should understand that significant changes in interest rates can indeed have a knock-on, or will have a knock-on effect on the value of forward positions. And we're making here a slightly strange assumption, and that is that interest rates have changed meaningfully, and the stock market doesn't care, right? That's, of course, in practice, a bit of a stretch, but it's nice to just show the isolated impact here of a change in dividends. So let's assume that we had a client investigating about this product this morning. We calculated the 12 months forward price at 99, and the client accepted the condition, and the client bought this forward. So client bought forward at 99. So just for simplicity, we haven't put bid-offer around it or anything like that.

Ignore transaction costs, all of this stuff, right? So client bought from us forward at 99.

What we haven't done this morning, because, for whatever reason, is we haven't hedged this trade at all.

We haven't bought the shares on a spot basis, we haven't borrowed the money or anything like this.

And now let's assume that for whatever reason, at the end of the day, interest rates for 12 months are down to 4%.

Right? What does that change? It changes the fair forward price, because now if the spot price of the share is still 100 and the future value of the dividend's also unchanged at $6, now the fair forward price is 98. So basically, if we're now looking at our mark-to-market valuation PNL wise, we have sold these stocks on a forward basis at 99 this morning.

Currently, the market price for this transaction is 98. So we have basically a PNL impact of $1 per stock here. And the interesting thing is, despite the fact that we're talking about equity contracts here, this PNL is not driven by a change in the stock price, but that's driven by a change in interest rates, right? And I just wanted to put that out there because this is sometimes a little bit of a wake-up call that you do have other risks now on your portfolio when you're using derivatives rather than just the spot price of the underlying borrowing costs play a meaningful role. And also just to share that as well or make that obvious as well, what if-- and we can extend that example, or we can change that example to show the impact of another factor.

What if the company had declared, unexpectedly, a change in dividend policy during that day, so after we have made the trade and say, "Well, we're just only going to pay out half the dividend because we want to reinvest more money for this aggressive growth strategy that we're having." Then we will basically have to adjust the three months future value of dividends to roughly $3, and that would bring the fair forward price now to 102. And in this case, we sold at 99 this morning, and now the market price is 102.

So now we're facing a $3 loss per share, per contract here. And that is because the dividend payout ratio has changed.

Now, again, it's a bit of a stretch to assume there's been a significant change to dividend payments in the future, and the share price is not moving at all. But just to be able to show this in isolation, I'm just making this a very, very simplistic example here.

So bottom line is we now realize that the forward market is obviously extremely useful.

It's a derivative with all these advantages and considerations.

But one thing to keep in mind is that we're also introducing rates, or sensitivity to changes in interest rates and also to dividends into the mix. And that's just something to keep in mind.

Okay, so now let's move and look at the futures, and we're going to look at index futures here.

But of course, more or less all the points on this slide here really are true for any kind of future, if that is a single stock future or bond future or something like that. So we know what the difference is, right? A forward is the OTC version, so it's bilaterally agreed between two counterparties.

And that means we can, within the legal acceptable terms here, really negotiate whatever we want, right? So that's a clear benefit of OTC. It's much more flexible. You can tailor it exactly to your needs.

However, as you would expect, very often this tailoring comes at a cost. And another point to mention here, although not necessarily a very, very significant problem, is that we do require a valuation model for those sort of things, because as it's an OTC instrument, there is no central market price. So we traded something in the morning.

Now we want to see where's the PNL for this position at night.

We cannot just go to the exchange and feed the closing price and do our valuation with that, because that closing price simply isn't there because it's a tailored contract between us and the counterparty.

So we need to calculate the theoretical value of these contracts.

As we've seen, the formula is not necessarily super difficult in this sort of case, but a couple of question marks remain as we said. It's a requirement, and therefore we have to think about this valuation model. On the other hand, we have futuresWhich are basically the exchange traded version of forward contract, and that means that there is a degree of standardization.

For example, in terms of underlying.

You will not get a futures contract on maybe any underlying stock there is or underlying currency or whatever it is. There's only a certain range.

Then the contract size usually will be, not usually, is standardized. So one contract gives you a certain degree of exposure.

Now, if you're looking to transact a multiple of this contract size that's given on the exchange, then that's not a problem, just buy a multiple of contracts.

If you're looking for a smaller, then you might run into trouble.

There's also standardization with regards to expiry date.

So we have futures, for example, in the equity market on the index level that expire in March, June, September, December.

But if you're looking for a hedge until October, for example, then you can either choose a September contract or the December contract, but you don't have specifically the October contract that you might be looking for.

And there's settlement that is standardized, either cash or physical, dependent on the contract, the underlying, et cetera.

Now, what is clear is that futures, because of the standardization, feel a little bit less flexible, and they are.

But in exchange for that, we typically get superior liquidity, i.e.

the possibility to trade much larger amounts over a very short period of time without significantly impacting the market. And that's just because all the people that want to trade this particular underlying on a forward basis or a futures basis will have to funnel their liquidity, i.e. their demand supply, to a small range of contracts. And that, in turn, means we have a higher concentration of supply and demand on each contract, which then means a better liquidity situation than in many OTC contracts. And high liquidity, again, means lower spreads, the ability to trade larger number of contracts, et cetera.

Another advantage of futures is there is a central market price.

So if we want to value a position that we entered in the morning, we just have to compare the price we bought at with the closing price of the exchange, and then we see an example of how to calculate this later on, so relatively straightforward.

And then one point that is not, again, hugely relevant for most of us, but for some, it's actually quite interesting, the exchange usually provides quite a bit of data points here that may be quite useful to analyze.

And examples here are things like open interest and volume. Okay? And volume is usually very well understood. It's just the amount of contracts, for example, that is changing hands, and we usually measure this on a daily basis, right? So we reset to zero at the open, and then we're just tracking the amount of contracts that have changed hands during the day. So again, relatively straightforward and well understood.

Open interest is, when you look at this for the first time, maybe a little bit confusing if you just hear the definition, because it's something like, open interest shows the number of outstanding futures contracts that are held by market participants, and that's usually done on an end of day basis.

So I want to explain it slightly differently, and the second box there is much closer to this because what we're basically trying to do here with open interest is showing the amount of risk that is held in those contracts, and this is done by just looking at positions that have been opened but not yet closed and/or settled.

So best looked at or best explained, I think, in terms of an example, and that's what you see at the bottom of the slide here.

The assumption is we have a brand-new futures contract that wasn't available yesterday. You couldn't trade it yesterday.

Hence, by definition, volume and open interest must be zero. We're launching this contract today, and then the first trade ever in this contract happens, and that is Trader B selling 100 contracts to Trader A. Now, when you trade futures on the exchange, you will have to let the exchange know whether this is an opening of a new position or a closing of an existing one.

In this case, it must be an opening, for both because this contract wasn't around before, so nobody can close any position because there cannot be an open position.

So now we have the situation where Trader A has opened a long position of 100, Trader B has opened a short position of 100, and that means we have a total volume of 100 because 100 contracts changed hands via the exchange, right? Trader B to Trader A.

And we also have an open interest of 100 because Trader A is long 100 and Trader B is short 100. So this is basically 100 contracts that need to be closed, if you think about it, to eliminate the risk for Trader A and for Trader B. Now, in a second transaction that ever happens in this contract, Trader C sells, and this again must be an opening transaction because Trader C hasn't done anything yet, so they cannot have a position to close. So they sell as an open.

Trader B, however, buys back basically 50 of or half of their shorts. So for them, let's say this is a close.

So they are closing 50 of their shorts.

And that means, first of all, because 50 contracts are changing hands from Trader C to Trader B via the exchange, we see total volume going up by $50.

Sorry, by 50 contracts, so we're at 150 now. Interestingly though, the open interest has not changed whatsoever, and that's becauseTrader C has replaced Trader B. So Trader B closed half his shorts, but Trader C has opened the same number of shorts.

So now basically Trader A is still long 100, Trader B is short 50, Trader C is short 50, and that means open interest remains at 100. And now the third trade that ever happens in this contract is Trader A selling 50, and this shall be a close as well, because Trader A now gets rid of half of their long position.

And Trader C closes also their 50 short position. And so what this means is now the net effect we have, again, another 50 increase in volume because we had another 50 contracts traded, but the open interest now is not stable.

It has actually declined because what's the net situation in the market? We have Trader A being long 50 contracts, Trader B being short 50 contracts, so the open interest is 50. Now, the question is, whereas we can very easily see the relevance of volume, right, where you say, okay, volume gives me a good feeling for how liquid a contract is. So if I'm having a contract here that typically trades 2 million contracts a day and I want to transact 400 contracts, I can safely assume that this is going to be done reasonably quickly, right? If I, however, want to sell 400 contracts in a contract that changes typically or that trades in volumes of 50 a day, that's going to take me quite a while, right? So that's a immediate takeaway there, why is volume interesting. Open interest, sometimes a little bit less obvious why this might be relevant, but it has a couple of points that are of interest. But one thing I would like to point out, and I come back to the slides in a minute, I just wanted to look at the behavior of open interest.

And here we're looking at the September '23 and the December '23. But that's not for any reason, that would look more or less exactly the same if we looked at September '25 to December '25 or March '25 to June '25 or '26, I mean.

So this behavior is relatively consistent in S&P 500 futures contracts. And what we can see here that the September 2023 contract traded or had a zero open interest, more or less, on the 1st of May 2023, and it remained relatively close to zero until this point, which brings us to June 2023. And what's special about June 2023? Well, that's the expiry month of the June 2023 S&P future. So in that month then we saw the open interest in the September future going up very, very significantly, then flattening out a bit, sort of there was a bit of ups and downs, but the open interest remained surprisingly stable, I would say, over this three-month period. And then we see a sharp drop.

All right. But not only do we see a sharp drop in the September contract's open interest, but at the same time we're dropping in September OI, we're seeing a relative sharp increase in the December 2023 contract. And this is a behavior that's observable in many futures markets, and that's what we refer to as the futures roll.

And that's especially for equity futures, observable for rates, the short-term interest rate futures, not quite in the same way. Bond futures is similar behavior, but let's stick to the equity index world here.

What typically trades most liquidly is what we call the front month's contract.

There are several contracts outstanding, but not all of them are actually trading very liquidly.

So we have basically the front months, the next expiry months trading most liquid.

So from today's perspective, given that we're in April, the most liquid contract will be the June 2026.

September 2026 contract exists, yes, but there will not be any huge liquidity.

So we then have to discuss a little bit of why this is, and I do not necessarily have 100% conviction answer. But I would expect that the reason why the front month trades most liquid is a link to what we said earlier, and that is that these forward contracts have a certain sensitivity to interest rates and dividends. And when you are familiar with interest rate mathematics and you've seen the formula before, the longer the forward period is, the more impact a change in interest rate will actually have on the futures price. And if you're trading equity index futures, I don't think that your primary reason for doing so is that because you want to take interest rate risk. You will just want to hedge stocks and therefore might want to minimize.

And so choosing the front month's contract, I think, is very, very good practice, and that then leads to liquidity being concentrated there, and then the whole thing becomes an almost self-fulfilling prophecy.

What it does mean, though, is that, and we're going to talk about this a little bit later as well, is that when you hedge today and you're using the June future and Junecomes, you have to decide whether or not you want to keep your hedge open, and then you will just have to roll into the September contract, or whether you want to close your hedge, and then that's fine.

But that's, I think, one of the things to just consider, and we're going to see this in a minute.

Okay, so I said I was going to go back to the slides, and here we are.

So now we're going to look at a concrete example and go through a couple of those things that we have already mentioned.

So we're looking at the E-mini S&P 500 contract.

What we can see here is that, and here this is something that needs to be on with an index, because the index itself is not directly tradable as an underlying. You cannot buy one index of the S&P. That doesn't really exist.

You can buy mutual funds and ETFs and future contracts and stuff, but you cannot buy one index as a piece. You can buy, of course, the 500 stocks in different weightings and stuff.

But, and therefore, when you create an index futures contract, you have to obviously create a link between the level of the index and the actual size of the position that one of these future contracts basically translates into.

And that's done for this particular future contract by this formula here, where the CME has defined that every index point is $50. The tick size, i.e., the smallest possible price change, is a quarter of an index point.

And as each index point is worth 50, the value of each tick is in a quarter of 50, and that's 12 and a half dollars.

There's consecutive contracts there for each quarter.

These things change every so often, so don't want to spend too much time, but the third Friday of the contract month, March, June, September, December, that's something one should remember in equity index cases, how many contracts are listed at any point in time.

That, as I said, changes. But remember March, June, September, December and third Friday, that's relevant, and cash settlement is also the norm. So now on the right-hand side, we see an example.

Again, the mechanics don't change.

Hence, I haven't updated this. But of course, now the S&P trades significantly higher. But we can use the example nonetheless. So, the last price of the future contract was 4,565.75, and then if we just multiply that with $50, we will get $228,287.50.

Meaning that basically buying one futures contract at this price is economically similar to investing 228,287.50 in the S&P 500 index. So that's something to remember because we're going to see this again when we're calculating hedge ratios. But before we do that, a very simple example on how we can calculate the P&L for a position. So the assumption is we bought 100 of these contracts at the last price, so at 4,565.7, and then we sell it at 10 index points higher.

What is our profit? And that's actually relatively simple to calculate because we made 10 index points.

Every index point is worth $50, and that is per contract, so we then need to multiply that with 100, and that gives us then, if I'm not mistaken, $50,000 P&L. Ignoring transaction costs, anything like that. So this is really simple.

That's prices that we can see on the central market, on the exchange. So we don't need any valuation model here.

It's very simple to calculate this P&L effect.

So, these contracts are obviously used by many different participants for a variety of reasons, but one that's particularly intuitive, I think, to grasp is the hedge of a portfolio. So let's have a look at an example here, and we're going to start with the text of the example here.

So we have an investor that has currently $100 million invested in S&P 500 companies.

So that's the current market value of the portfolio, and now they're looking to hedge. They want to take the market risk off, and that's maybe because they're expecting a correction or whatever to come, so they just want to de-risk their portfolio. Now, question is, how can they do this? The perfect way to do it, let's put it this way, from a risk point of view, will of course be to sell all the different shares that you're holding on your cash portfolio and then go into cash.

However, depending on the number of stocks, and you don't necessarily want to do this, so you overlay your existing risk position with an offsetting risk position, i.e., you're going to sell some future contracts against this equity portfolio.

Because the theory is, if there's a correction, yes, your portfolio value is going to go down because the shares will become worth less, but the futures that you've sold, you can now buy back cheaper, and that will then offset the losses on your portfolio. And then the only question you have to answer is, how many contracts do I need to sell? So we have given you all the numbers here.

We know that we have $100 million to hedge.

We remember from the previous slides that each contract effectively is like selling $228,287.50.

So now we divide one by the other, and we get a number of 438 contracts now. Rounding.

I think it's relatively obvious what one would do here, but there's some residual risk that you might be willing to take.

Okay, so that'sThe easy way to calculate the hedge ratio here, just divide the notional that you want to hedge by the notional per futures contract.

Ignores one thing though, and that is that the constituents of this cash portfolio might deviate actually quite significantly from the index construction or the weights. So we might have much fewer companies.

We have maybe just selected a certain type of company with a certain type of risk behavior.

And so we're hedging now a portfolio that has parts of the index in it with the index itself, and that means there's not necessarily a perfect correlation between those two. And we now want to think about ways in how we can improve our hedge, at least conceptually. And one popular way of thinking about this is that you are just analyzing the beta of your portfolio, and you just basically compare how similar, in lack of better terms here, does my portfolio behave to the actual index that I'm looking at here? And that's a concept of beta.

It's comparing returns of the cash portfolio with the underlying index, and then just gives us an idea whether or not we're using in the same direction, which usually is the case, but then also a relatively higher volatility or lower volatility. So just a set of numbers here.

Beta of one means your portfolio moves in the same direction and moves in line with the broader market. So index goes up 1%, your portfolio gains 1% and vice versa. Beta of two, you're moving in the same direction, but at twice the speed. So index goes up 1%, your portfolio should gain 2%, but index goes down 1%, your portfolio should also be expected to lose 2%. Beta 0.5 is a little bit more defensive. You're moving in the same direction, but at half the speed, and that's the general way to interpret beta.

And now, just to include that in our example.

So let's say that the portfolio beta of this portfolio here of our $100 million portfolio has been calculated to be 1.2, which means we're moving in the same direction, but with 20% more volatility in a way.

And that means when the index goes down 1%, then we would expect our portfolio to go down by 1.2%. And then if that is a relationship that we think is established, we should adjust our hedge accordingly. And the way we do this is we're taking our 100 million portfolio value divided by the notional of one contract, but then multiplied with 1.2.

So effectively what we're doing here, we're hedging 100 million cash portfolio with 120 million worth of futures.

That's 1.2 times the notional. And so if we're now having the situation where the index declines by 1%, we get 1% of 120 million, that's 1.2 million. And if we then at the same time see our cash portfolio declining 1.2%, then that's 1.2 million as well. And then the hedge should work out, and so we have to now sell a slightly higher amount of futures, one point times two times the amount to be precise, and then the hedge should work.

The big thing to be aware of here, though, is that portfolio betas are calculated using obviously historical analysis.

We're using data from the previous months, years, whatever.

And so we're making the assumption here that this relationship that our portfolio has shown relatively to the index in the past will continue in the future. And that is, of course, not guaranteed.

So there is still a little bit of risk then to be aware of. And that then brings us to wrap this up, this section here of futures to advantages and considerations.

We've talked about some of the advantages already.

Extremely high liquidity, that means narrow bid-offer spreads, deep markets, i.e. large transactions possible.

We have transparency in terms of where the price is.

We see we have the order books visible.

We have also practically zero counterparty risk because this is centrally cleared with variation margin, initial margin, et cetera. And that's an advantage we haven't really mentioned specifically, but I'm sure you're all aware of this.

They have quite significant leverage.

So if you were to invest or to buy one futures contract that's worth 287,000 or 228,000 something, the margins that you would have to post down the initial margin would have been a very small fraction of this amount. There are, however, also some considerations.

The biggest one I think we have just discussed, and that is that there may be as well a difference between the portfolio constituents and the index you're using for hedging.

And if that is the case, then of course it becomes relevant what your beta is, and that beta might not be stable. There is a little bit of interest rate and dividend exposure, as we have discussed at the beginning. And then there's also a little bit of other risk involved in terms of basis and roll. The future roll we have already discussed, and that is when your contract comes to expiry and you want to maintain your hedge, you will have to close the front month's position and then open the following contract. And that is obviously something that could result in costs if there are either unfortunate circumstances or just not transacted carefully. So there are, of course, a couple of things to keep in mind.But I wanted to use the last 10 minutes of this session to just give you a brief intro to equity swaps.

And loosely speaking, the definition of an equity swap is relatively straightforward.

It's an agreement between two counterparties to swap the return of an equity index or a custom basket, or even a single stock for an interest rate. This could be one of the near risk-free rates like SOFR, SONIA, et cetera.

Sorry, TONA, not SONIA.

Or it could even be an IBOR rate like EURIBOR plus minus a spread. And that's not too dissimilar from a interest rate swap fixed to floating in a way, or generally swap means we're exchanging two streams of cash flows.

But of course, what we're exchanging here differs rather significantly. So let's have a quick look at this.

And this definition is accurate, but I think it may not necessarily be very helpful in understanding what exactly is happening here. So I would suggest think about it as follows.

The equity receiver, so this party here that we refer to as the equity receiver on an equity swap, usually through the equity swap is put in a position that's almost identical to the economics of being long a stock. Right? And now think about it. When you buy a stock, what is your risk that you're taking? Well, you're losing money when the stock price goes down on a mark-to-market basis. Your income sources are increase in share prices, and of course, also dividends.

So that's basically the economic exposure that I just claimed is going to be created by this equity swap. So let's just look at how this works. So we have these two counterparties, equity receiver, equity payer, and there's an agreement between them. The equity receiver agrees to pay a reference rate plus a spread, and in return, will receive the equity upside and the dividends and will also have to pay the equity downside. Lots of boxes and arrows, but conceptually, it's actually relatively straightforward. So let's assume we're a market maker and we're taking the side of the equity payer. What is it that we have to do? If the share price goes up over the term of the swap, we have to pay the upside.

If there's a dividend payment, we have to pay the dividend to our counterparty. So how can we make sure that we get the equity upside delivered to us, and that we also get the dividends? Well, we can again buy shares in the spot market, right? What we have to do again is we have to borrow the money, and therefore, we would incur interest rate cost. This time, we're not baking this into the price, but we're actually taking this into the contract, and we say to the equity receiver, "You know what? You just pay us a certain reference rate here for every day that you're in this swap plus a spread, and then we just calculate that on regular intervals how much money you owe me, and then you pay me after three months," or something like that.

So we borrow money, and then we charge the costs onto the equity receiver, basically. So now what do we have? We have bought the share spot.

If now the share price goes up, we can sell them at a higher price.

That gives us the equity upside that we need to pay to the equity receiver.

If the stocks pay dividends, we get the dividend because we're holding the stock, so we can pass them on to the equity receiver.

We needed to borrow money for being in this position, and the good news is that we get the interest that this will basically, or the interest cost that this will create for us from the equity receiver. The only thing we haven't explicitly talked about is what happens when the share price goes down.

Well, we have bought them at a certain spot price.

If we can only sell them at a lower price, that means a loss for us.

But we don't have to worry about this because this equity downside is what the equity receiver is actually going to pay to us.

And so what have we achieved? The equity receiver gets dividends.

The equity receiver benefits when the share price goes up and loses when the share price goes down. That is economically like being long the stock. The key difference between the equity swap and the forward, as we discussed, is here that we are not baking the dividends into some sort of forward price using some dividend assumptions, but here the actual dividends will be transferred from equity payer to equity receiver. So the one thing that is different between equity swap and just physically being long the stock is that an equity swap doesn't give you physical ownership of the stocks, or legal ownership of the stocks, and that means you won't have the voting rights that normally come when you buy a stock in the stock market. Right? And that's it.

And that makes it a very versatile tool.

So now we can obviously also we should mention that clients cannot just take the position of an equity receiver, they can take the side of an equity payer as well.

Then the dynamic changes a little bit.

We have to sell short the stock, we need to borrow the stocks, we have to pay borrowing fees, and so on and so forth.

But generally, the idea remains more or less the same. And equity swaps are also OTC instruments, meaning we need to sort of discuss some key aspects of the trade here before agreeing on the reference rate and spread and things of that nature.

So first thing is, of course, the notional, right? What's the underlying amount of money? What's the risk we're taking here? How many shares orHow many millions in notional? Because we're calculating the equity upside and also the interest rate in percentages, and to turn that into an actual cash flow, we need to calculate, or we need to have a payment calculation basis to start with. We need to discuss the tenor.

For how long should this swap basically run? Is it a year? Is it five years? Anything like that. One thing to point out here is that equity swaps usually have the mutual right of early termination, meaning even if the swap was originally agreed to be one year in length, both counterparties can say at literally any point in time, "Okay, let's close this thing," and just exchange all the payment, all the return components that have accrued up to this specific point in time.

Then in theory, there's different types of return swap. The vast majority of equity swaps is trading on a total return basis, meaning price, return, and dividends. In theory, what we could do is just trade a price return swap, then saying if the equity receiver would say, "No, I don't need dividends, you can keep them." So they are basically willing to give up this cash flow, but of course, they expect something in return, so we would probably have a discount to give on the reference rate then in exchange for that.

I don't think it's very common to do that, so stick with the total return example. We then have to think about the reference rate.

Is it SOFR? Is it EURIBOR? Whatever it is, how often will we reset it, and all those wonderful things, and what is the spread that actually the bank will charge for the service? And then how often do we want to reset? That means how often do we actually meet? If that was a 12-month swap here, how frequently do we want to meet and agree on how much interest rate has accrued, and how much equity upside, and how many dividends, and then exchange those payments, and then basically reset the swap back to zero value. That could be done at the end. That could be done multiple times during the trade.

That is obviously a question also on counterparty exposure, et cetera.

That's for another day. And the same topic, the margin reset is also thinking about counterparty exposure, and that is the initial margin.

Should we just base this on the initial notional amount, or if something happens over the next 12 months and the share price goes up in a meaningful way, should we maybe reset the initial margin to a higher amount? So those are just a couple of things that need to be discussed. So obviously a very interesting instrument to have a closer look at.

Thank you so much for your time and participation today and for all those great questions.

I tried my best to answer all of them as I was going along. But thank you so much.

I hope you found it beneficial, and have a good rest of your Friday. That's all I wanted to share with you here today.

And have a great weekend, take care of yourselves, and I hope to see you again very, very soon on one of our sessions.

Take care for now. Bye-bye.