Good morning, good afternoon, potentially good evening, and very warm welcome to this Felix Live session on option mechanics.

My name is Thomas Carswell, and I have the honor to take you through this session over the next 60 minutes. And then, of course, one of the first questions everybody always has at the beginning is what exactly are we going to talk about? Now, we are going to take a closer look at the world of financial options here today, and that means we're going to introduce the main option terminology. We are going to talk a little bit about the moneyness, the different exercise styles that exist, and then we will take an intuitive look at the option premium. You won't see a Black-Scholes or a binomial tree here, but we want to make sure you have a solid understanding of what the principal drivers of the option premium actually are. And then if we still have some time left at the end, that of course depends a little bit on the questions that you send in, we will also look at some of the more common option strategies, but only scratching the surface here, of course, because there's quite a few of different use cases and therefore different strategies that you will find in the reality.

We're going to start with a quick review of the terminology here that financial options generally come with, and here the classic definition of what a financial option is, is on the slide here. That is a contract that gives the buyer of this contract the right to either buy the underlying asset, then it's called the call option, or to sell the underlying asset, that refers to a put option.

And they can sell the underlying at a predetermined level.

That's what we call the strike price or just the strike.

And this right is obviously limited in terms of time you'll have it for, and that means that this right needs to be exercised at or before a specified date in the future, and that's what we call the expiry date.

And so if we basically look at the example here at the bottom, we have given you a 12 months call option, the $105 strike, and the premium was $7.90. Now let's hold back on the premium for a while.

We just kind of want to dissect the first part, two points here.

What do we have if we bought that call option? Well, basically for the next 12 months, we now have the right to buy the underlying asset, whatever that is, let's say it's just some stock, at a price of $105. So then the question is, when are we going to use this right? And let's fast-forward 12 months to the expiry date of the option and just have a look at different scenarios. So if we were now in 12 months' time realizing that the price of the asset, the spot price of the asset is 105, we're sort of indifferent as the option holder whether or not we're going to exercise it because the option gives us the right to buy the stock at a price of 105. The market price is also 105, so the option doesn't give us an economic advantage. It doesn't give us a disadvantage, so basically from a value point of view, the value of the option then is zero.

If, however, the underlying price was at 110, then of course that then exchanged quite dramatically because now we have the right to buy the asset at 105, but it trades at 110 in the spot market.

So depending on the type of settlement here, physical versus cash, but stick with physical settlement here for simplicity, what we could do is exercise the option, buy the stock through the option for a price of $105, and then sell it immediately at 110, and that would give us then obviously a profit of $5. I said we're ignoring the option premium for a moment here, so let's just focus on these red dots rather than the hockey stick that you can already see here, right? And I'm removing this one. And then at 115, you can do the same calculation, right? We buy at 105, we sell at 115, that gives us 10 ignoring transaction costs.

At 20, we're getting 15, and so on and so forth.

We could extend this line as far as we want all the way up to infinity, right? Now let's look at what happens, though, when the underlying price is below the agreed-upon strike price, and that, of course, would be an example of strike price or underlying price being at 100. So what happens? You have the right to buy the asset at 105.

It does trade at a $5 lower level in the spot market, so that basically means we're not going to use this right because using the right would put us in an economically worse situation than just letting the right expire, i.e.

in this case, we're simply going to move away, meaning the option has zero value for us, and the same then at even lower prices.

So that's basically the red dot symbolized what is the value of the option contract, the intrinsic value. We're going to talk a little bit more about that in a minute, but the immediate value of the exercise right at that particular point in time. So, and the fundamental difference between options and the Delta One world, i.e.

forwards, if we just kind of use it as a direct comparison tool here, is that the option only gives you the right to do something when you buy it, but not the obligation. And that's kind of like where we can very nicely visualize the difference between an option and a forward contract.

So let's say that we, instead of going long a call with a $105 strike, we entered into a forward, and that forward was struck at a forward price at 105 as well. So basically, instead of having the right to buy the underlying at 105, we now have the obligation to buy the right.

And let's use blue to draw the P&L profile of such a transaction.

And of course, at 105, we still have zero P&L because we're buying at 105, the asset trades at 105 is zero value. But at 110, we're also having $5 profit because we're buying at 105, selling at 110.5, and so we can extend this.

So basically, this blue dotted line gives us the P&L of the forward to the upside.

The fundamental difference, however, between forward and option now comes when we look at the left-hand side, i.e. what happens when the price of the asset has declined to 100, for example. Well, in case of a forward, we do not have the right to walk away. We have to buy the shares at the agreed-upon forward price.

So we will then buy something at 105 that trades at 100 in the spot market, and therefore we're generating a loss.

So if you now compare the blue dotted line with the red dots, something becomes relatively obvious, and that says that the option with not taking the premium into consideration would have a clear advantage over the forward because you have the same potential profits, but you have zero potential losses.

And that is then why intuitively the option cannot trade, or cannot come for free. And that's exactly the idea of the option premium here conceptually, right? Because there simply wouldn't be a forward market if you could buy this red-dotted call option here for free, right? Why do you ever take a forward position? But if you kind of turn this around, who would ever sell an option if there's no scenario in the world where you would earn some profits, but there's plenty of scenarios where you can lose a substantial amount of money.

That is extremely undesirable from a risk-reward point of view.

So there wouldn't be option sellers if there wouldn't be any sort of compensation for the risk asymmetry that this red-dotted P&L profile here really shows. And this compensation for this undesirable position to be short an option on the face of it is basically then the option premium. So the option premium here was given with $7.90, and what this then means, basically, we have to adjust all our red dots here downwards by $7.90 because in every scenario, what we haven't considered is the $7.90 upfront payment.

We're ignoring time value of money here just for simplicity, but you get the idea, right? So basically, this needs to be adjusted downwards by $7.90. That is then basically the option premium that we paid upfront. So what we could understand the option premium as is basically, as I said, the compensation for the option seller for taking this asymmetric risk profile here, i.e. there's a limited amount of money that the option seller could make, and there's, in theory here, in case of short call, an unlimited amount of or an unlimited loss that the option seller could face, and the option premium ...

is basically adjusting for that. So if you now compare the dark solid line here with our forward, which I'm just now drawing in as a solid line as well, what you do realize that there are scenarios where the forward outperforms the option, but there's also scenarios where the option is a much better choice than the forward, right? And that is now basically the explanation as to why there are both contracts, forwards and options.

It really depends, if you just look at these products in isolation, what your risk appetite is, how much you want to be protected to the downside, et cetera.

So we start to understand what the option premium should reflect.

Before we go there in more detail, let's have a quick look at those four general positions that one could take. We looked at the long call, i.e., we bought the call, the right to buy the asset, and we had $105 strike, $7.90 premium, and therefore we understand why this hockey stick, the green line looks like it does. And then basically to derive the P&L profile or the hockey stick of the short call position, i.e., thinking about the seller of the option, all we have to do is basically mirror this horizontally at the zero line because basically the buyer's profits are the seller's losses and vice versa, right? So basically from a seller's point of view, the best outcome is here, obviously, that they keep the premium of $7.90.

Worst outcome is they lose a substantial amount of money depending on where the underlying price is at expiry. Now, of course, this assumes no hedging whatsoever.

So this would be just a naked short call payoff, right? We sold the call, we haven't done anything to mitigate the risk, and then the market goes up, we're losing substantial amounts of money, which of course in reality from a market maker's perspective is hardly ever going to happen because we are very actively addressing those market risks that we're taking then.

But anyway, just for simplicity, we assume no hedging.

And then the other two positions one can take in the option market is, of course, you can buy the right to sell the asset, which basically now means you're taking the opposite directional risk because the right to sell something at a pre-agreed price is basically giving you protection to the downside, meaning that the option contract itself is going to have positive value when the market turns lower. So we're now sort of looking at $105 strike put, right? We're still looking at 12 months here, and we're assuming for simplicity that premium was also $7.90 here, and then you see pretty much the same principle. We have $7.90 maximum loss.

That is when the underlying price at expiry exceeds the strike level or matches the strike level because then if I have the right to sell at 105 and the spot market pays me 110, I'm simply not going to use my right to sell, right? I let the right expire and I sell in the spot market and get $5 more.

I paid $7.90 to get the option in the first place, and so I'm realizing a loss of $7.90. However, the moment the underlying price drops below the strike level, I will start exercising my option.

So if I'm just taking one random number here, say let's look at the payout, or if the P&L at $90. Basically, the way this works is we're making from exercising the option that gives us $15, right? We have the right to sell at 105.

The market is trading at 90. That means we have a $15 economic advantage here.

We have to also consider that we paid $7.90 for it.

So basically $7.10 would be the P&L here.

So just putting that there, $7.10.

And so it's more or less the exact same than the long call, but this time we'd have just to mirror it along the vertical at the strike level because it's from a directional point of view, the exact opposite risk that we're taking, right? And then we talk about how to get from a long option position to the short option position in terms of payoff. So we have to once again mirror it horizontally on the zero line because also for a put the buyer's profits are the seller's losses and vice versa. So that are the four general option positions, and that is almost giving us complete coverage of the key terminology there is in option world.

One thing that's missed or two things missing are the option moneyness and then also the different exercise styles.

So, let's start with moneyness because that's often used in markets to not necessarily describe specific strike levels, but categories of strike levels.

So, when we're just communicating what's happened in the option world, then very often we don't communicate in specific price terms, but we say that in-the-money options were in high demand or out-of-the-money options saw their prices being lifted or whatever it is that you hear.

And so then it's really useful to understand what does in-the-money, at-the-money, out-of-the-money means. And I think the good news here is on the face of it at least, it's relatively straightforward because all you need to know to answer or to find if an option is in-the-money, at-the-money, out-of-the-money is does it under current circumstances make sense to exercise the option or not? If the answer is yes, then we call this option an in-the-money option.

And basically, coming back to this original definition, all that means is if the market price of the underlying stays where it is until expiry, this option would be worth exercising. So example here is you have 105 strike put and the underlying trades at 90. So that means you have the right to sell at 105, the underlying price is 90. Under this circumstances, you will certainly exercise the option. It's an in-the-money strike.

At-the-money is then basically when you're indifferent.

So that is usually the case when the strike equals the current market price, $105 strike call, underlying trades at 105 or $105 strike put underlying trades at 105. Both cases you're indifferent because the option itself doesn't give you any economic value. So that would be an at-the-money option.

An out-of-the-money option, however, is then basically the case when the option will not be exercised under current circumstances because it would economically be a disadvantage. So example here, you have 105 strike call and the underlying trades at 90. So why would you use the right to buy at 105 if the asset is available for $90, right? That clearly doesn't make sense.

You would pay $15 more than you have to.

This option will not be exercised under those circumstances, and that is then qualifying it as out-of-the-money. So these are the terms.

Now there's a little bit more complexity to it, though, and I've so far just said market price here on the slides, and this is deliberately vague because we have not really identified which market price precisely because there are different ways of looking at this.

Before we do that, I would like to go and talk briefly about the option exercise styles, and then we're coming back to the previous one.

And this is sort of also sometimes known as the option geography because for obvious reasons, right, we call those or we distinguish here the American from the European and then also in some asset classes for rates for example, we also have the mutant option. So what does that all mean? American option is arguably the most flexible type because it gives the buyer the right to exercise at any point in time between basically the time they bought the option and the expiry of the option, right? So if you buy a 12-month American call, for example, then every day over the next 12 months business days that is, you have the right to exercise the option.

Points to remember though, it can only be exercised once, and once you exercise, you basically can no longer benefit from The option contract.

And the second point worth noting is that, of course, rather than exercising an option, you can always sell it. So you bought an option, it has gone up in value, you will be able to sell it, and through that, monetizing your P&L there. Okay, so the next one, the European option, on the face of it offers a lot less flexibility, and that is because this type of option can only be exercised at the expiry date.

So 12-months European option means you cannot exercise it at any day.

You can only exercise it at the expiry date, which is 12 months away.

But, and this is why it may actually sound worse than it is in direct comparison to the American option, you can, of course, sell an European option before expiry as well. And then Bermudan options, I'm not saying that's sort of in the middle because it isn't, but if you look on the world map, you kind of sense that Bermuda sits between Europe and America, and therefore, the option exercise style does the same. It's kind of a mix of the two, because a Bermudan option can be exercised on more than one date, i.e., not just on the expiry date, but it cannot be exercised on any specific or at any point in time either.

There are specific points of the life of the option on which the option could be exercised. Could be a 12 months Bermudan option that can be exercised quarterly, so at the end of the quarterly period, so end of the first three months, second three months, et cetera.

That is then from a flexibility point of view, somewhere in between.

But same like American option, it can only be exercised once, and you can no longer benefit from any price moves if you had exercised it.

And of course, it can be sold as well prior to expiry.

So with that knowledge, now let's go back to the option moneyness because what hopefully has become clear is that we need to be a little bit careful when we're determining the moneyness because of the different exercise styles.

So here's the situation. Let's assume we have a counterpart calling us and they ask for a 12 months call option with an at-the-money strike.

And then the question, of course, is what is that strike level now going to be that we're going to use for pricing this particular option? So it depends a little bit, I guess, on what the option exercise style here would be.

If it was an American option that the client has asked for, you could argue that the best price to compare the strike with and to determine what's the at-the-money strike would be, in fact, the spot price.

Because that's where the market currently is, and that will sort of determine whether at the moment, you as a holder of an American option would exercise the option or not.

And so there you could say, okay, this is an American option, so the at-the-money strike should be set at the current spot price of the asset. If this was a European option, you could argue, well, a European option cannot be exercised right now.

It can only be exercised in 12 months' time, and the best risk-free recreatable sort of price I have for a 12 months in the future is a 12-month forward price. So then you could say, I should not, when determining the moneyness of the option, compare the spot price and the strike.

I should compare the strike with the forward price.

And if the forward price in our example was 105, then for me, if I have that way of looking at things, the at-the-money strike would be 105.

So let's say I'm pricing this with $105 strike, and then we never in our conversation here between the two counterparts explicitly determine the strike level, and then my counterpart believes they bought a $100 strike option. I believe I sold a $105 strike option, and of course, these two rights are not perfectly aligned. And when we're exchanging trade confirmations, then it will turn out that we haven't traded the same thing, and then, of course, it becomes somewhat tricky. So I highly warn against just using this conventional terminology for basically real trade discussions.

Take these two seconds to confirm this strike in numbers.

So a client might call you and say, "I want an at-the-money strike." And you say, "Okay, I see the at-the-money strike currently at..." And then you take the 100 or 105, depending on what exercise style it is or what's the convention in your market is that obviously is also an important nuance there.

But generally sort of confirm back, I see the strike at 100 or at 105, and if the client or your counterparty then agrees to that, then at least it's been confirmed and we're in a safe environment here.

So keep that in mind because an option strike might be, as in this example here, if the strike is 105, on a forward basis, that's at-the-money, but on a spot basis, that's out-of-the-money. And so here it does make sense to just add the words spot and forward to describe that in a little bit more detail, just to give some clarity there. Okay, we talked about this.

Now time to look at the premium, and here I want to talk about the fundamental concept.

And I think we already laid the foundation here.

We already introduced the fact that the reason why the option premium exists is that option payoff profiles are fundamentally asymmetrical.

So as it says here then on the sub-bullet, basically, this means that the risk and opportunities are very different when you compare them across buyers and sellers. And so without the option premium existing, everybody would love to be long options, but nobody would ever go short an option, so the option market would simply not exist.

And therefore, the premium is there to make or take care of this risk asymmetry, and compensate, obviously, the option seller for taking that in other ways, less preferable position.

Okay, now that's the reason why the option premium exists.

Now let's think about what the option premium conceptually should cover. And here, I think it makes sense to understand the option premium as sort of the expected loss of the option seller.

So basically, if I wanted to now come up with an option premium, I have to answer these two questions, because that's what I need to determine my expected loss. And the two questions are, what are the chances that the option will be exercised against me when I'm selling it? And if it's exercised, then how much do I have to pay? And I have the probability that I have to pay times what I have to pay gives me my expected loss. So to bring this into a very simplistic framework is where we can start with a coin flip or coin toss.

So I'm taking a £1 coin, and I will invite you to this game here to guess the outcome when I flip this coin.

And then, of course, there is, and this is at least common belief, two outcomes, heads and tails. There is, of course, also the possibility, which is not zero, that it lands on the side, but let's leave that aside here. That's definitely a fair tail outcome here.

So anyway, heads and tails. And let's assume that £1 coins are sort of designed in a way that their weight distribution is in a way that there's no preference here from a static point of view for the coin to land on heads or tails. So that would mean we have a 50/50 chance of both outcomes to appear. And if I now say the game works as follows I flip the coin, you tell me what is going to be heads or tails, and if you're right, you keep the pound. Then the question is, how much should I charge you for that? So remember, that is basically thinking about my expected loss, and then I need to answer the two questions here.

What is the probability that the option will be exercised, i.e.

what is the probability that you get the outcome right? And we said that's a 50/50 chance here, so you have a 50% chance of being right.

And if you're right, I pay you one pound, basically meaning my expected loss is half a pound or 50p. Right? And so that's basically what I should charge you upfront to play this game. And then if we play this game an infinite number of times, law of large numbers suggests half the time you would win, right, and you paid me 50p to play the game. Then you get £1 from me because you won the game. So that means half the time you're pocketing net 50 pence.

The other half of the time, you get the outcome wrong and I keep your 50p and I pocket 50p net. So none of us will be better off in the long run.

That's a sort of conceptual starting point, right? Answering these two questions.

And that works well for a coin toss because there's only really, I could argue with two or three, but let's not go there. There's a very limited number of outcomes.

Okay? If you now extend this context or concept into finance, very easy to see that complexity increases extremely fast because if we're now moving away from coin toss to stock prices, then the only thing we can really say with absolute certainty about a stock price at any point in the future is that it's going to be somewhere between zero and infinity. That's all. Right? Nothing more with 100% certainty.

And that of course, means we cannot even finish conceptually to determine the number of possible outcomes, right? Some of them may feel that it doesn't really make a lot of sense because if this is like a two-day time horizon we're looking at, it's very unlikely that Apple is going to trade at infinity tomorrow.

But also, we cannot necessarily say it is impossible for it to happen if you get the idea at least. So that is then the real challenge.

And so what we need to then answer these two questions is obviously we need to come up with ways that we can sort of apply some useful tricks there that allow us to answer these two questions, and that's where the option pricing models comes in, right? That's really what Black-Scholes and binomial trees and Monte Carlo simulations are really all about. They're trying to find or to use statistical techniques to help us find the answers to those two questions and then wrap it up in an option price. So that I think is really the conceptual idea behind option pricing. Now that is of course an extremely complex subject or at least can become very complex soon. And Nobel prizes have been given away for coming up with models which are now seen as maybe not all encumbent, but still being a useful guidepost there for option valuation.

So we're not going further into the pricing aspect, but as I said, what I want everybody to take away from here is to have a good understanding on when would an option premium be high and when would it be low.

And of course, conceptually, we are already there because for a high option premium, that means my expected loss needs to be high, and this could be driven by a high likelihood that the option is going to be exercised.

It could be also driven by a high amount of money that I need to pay when the option is exercised. And of course, high values of both, that's the ultimate pain.

So then the expected loss will be very high.

But it does make sense, I think, to get a bit of a healthier understanding what drives now both answers to these questions and what drives option premium is to look at a concept that is very well established out there and is taught a lot as well. And that is we're looking at the option premium really as a sum of two parts. And there's the intrinsic value and there's the time value of an option.

And the intrinsic value, I would argue, is relatively simple and intuitive to understand because what we're looking at here is basically the value the option currently has under current market circumstances.

So when you look at American options, for example, because it's a bit easier to conceptualize that. So let's say we have a 12 months American call, and the strike is 105 and the spot price is 110. Then of course the question is what is the current economic value of the option when exercised, right? And here in an American call option, I can exercise immediately, meaning I can buy the asset through the option at 105, I can sell it in the spot market at 110.

We're ignoring transaction costs for simplicity.

So the intrinsic value here-- Well, let me just spell this.

Intrinsic value is $5, right? So that's the idea.

We don't have to PV anything here because that would be immediate realized profit. So that is something we would gain right now by exercising the option right here, right now.

Now, what is worth saying, the reason why it says present value here is that in case of a European option, you cannot really compare the strike with the spot.

You would have to compare the strike with the forward, and then you would have to discount that difference back to today, so the PV is just capturing that part as well. Important also to note is that the intrinsic value is only there when the option is in the money.

If we flip this, right, the strike is 110 and the spot is 105, that doesn't mean the intrinsic value is -5.

That means the intrinsic value is zero. Why? Because we have the right to buy the asset, and under these circumstances, spot lower than strike, we're simply not going to exercise the option.

We're going to walk away. So the intrinsic value is floored at zero from the buyer's perspective, but it can be of course, positive dependent on the relationship between or the distance between the strike and the spot price.

So at the money and out of the money options have an intrinsic value of zero.

In the money options have an intrinsic value of non-zero as we've calculated above for our American option. You've seen it's relatively straightforward.

And the importance of the intrinsic value is, and I think again that is hopefully intuitive, it's a minimum price boundary for an option, right? So if in our case here the American call had an intrinsic value of $5, we will not get it below $5 in the market, right? It would not trade below $5. So if it was trading at $3, for example, if you could buy this call option for $3, we would all do this, right? We would all buy the call, i.e., the right to buy at 105, then exercise immediately and sell the asset at 110, capturing $5 profit. We paid $3 premium, so $2 risk-free return. We'll buy as many options as we can, exercise and sell the stock.

And this would then of course drive up the premium of the call option and potentially also the spot price would be pushed down until these two things are aligned, i.e., the intrinsic value matches the option premium at least.

So that is why you can see the intrinsic value really as the minimum price an option should trade at. But not only will the option not just trade below its intrinsic value, it also is very unlikely that an option trades exactly at the level of its intrinsic value, especially when there's still a reasonable amount of time left to expiry because there is a second component and that's called the time value. And of course, conceptually, I think it's relatively straightforward, right? You sell, let's say it was 12 months, we're going back to our 12 months American call And the strike is 105, we said, and the spot is 110. And so the intrinsic value would be the $5 we had. But you won't sell this option at $5 because there's still 12 months left. Right? And over the next 12 months, all sorts of things can happen. And this is now where I think there's often a little bit of confusion as to what the time value really represents.

And most commonly, from an intuitive point of view, it's explained to as, well, if you're selling this option, you do have the risk that the spot price goes up and that you actually end up paying more than the $5 intrinsic value.

And so you should charge a little bit more to have some sort of a cushion there.

And while this is wonderfully intuitive and also common sense, if you like, I don't think it's necessarily really the best way of looking at time value, because as we will see on the next slide, there's a behavior of the time value that we cannot really explain with this argument.

So I think, therefore, it should be, or it's a better way of thinking about time value as the compensation that the seller of the option will ask for, for taking uncertainty whether or not the option is going to be exercised.

And now the question is, why is that something that the seller needs to be compensated for? Well, and I'm just going to go and open a board here, which is giving me just a little bit more space to write. So, if we think about-- and actually, as I'm using Zoom now, I need to actually change the screen I'm sharing, so let me do that. Okay.

So the point is, right, if we compare the option with a forward, then it becomes relatively clear. So, let's say we have sold the stock 12 months forward. Right? And that was 105. Let's say our spot price is 112 months interest rates are 5%, and there's no dividends paid by this underlying stock.

So then the fair forward price is 105 because all we have to do to hedge this position, not saying that's always what we do, but we could step one, buy spot at 100.

Second step, borrow the money at 5% for 12 months, and then in 3 months pay back-- Sorry, in 12 months. Not sure where the three months came from.

Pay 105 to lender, and also in 12 months, receive 105 from counterpart. So basically, we are perfectly hedged here. If the asset now goes up to infinity, it doesn't matter.

We bought the stock already. We're going to deliver to our counterpart.

They give us 105. We pay back to the lender.

If the stock has crashed to zero, we're still going to sell 105 to our counterparty, and we're assuming counterparty, or we're excluding counterparty risk in our thought process. So of course, it's not always that easy.

But generally speaking, from a marketer's perspective, we're hedged. Right? So no matter which outcome happens, we're hedged, and therefore forwards are perfectly hedgeable. Now, again, if we do this or not, that is everybody else's decision, but conceptually it can be done.

So this thing is relatively easy to hedge. Now let's switch to a call option.

Right? 12 months call that we sold, and let's say the strike is 105.

So again, now think about what's our pay trade here, right? So if we have sold this call, obviously, the risk is in 12 months, spot above 105. Okay? Because then we have to deliver the shares to whoever the call buyer is, and we will get only in over the commerce $105. So how could we potentially hedge? Well, I could, for example, buy-- Let's say we have sold 100 call options.

I could buy 100 shares 12 months forward.

And as we just said, the fair forward price was 105. Right? So then basically what I have done is I have hedged a call with a forward and say, "I'm going to buy 100 shares at 105 in 12 months time, and then if the market has gone up, I have no problem because I bought the shares at the price that my option buyer is going to pay me, and that was a strike of 105." The problem is, though, that while this protects us against rising prices, the issue is what happens when the price in 12 months is actually below 105.

So let's say price in 12 months at 90. So now the following happens. The option is not going to be exercised.

That means we're not selling our 100 shares at 105 into the option. Some might argue, well, okay, nothing to worry about because we still have shares, we can now sell them in the market.

But the problem is, all we get now is 90.

But we're paying 105 for it through the forward contract because that's not a right to do something, that is the obligation.

So we have to buy the shares at 105, and then we sell them at market prices of 90, and that means we have lost $15 per share from our hedge.

And that, of course, is not really sounding like a good hedge.

So the problem with hedging an option is not really the risk that we're ending up in the money, deep in the money, but it's the risk that coming from the fact that we do not know whether we're going to end up in the money or at the money or out of the money.

We don't know if we need the shares.

And that makes hedging then conceptually very difficult.

And I don't want to go into delta hedging because we only have an hour and we're going to run another session on this, so look forward to that.

But the idea is hedging an option will sort of require a dynamic approach. We cannot just when we sold 100 options, go and buy 100 shares.

That's not going to work. We need to think about what's the likelihood of us needing those shares, and then we're just going to put some sort of fractional hedge in place, and we're going to dynamically adjust this.

But the dynamic of that is obviously we have plenty of buying and selling to do, which of course is costly, and that basically then means that we should charge for this risk that we're taking because we do not know if the option will be exercised or not. Okay, so that hopefully paints an intuitive picture here for the time value. And then with that in mind, then we have to think about where does uncertainty, whether or not this thing is going to be exercised really high under which circumstances.

And that is again, then I think, relatively intuitive.

The first thing is, of course, time matters. Right? Over a one-month period or one-day period, I probably find it much easier to get an estimate whether or not an option will be exercised tomorrow than over a 10-year period, right? Because there's a lot more uncertainty over 10 years than there is over one day, at least arguably. The second thing is the volatility of the asset. Is it an asset that is very stable in price? Then there might not be that much uncertainty around whether or not this is going to be exercised. But if it's something that swings around quite a lot, then there's a good chance it will be exercised being deep in the money.

But there's also a good chance that it's going to be deep out of the money.

So there is just, again, higher uncertainty.

So time value from that understanding should be driven by time to expiry of the option and then, of course, by the volatility of the underlying asset. Right? And so this is a great starting point, and I would argue so far, this would have worked very well also with this argument that this should be for the risk that we might see intrinsic value going up.

But where this doesn't work anymore is when you look at the way option premium then actually behaves or option premium behaves.

So what I've done here is I have looked at a three months option Right. A three months call option was $100, right? And I have calculated this using, obviously, interest.

Well, interest rates were zero, I believe dividends were zero, so strike equals forward, that kind of stuff. And then I've basically calculated the value of the option at three different points in time across a range of scenarios.

So basically what we're doing here is we're shifting the time, i.e., we're starting the green line with the option premium at spot, so that's three months to expiry.

Then we're looking at this two months to expiry, so one month later, two months later, and then at expiry. So we're basically allowing time to pass, and we're also looking at the value of the option across different underlying prices.

So starting at 80, going all the way up to 120.

But let's say with at the money point now, and so here, basically what we see is that the three-month option has the highest price, two months option slightly lower, one month option even lower, and at expiry, the price is zero.

And that basically this movement here is basically representing the passage of time, right? So we're keeping volatility here unchanged.

That's something I need to probably say.

So we're not looking at a change in vol of the underlying.

We're just saying, okay, let's freeze everything and just allow time to go by.

And then what happens is the time to expiry shortens, and as time to expiry is one of the principal drivers of the time value, what we do see here is that the time value goes towards zero. Because remember, when the strike price is 100 and the underlying price or the forward price or the spot price is also 100, which it is because I've set interest rates and dividends to zero here, then there is no intrinsic value. Intrinsic value by definition is zero for an at the money option, as we said earlier. So the difference between the light blue line and the green line is purely time value. Also, at expiry, which is basically this here, the option does not have time value anymore simply because there's no time left, there's no uncertainty.

We know whether or not the option is exercised, we know the intrinsic value, and basically this blue line is, and light blue line, is only the intrinsic value of the option. Okay, so you might now say, well, this still works with the slightly limited version of a time value explanation, and you're right.

Just looking on things from an at the money point of view, this works.

But when it doesn't work, when we start looking at what happens when the option goes deep in the money. So we're looking at strike of 100, and now we have the underlying trading at $120. So the option is deep in the money.

And what we do see is that these lines here are starting to converge, right? So suddenly the premium of a three-month option is not really meaningfully higher than for a two months and for a one-month option, and even the option that is already at expiry has a very similar option premium.

So what seems to happen here is when the option moves deep in the money, the time value disappears. And that's kind of difficult to explain with things cannot get worse because I would argue, just because we're sitting at $120, it doesn't mean that over the next 12 months we cannot go to 125, right? There still is a risk that we can climb up further.

There's no cap, at least necessarily.

So that doesn't work.

And so this is where the simple way of looking at time value, in my mind, falls apart. But it works beautifully with our way of explaining.

And I would suggest think about extreme examples, right? Let's say we have an option that expires five minutes from now.

The strike is 100 and the underlying spot price is $10,000.

So let's think about this, right? The asset we have the right to buy at $400 currently trades at 10,000, and there's five minutes left to expiry.

Yes, legally, in the contract it says this is an option. We've bought the option.

We have the right to buy the asset at 100.

Practically though, we have the obligation to buy the asset at 100 because we would give away $9,900 per share for not doing this, right? And so we have the obligation to basically buy the asset in five minutes from now.

So while legally it's still an option, practically it's kind of a five-minute forward contract. We know we're going to exercise.

We just don't know how much money exactly we're going to make because that depends then on where the spot is the point of exercise, right? So that is the kind of thing that I want you to keep in mind.

An option is always an option from a legal point of view, but there are scenarios in the market under which an option isn't really optional anymore. It's the obligation.

Or if we go to the other side, if we do the other extreme example, we have the right to buy at 10,0000 and the underlying trades at 100, and the right expires in five minutes. That's nothing, right? This is not-- Yes, we still have the right to exercise, but we're not going to do it, right? Because why would we buy for 10,000 if we can get it for 100? That doesn't make sense, right? So then again, it's no uncertainty, right? This thing is simply not going to be exercised, so it doesn't have time value.

There's no uncertainty whether or not the option will be exercised.

And I think that then works reasonably well explaining the overall premium dynamics. And with that, not looking at any pricing model whatsoever, we already have built a decent understanding on what drives prices of plain vanilla puts and calls. So let's start with-- Or let's just kind of do this for the call first and see what should conceptually happen to the price of a call option when, for example, the spot price increases. Now, what does happen? Assuming, and we're just kind of taking the very intuitive look here, we have an in the money call option, and now the spot price increases even further.

That means the option now goes even deeper in the money, and then, of course, we know this will increase the intrinsic value, and that will bring us the option price of a call. Now, time to expiry.

That, of course, is usually not increasing over time because time goes only one way. But if you were to compare a three-month option with a six-month option, then of course, you would have guessed it right.

All else being equal, the six-month option should be more expensive than the three-month option, and that is basically the time value at work here.

Next thing, volatility. If volatility of the asset is higher than of another asset, that option should also be more expensive because volatility drives time value and therefore, no surprises here. And then, of course, interest rates and dividends, that has something to do with forward prices.

So, let's leave it here, but just very briefly, if interest rates go up, forward prices go up, and that would basically push up the moneyness for an European option and therefore would increase the intrinsic value and therefore lift the call price. You can think about it different ways, but I think that's the kind of easiest way. And so if the dividend goes up, forward prices come down.

So this is more like a function of the forward rather than the option itself.

Quickly on puts, just to wrap this up, and this is the final slide that we're going to do here.

Because we have the opposite directional risk on a put, as we have basically explained at the very beginning of this presentation, the relationship between spot price increase and impact on the option value is exactly the other way around.

So if a spot price increase increases the value of a call, it decreases the value of a put simply because it's less likely that we're going to exercise now.

But it also affects the intrinsic value.

Time to expire and volatility, though, have the exact same impact on call and put options simply because it's about time value.

It's not about the intrinsic value.

It's just about uncertainty. And in the same way that it does for calls, increase in time and increase in vol of the underlying will increase the uncertainty of the option seller with regards to will the option be exercised, i.e., do I need the shares or do I not need to hedge the underlying position. And that, ladies and gentlemen, is all I wanted to share with you here today. Thanks for all the questions that you have asked there.

I answered all of them along the way. Really appreciate your time here.

Hope you found it beneficial. And yeah, look forward to having you on board in our next couple of sessions. And please, please, please remember to just put in the feedback any follow-up questions you have or any other topic you would like to see covered. Always looking for ideas here. And that was it.

Enjoy your rest of your Friday. Have a fantastic weekend.

Take care of yourself, and see you hopefully very, very soon. Bye for now.