Hi, good morning, good afternoon, good evening everyone, and welcome to this Felix live session on Option Mechanics.

My name is Thomas Krauser, I'm head of financial products here at Financial Edge, and I have the honor to guide you through this introductory session today.

So let's have a quick look at the agenda.

We're going to start with an overall introduction to options.

We're clarifying some of the key terminology.

We're gonna look at the four standard option payoff profiles, and then we're going to take an intuitive look at the option premium.

So this is not gonna be about pricing formula, et cetera, but we want to evolve or develop this healthy understanding about what are the premium drivers and how premium should behave under certain circumstances.

And then towards the end, if time permits we might look at a couple of elementary option strategies as well.

But without further ado, let's get started.

And as I said, we're gonna start with the general definition and a lot of terminology clarification.

So let's start with the question, what actually financial options really are, and the classic definition of financial options you'll find here in the top right corner of the slide, those are financial instruments that give the buyer of this instruments. So the option buyer, the right to either buy the underlying asset or sell the underlying asset. These are two different types of options, right? The right to buy is given to us by a call option. So if we buy a call, we have the right to buy the asset.

If we buy a put on the other hand, we have the right to sell the asset.

And then there's a couple of conditions. Obviously we have the right to buy or sell the asset at a predetermined price that is agreed at the time, well, usually, at least at the time we trade the option that is what's referred to as the strike price.

And this right has a sort of time limit attached to this before because we can only really exercise this right either before or at a specified date in the future, and that's known as the expiry date. Those are two different types of options. We're gonna talk about that in a little while, but generally there is the attached time limit to this.

So if we are looking at an example and we're ignoring the premium for now just looking at these two boxes here, we have a 12 months call option, and the strike of this option is $105.

What does it mean? We have basically bought the right to buy whatever the underlying asset is. Let's assume it's a stock because it's the most intuitive asset class here.

So we can buy whatever the underlying stock is here at a price of $105 within the next 12 months.

So regarding, as I said, which type of option it is, we can either exercise this option any point in time over the next 12 months, or we can exercise it only once in at the expiry date in 12 months, as I said, a little bit more on that later.

But you see basically the in for, for this clarification of what an option contract really is.

This distinction is not really important for us right now.

So let's go through this and think about the options value, excluding the premium payment for now.

So we have the right to buy the underlying at $105 in 12 months time.

Let's say, let's have a look at what is the value, what would be the P&L ignoring the option premium B across different asset price scenarios. So let's start with the strike price.

Let's assume that in 12 months, the asset price actually trades at 105.

We now have a contract that gives us the right to buy the asset at 105, and the market price of the asset is also 105.

It's hopefully intuitive that this option doesn't really have any value attached to this because the price at which we can buy the asset through the option and the price through which we can buy the asset in the market is identical.

So there's no advantage, no disadvantage in the option. So the value of the option, that point will in fact be zero.

How does that change when the asset price has increased? So let's look at an asset price of 110, for example, we have the right to buy the asset at 105.

It trades now at 110 in the market, regardless if this is cash or physically settled, if we ignore transaction costs, hopefully it's intuitive to see that the value of the option is $5.

And you can think about it conceptually as a following.

We can buy the asset at $105 through the option.

We can immediately sell it at 110 at the available spot price that should ignoring transaction cost, bid offers, et cetera.

Leave us with $5 in our pocket.

So at an asset price of 110, 5 should be the value of the position, again, ignoring the option premium.

At 115, we can do the mass again, or we can just basically say, okay, 115 minus 105, that gives us a P&L of 10, and so on. So this is basically a linear increasing line.

What happens on the other side though, and this is where options fundamentally differ from all the other types of ERs, forwards, futures and swaps, and that basically is all driven by this one word here at the top.

Options give the buyer the right to do something, not the obligation, because forwards futures and swaps are all obligations for both counterparties involved.

If you agree to buy an asset on a forward basis, let's say at a forward price of $105, that means you will have to go through with the transaction If the price has increased beyond the forward price or if the forward if the spot price has declined below the forward price, that doesn't matter.

You will have to buy the asset or the derivative will be settled, against the price of 100 or forward price of 105.

Now, in the case of an option, if you buy it, you only have the right to do something, right? And so let's think about what we would do as the holder of this call option.

If the asset would trade at a price of let's say, 100 dollars in 12 months time.

So we have the right to buy the asset at 105, it trades at the same time in the market at spot at 100 dollars, would you use the right to buy the asset at 105? I believe not. You would let the option expire. You would simply walk away because there's no reason to pay 105 for an asset if you can buy at $5 cheaper in the market.

So at that point, you would simply walk away, you would take advantage of the fact that you don't have the obligation to buy the asset.

You just have to the right to do something. And of course, you will only ever exercise this, right, if in fact it was an advantage for you.

And so then we can say that basically all the prices to the left of the strike price here, because we're talking about a call option, would lead to us just walking away so we can draw the jotted line there. And so this dotted line is now effectively the value of the option, excluding any option premium here at expiry depending on where asset prices are.

And so just by looking at this, you now see as to why in compare or in contrast to forward future swaps, an option needs to have an option premium. Because if there was no adjustment by any means, then we would of course all really enjoy being long an option simply because the minimum or the worst case according to the red dotted line here would be a loss of zero.

And the best case is a quite considerable amount of profits, right? That should not be, or well, if that was in fact an instrument that was available, then of course there would be a lot of buyers for this instrument.

I would assume though, that you would struggle to find a seller.

And remember, if you wanna buy an option, somebody else needs to be willing to sell it.

And of course, if you look at this profile, who would be happy to take a position where the best outcome is zero and the worst outcome is quite a significant loss that doesn't really make any economic sense.

Hence, there is a requirement that you hopefully now intuitively agree for an upfront payment to convince any other counterparty to be willing to sell this option.

And here in the orange box, we just capture that in one sentence, and that is that risk and opportunities differ quite significantly for buyers and sellers, right? If we don't have the premium, then the buyer can lose nothing, can make a lot of money, the seller can make nothing and can lose a lot of money.

That's quite significant difference.

And the option premium basically compensates for this asymmetry, right? And so in this example, the premium was $7.90.

We're gonna talk about what drives this premium later on.

But for now, let's just take this as a give number.

And that basically means for us to get the 12 months $105 strike call option in the first place, we had to make a payment of $7.90. That's not a deposit that you will get back at the end, that's just the price you pay for this product option.

And regardless if you exercise the option later or not, these $7.90 will be gone at the end of the options live.

So what then hopefully makes sense is that in, you know, obviously considering the option premium and ignoring time value of money here for simplicity, we basically have to shift this P&L or value profile off the option down by $7.90. Because at a strike or at a good price of $105 in 12 months where we said we're indifferent, we're not really exercising the option, we we're not against it because the value of the option itself will be zero.

Yes, the value of the option will be zero, but remember we paid $7.90 for it to get the option in the first place.

Hence, we will actually from a P&L point of view have an impact of negative $7.90 because the option we bought has now expired.

We haven't generated any income through the option.

So $7.90 are gone. And then of course if we go and see what happens when the price of the asset goes up, 110 was the first stop earlier.

So we said at $110, the option value is five, right? Because you have the right to buy at 105, it's trading at 110.

So $5 could be extracted out of the optional contract, but remember we paid $7.90 for it.

So again, ignoring time value of money, the P&L and l would actually be negative of $2.90.

And then actually the break even point for this, particular option would be the strike price, 105 plus the option premium paid because that's what we need to earn back, and that would lead us to $112 and 90 cents.

So at that point, we're making $7.90, exercising the option that then basically nullifies our premium payment.

That's the breakeven point.

Anything beyond that would be a profit.

So basically by just making this adjustment by the premium, we're compensating the seller.

For this you know, slightly, not slightly for this actually quite different risk and reward profile, and that is basically incentivizing counterparties to be on the option seller side as well.

Okay, more on that, as I said later.

Let's have a quick look at the four general option positions that you can take.

Remember, there's two different types of options, at least in the conventional option space, right? We're ignoring the exotics here for now.

So there would be obviously the call option that you can either buy or sell, and then there's a put option that you can also buy or sell.

And the long call position here, upper left Corner, we have just basically looked at, right? So we have said here $7.90 premium, just putting it there one more time.

We have a strike of $105 and that was the reward or the P&L profile that we have sort of developed on the previous slide. And then to get from the long call profile to the short call profile is pretty straightforward because it's just the opposite, right? Because the, you know, winners or the bias profits need to be the sellers losses, vice versa.

And we're assuming here no hedging and this is just a single option position where no transaction has been made addition to that.

So we're just really taking an isolated look here. And then what's the P&L profile of this short call position? Well, we've sold the call.

We got the premium of $7.90 cents here to start with, and the best case for the option seller then without any hedges et cetera, would be that the option expires worthless.

That will happen at any asset price below and all the way up to $105.

Once the asset price surpasses or increases beyond $105, the option will be exercised.

That will result in a payment that the option seller has to make to the option buyer.

That's of course not gonna lead to losses immediately because the option seller has re received the option premium that they can afford to lose before the P&L actually turns negative.

But again, break even point $112 and 90 cents here, that's when the amount that the option seller has to pay into the option through the settlement increases or is higher than the option premium that they have initially received.

And that's when the whole thing will become a net loss.

Again, we're simplifying here, we're assuming no hedging has been done no other positions, et cetera.

Good. So let's have a look at the long put.

And we are assuming the same parameters here.

We're saying, okay, the strike price is $105 and the premium is $7.90, but now we're looking at a put, which means we're looking at the right to sell the asset rather than the right to buy the asset, which then basically switches the whole thing around because the buyer of a put now benefits from the price of the asset to decline, and it needs to decline below 105 because the right to sell is for a price of 105.

And if the asset price now was 110 indeed in the market, why would you use the right to sell at 105? You would sell the asset in the spot market for 110 and you would walk away from your put option. However, if the asset falls in price below 105, so let's look at the price of 100 for example, that situation changes.

You have the right to sell the asset at a price of 105.

If the spot price is 100, you would exercise the option that should give you ignoring all sorts of, costs here. For simplicity, $5.

Remember, you paid $7.90 to get the option in the first place.

Again, leaves you with a negative P&L but of $2.90 cents.

But if the asset price continues to fall, you go reaching breakeven, you're passing through breakeven, and then at some point you will actually, be a net or will achieve a net profit.

And so that's why the hockey stick profile looks fairly similar to the long call. It's just basically the mirrored version of it.

And then the short put again is just the opposite of the long put profile.

So the seller of the put best case will keep $7.90 cents.

and that will be the case when the asset price does not drop below $105.

However, if that was about to hap or if that was actually to happen, that's when the option will be exercised and whatever the value that has to be paid in the option is will reduce the option premium received initially, and that could obviously turn into net losses all in all taken together as well.

So these are the four classic option positions of these hockey sticks at least.

And, hopefully this all makes intuitive sense.

Now, let's talk a little bit more about, terminology here before we then actually start looking into, the option premium a little bit more.

And it's fair to say that when practitioners communicate about options, very rarely, I mean, if you're, if you're negotiating a particular trade with your counterpart, you would of course mention the strike specifically you use a number to describe it, right? But very often when we're just talking about options and, and how the option landscape or you know, has behaved in a specific day, then rather than referring to the strikes by a specific number, we often use sort of categories to describe what kind of option we're describing.

And those three categories that we're distinguishing here also refer to as the option monies.

And the three categories that one should be aware of are in the money ITM then at the money ATM and out of the money OTM.

Now, which option falls into which category, and it's actually a pretty straightforward interpretation, at least when you start thinking about it.

And what you need to do is you need to compare the strike price of the option with the current market price of the underlying. Now we kept this deliberately vague here because obviously this could be either spot prices could be a forward price, and we need to discuss that and we will later on. But for now, let's just kind of keep it generic and say that's the market price.

And then an option is in the money when the strike off the option is actually more favorable than the current market price.

And then example here is shown on the slide, for example, $105 strike put option when the underlying trades at 90 i.e. you have the right to sell the asset at 105, and the asset price currently is 90.

Clearly the price of the, or the strike price of the option because you have the right to sell is more beneficial or is advantageous over the spot price.

And that sort of means that under current circumstances, if nothing changes, if prices stay as they are, we would exercise this particular option right at the money options are then basically options where we simply are indifferent, right? And that is the case when the strike off the option equals the current market price.

Back to example here, let's say we have a hundred strike call when the underlying trades add 105, we have the right to buy the asset, add 105 through the option, and we can buy it at 105 in the market.

So there's no advantage, no disadvantage on the option that is basically then to say we're in different, whether or not we're gonna exercise it, assuming nothing else changes.

And then out of the money options is then basically just the opposite of in the money here is the case that the current market price is actually better than the strike price, at least in, in comparison, right? And that would be the case.

For example, if we have $105 strike call option and the underlying trades at 90, so that means we have the right to buy the underlying at a price of 105, but if we can buy it at 90 in the market, then why would we use the option to do so? So basically think about it as undercurrent circumstances.

This option would not be exercised if conditions don't change.

Now, here's the thing, and that's we, as I said before, we keep it well we kept it deliberately vague.

The problem is that the money of an option can be determined in two different ways, right? We can determine it in reference to the spot price.

All we can determine in reference to the forward price.

And if we have a 12 months option and we're going back to our 12 months call then, and I said it before, depending on what type of option that is, we can either exercise this every day over the next 12 months or, in 12 months time only.

And then in both cases though, we could still compare it against spot and the forward price of the 12 12 months forward price, right? So, which is then obviously showing the necessity as to why we are communicating, not just in terms of add the money in the money out of the money, but probably also add whether this money is determined against the spot price or the forward price. Because remember, forward prices can potentially be quite different from spot prices.

And an example as to where an inefficient or in inexact communication would cause trouble potentially, is something we've printed here on the slide.

And as we have a 12 month call option and the strike price is 105, so this one we already are pretty well aware of this or familiar with that option, but let's say that the spot price of the asset is a hundred and let's say the 12 months forward price of this asset is 105.

So then depending on whether or not we determine money on spot or on forward, we get two different results. Because on a spot price or when we're comparing against spot, we have the right to buy at 105 and the spot price is a hundred, that means we have an out of the money strike, but we should precisely call it out of the money spot because it's out of the money in reference to the spot price.

If we look at it against the forward price, then we have to compare the strike with the forward price, which was given here to us at 105.

And that means on a, you know, if we're looking at it at a, on a forward basis, then this would actually be an at the money option, hence better to call it an add the money forward.

Now, of course, you know, conventions are different across different market asset classes, et cetera, whether or not we're using spot or, or forward.

So it's always worth obviously checking with the subject matter experts, which convention is predominant here.

There is something, you know, else to say because in some ways it should really depend on the option exercise style.

And this is something we've already referred to earlier, but we haven't really called it by name.

So here's three different styles, although, you know, let's just start with looking at the two extreme cases if you wish, the American option and the European option.

And of course it has nothing to do with the location where the option is traded, it's just describing different type of exercise styles.

So let's start with the American option.

What is the style of exercise rights? We have an American option is an option that can actually be exercised at any time after and including the expiry date. So if we have our 12 month call and we bought it today, then over the next 12 months, literally at any point in time, well in during business hours, of course we can decide to exercise this option, right? So that could be tomorrow, that could be in 12 months time, any point, well not tomorrow, Saturday, but you know, Monday.

And then in 12 months time, at any point in time we could decide to exercise and let our counterparty know.

The important thing to remember though is it can only be exercised once.

So if we were actually to exercise on Monday, we will obviously get the value of the option at that particular point in time, but we will no longer be able to benefit from future price changes because the right to buy the asset at the price of 105 now has ceased to exist.

Alternatively, if you don't want to exercise the option early, you can obviously sell the option at any time before expiry as well.

Okay? So that's the American option. If we're now looking at the other extreme, the European option, that option can only be exercised at the expiry date.

So not on Monday, not on Tuesday, just in 12 months time, okay? When the option is about to expire, and it feels hopefully intuitively as if the American option is much more flexible and therefore, you know, has an advantage over the European option.

And whenever something has an advantage in, you know, in in efficient markets, it should have a disadvantage of some sort and that will probably be the premium here. And that leads to the conclusion that the American option premium should be higher then a European option premium.

Well, assuming the options are otherwise identical apart from the exercise style.

Now that is I think we can all agree that an an American option probably will not be cheaper than a European option, but often it's also not really the case that it's much more expensive.

That obviously depends on many, many different circumstances and it's going beyond the purpose or the scope of this session here today.

But I just want to let you know that in many cases the American option isn't necessarily much more expensive than a European, but it can obviously be, and one of the reasons as to why this can be explained is that both options can only be exercised once and also both options can also be sold at any time before expire. And that turns out that only under the specific circumstances in early exercise of an American option actually makes a lot of sense anyway.

So these are the two extreme cases, and then we have added the ber muttin exercise here as well. This is something you find, for example, in the rate space where we have Bermudan swaptions, and a Bermudan option is something that's sort of in between American and European options.

That means it can be exercised on more than one specific date, but it cannot be exercised at any date before expiry. So for example, you have a five year option that can be exercised at the end of each year and therefore not any day, but also not just at the final expiry date.

It can like all the other two things options that we've seen only be exercised once and if exercised early i.e. before the final original agreed upon expiry date, then no further participation in beneficial price moves is possible.

And also like other options, it can be sold before the final expiry date.

Okay? All that in mind.

Now we're gonna move and discuss the option premium.

And as I said in the introduction, we're not gonna look at pricing models here.

We just want to develop a fundamental understanding on where option prices generally come from.

I.e. what is the premium trying to express, and then we're gonna develop this into some basic understanding and what drives the option premium i.e. when should an option premium be relatively high? When should it be relatively low? And the starting point, we have already spent some time discussing on the first slide that we saw here today.

And there we basically drew the conclusion that an option payoff profile is of course, highly asymmetrical, right? It's this hockey stick. And that, you know, basically was then translated into that risks or to the conclusion that risk and opportunities for option buyers and sellers differ quite significantly.

And that then was followed by our conclusion that if there wasn't an option premium, everybody would only like to be a long options because they would be preferable you know, to short positions.

And the premium of the option is sort of basically then the equalizer that will incentivize people to sell optionality as well.

Next question that we want to address right now is what I mean, we understand what's the reason or, you know to exist for the option premium.

We wanna sort of start thinking about, okay, what does the option premium actually reflect it's compensation, okay, but what, you know, what sort of would be our approach to fundamentally understand what the option price drivers are? Well, the first point is to say that, you know, conceptually an option premium should reflect the expected loss of the optional seller, okay? That obviously might require some explanation and it's probably worse stepping away from financial options there and just use a simple experiment there that's used a lot of times in finance, right? And that's a typical coin toss, okay? The reason why it makes sense to explain option pricing, or the idea behind option pricing is that this is a fairly, standardized and simple in a simple exercise or experiment and it has a relatively small number of possible outcomes, right? And of course, if I ask you the question how many outcomes there are in a coin toss, you will say it's three, right? Because there's heads, there's tails, and of course the coin can land on its edge that of course we can probably also agree is a fairly low probability outcome.

So for simplicity here, let's just erase that possibility in reality, probably we can't, but you know, for the sake of simplicity, we'll just say there is only these two outcomes and you probably have to flip the coin quite a few number of times before it actually lands and stands on the edge.

But anyway, so assuming this is a fairly or a coin with in sort of fair weight distribution then you would assume that the probability for each outcome here is actually 50 uh percent, right? And if I now would invite you to play a game with me here where you guess the outcome of our coin toss, before I toss the coin, and then if you guessed it right, you can keep the coin and if you guessed it wrong, I can keep the coin and also I keep the premium or the price that you paid me for being able to play this game, then we should, you know in the long run, none of us should be better off.

But the question is how much should I charge you for this game, right? So now let's think about this.

We can use the expected loss calculation here. An expected loss is basically nothing else but the probability of losing, which is 50%, right? You have a 50% chance of guessing the outcome rate.

And then if I actually lose the game, if you guess the outcome rate I am, and let's say we're using a one pound coin I'm gonna lose one pound, so I have a 50% chance of losing one pound, and that means the expected loss is actually 50 10th, right? And that's what I should charge you upfront.

And then if we play this game an infinite number of times, then law of large numbers suggests that half the time you win this game, so you basically keep my one pound coin, but I I'm actually not losing one pound per game, I'm losing 50 p per game because you paid me 50p initially, then you get £1 back. That means I'm out of pocket by 50 pence.

The other half of the times you get the outcome wrong.

And I don't pay you, I don't give you my one pound coin, but I keep your 50p.

So half the time I win 50p other half of the time I lose 50p.

And that means none of us will be better off if we play this game an infinite number of times.

And that's the idea behind the expected loss.

If we would obviously play this game an infinite number of times the outcome or this is a fair price for this game to be played.

Now, that's a fairly, you know, intuitive example hopefully, and that gives you some sort of general thought about how option pricing works because to calculate the expected loss, you basically have to answer two questions.

First question is, what is the probability that the option in this particular case will be exercised? So what is the likelihood of the option option seller having to make a payment? And the second question is, if the option is exercised, then how much will the seller have to pay out? And this is where obviously reality is much more complex for financial options than our coin toss example. Yeah, because in a coin toss, you know, you either have two or three potential outcomes.

If we're thinking about an option on a stock, for example, the only thing we can say with absolute certainty about a stock price at any point in time in the future is it's gonna be somewhere between zero and infinity, right? And that means we have an infinite number of possible outcomes.

So, you know, if we wanted to just do the approach that we've done here and say, okay, let's define the possible outcomes first and then assign probabilities, we would never, ever be able to finish step one simply because an infinite number of possible outcomes.

Yes, of course probabilities will be fairly zero or fairly close.

Fairly small apologies.

But you can see the point that in theory we wouldn't be able to finish that list of potential outcomes, right? So what then needs to happen is we need to simplify this approach to make it, to make it work or not simplify, but you know, to make it, to make it work.

We need, uh, help with statistics and, and for example, this is where then all the option price models, um, come into play. Because basically what they're trying to do is to answer these two questions, probability of option being exercised and if exercised, how much do we have to pay based, for example, on certain distribution assumptions or certain modeling techniques.

But the aim is obviously to answer these two questions.

Okay? That is the general concept on what should drive an option premium.

Now let's have a look at sort of a little bit deeper look and start to think about, okay, when should an option value or an option premium be relatively high? When would it be relatively low? And I think it makes sense to sort of understand to, to get a feeling for this to understand the option premium really as a sum of two components.

The first one is the intrinsic value, the second one is the time value.

Now the intrinsic value is actually relatively straightforward in to interpret or to, and also easy to calculate because it's basically the present value of the difference between the option strike and the price of the underlying when the option is in the money.

So let's go for simplicity and have a look at an American option, right? So we have a 12 months American call and the strike price, you would guess it right, is 105 dollars, okay? So if we own that call and the spot price of the underlying is 110 to send us as an example here, then what would be the intrinsic value? First thing is of course, this is now an in the money option, right? Because we have the right to buy at 105, the asset price is 110, that option under card circumstances makes sense to exercise, right? So that is an in the money option.

And now we can also calculate the intrinsic value because that's the difference between the option strike and the price.

So here we now have 110, that's the price of the option minus sorry, of the asset minus 105. That's the price at which we can buy the asset at through the option.

And so that means the option currently gives us a $5 advantage.

Now that's, it says here, present value, but there's no present value that we need to do here because it's an American option.

We could actually exercise this option right now and then we won't get these $5 immediately. There's obviously a T plus 1, T plus 2, whatever, but generally that is immediate in, in financial markets.

So we could extract that value of $5 immediately, hence there's no discounting.

If that was for example, a 12 months European call, then we would have to take a slightly different approach, but then we should also use the forward price, subtract the strike from it, and the difference should be discounted, et cetera. So it gets a little bit more complex, but generally you know, the fundamentals still hold.

So in the money option as we've seen, has a positive intrinsic value.

How about an at the money option? Well, if we're looking at an American option, the option would be at the money when the asset trades at $105 in the spot market.

That means here now we have not 110, but 105 as an asset price that doesn't give us a price of intrinsic value of five, but that gives us an intrinsic value of zero.

That would be the case of an add the money option.

If we now change the spot price further and change it to let's say a hundred, right? Then that is no longer an at the money option, but we're looking at an out of the money option.

And then here the calculation would be a hundred minus 105, but it's not gonna be a result of minus $5 simply because in this case, we would simply walk away from the option, we would not exercise it.

So we don't have any negative intrinsic value because we will just walk away.

We will never pay these $5.

Simply let the option expire, conclusion, add the money, and out of the money options have therefore an intrinsic value of zero.

Now of course you look at this and think, well, okay, what's practical relevance? And one important thing about the intrinsic value is it's actually the minimum price boundary towards the option premium.

Because imagine, and we're going back to our original case here.

So the spot price was 110, that means we are talking about an in the money option, right? And we're talking about an option I'm just changing the calculation back.

So now this would be a $5 intrinsic value.

Let's assume that this option, the way it's presented here on the screen, would trade in the market at a premium of $3.50 right? So what would you do? Of course you would go and buy as many of these options as you possibly can pay $3.50 per option contract for it, exercise immediately, and basically get $5 return.

That is, you know, more or less risk-free profit.

And that should not really be possible in efficient markets.

And the moment we all start buying these call options for premiums 3.50 in large volumes, of course the price of the option will go up until this arbitrage no longer makes sense. Of course, we're assuming here no transaction costs in the usual simplifications, but you get the idea, okay? So that is the minimum price boundary condition, that has been established.

Now, going back to an in the money sorry, an out at the money or out of the money option.

So let's go and say out of the money option right here, it was the spot price was trading at a hundred, not 105.

So we can forget about all this and then the intrinsic value is zero.

But is a premium gonna be zero? And of course the answer is no, because this is a 12 months option.

And while it might not have any intrinsic value right now that cannot, it can change.

And it's actually fairly likely that it might change at some point over the next 12 months.

So nobody would sell this option at a premium or four premium of zero just because it currently does have any intrinsic value.

So let's say the premium for this, and I'm just making this up, was $2.

Where does this $2 come from? Because it's clearly an intrinsic value zero option.

But remember we said the option premium has actually two components.

One is the intrinsic value, and then the second one is what we call the time value.

And the time value is basically what's left.

If you take the option premium, subtract the intrinsic value from it, and the rest is the time value of the option.

Now, why would this option be, or why would there be a time value attached to this option? And we sort of already indicated as to, or, you know, the reason as to why this is, and that's because this is a 12 month option.

And just because it doesn't have an intrinsic value right now, we cannot draw the conclusion that it will never have an intrinsic value, right? In fact, there's 12 months over which the price of the underlying can move up, down, whatever.

And then the option can build in intrinsic value.

And a fairly intuitive way of interpreting time value often is to say, okay, the time value is the compensation that the option seller charges on top of the intrinsic value to get compensation for taking the risk that actually the intrinsic value in the future might increase.

And that's of course negative for the option seller.

So right now, the in, in our, in the money option case, the option has, or let's say it was the out of the money option the intrinsic value of the option is zero.

But what if now over the next 12 months, the spot price goes from a hundred where it is right now to 120, and then the option expires, that means it expires. The option has an intrinsic value of $20, which we as the option seller have to pay, right? So we should sort of compensate for this risk when we're selling the option.

So we shouldn't just charge the current intrinsic value, but we should also think about the risk.

You know, that generous is, is resulting from the fact that the option still has 12 months to live. And as I said, it's very intuitive to think about this, this is compensation for technical risk that the intrinsic value might get worse from today's perspective.

And while this is intuitive and often works, you run into trouble sort of explaining certain types of behavior of the time value.

And one of them we're gonna see on the next slide if you stick to that way of interpretation.

So my advice is think about the time value more as the compensation that the option seller requires for really taking uncertainty or taking the risk that results from exercise uncertainty.

In other words, the risk that we're having as an option seller simply because we don't know if the option will be exercised or not.

That is a little bit abstract.

So let's go and have a quick look at an example here that, that hopefully will make this a little bit more intuitive.

And I'm going on a whiteboard because I needed a little bit more space.

So before we're looking at options, let's start with a forward contract, right? So we have a 12 months forward contract here that we've been asked to calculate or quote a price for.

Here's the information that we have.

The spot price of the asset is 100.

We have a 12 months interest rate of 5%. Let's ignore day count conventions, et cetera.

And let's also say this is a non-dividend paying asset, okay? So, we now are asked to quote a forward price here and we're quoting the fair forward price. So we're not worrying about bid offers, et cetera, et cetera.

We just need to think about how could we hedge this trade because let's, we're assuming the role of a market maker here now and we wanna facilitate this client trade without exposing our employer to market risk.

So how can we do this? Well, there's only one point in time where we know and assuming there's no forward market, of course, uh, where we know for sure where the price of the asset is. And that's right here, right now.

So the first thing we need to do to avoid all uncertainty is step one, buy the asset at spot, right? And that means we have to pay a hundred dollars per asset and this is money we don't have.

So step two is basically borrow a hundred dollars per share.

We bought obviously at a 12 month interest of 5% and then in 12 months, whoever lend us the money, we'll want the money back plus a 5% interest.

So 12 months from now we need to repay $105, but we have the asset that's now what we can deliver into the forward contract and we should then charge a minimal forward price here of 105.

So the idea is client wants us to or client wants to buy a 12 months forward, want to agree on the price right now, we buy the asset in the spot market, we fund the purchasing price through over the next 12 months, not to introduce any interest rate risk here into the mix.

And then the client in 12 months should pay us the price we have originally paid on the stock plus our borrow cost. And that is then here, the 105, as I said, that's th you know, fearful forward price if you wish.

So no bid offer, no service charges, et cetera included.

But you know, it's, it's, it's just about the concept here.

The beauty of this contract though is once we've done this transaction i.e. we bought at a hundred, we've borrowed the money and we're ignoring counterparty risk here. Now, for simplicity once again isreally doesn't really matter to us where the asset trades in a 12 month or the 12 months time horizon.

If the asset has rallied all the way up to 10,000, well we get 105 from the client, that's what we have to pay back to whoever lend us the money.

So we not having a negative P&L impact if the asset has dropped to a price of zero, remember it's an obligation to buy the asset from us, the client has.

So they will have to pay us 105, we take this 105 and give it to whoever lend us the money and they have their money back plus 5% interest.

So really regardless of the actual market price in 12 months we're not seeing any negative P&L impact, assuming obviously we've done these transactions and our counterpart doesn't default.

So that's the forward, there is no uncertainty around exercise.

The client is obliged to buy the asset from us and we're obliged to deliver the asset to them or instead of client, let's say counterparty, right? Because that's more universal.

And so this is a contract where there isn't really any uncertainty, right? So let's see how this would change if we're moving from a 12 months forward to a 12 months call option.

And let's say the strike is our usual 105.

So what happens if in 12 months from now the strike, sorry, the underlying price actually is higher than 105, right? So, and we've been through that example earlier, we said, okay, let's look at what the spot price is 110, then we are paying $5 into the option.

So the higher the asset price moves, the more we're actually gonna pay out into the option contract.

It's very tempting to say, well that's very similar to the forward right when the forward or when the spot price goes up and we have sold on a forward basis that that's why we're buying the stock in the first place.

So that we get that performance, from the hedge.

Now of course you could do the same thing, right? We could do basically buy the stock at the spot price, right? Borrow a hundred and then obviously fund it serve for 5% over the next 12 months, et cetera, et cetera.

So second, borrow the money, right? But the problem with this is if we bought all the stocks, so if client or our counterpart wants a hundred options, so we buying a hundred stocks now and then fund this through for the next 12 months, that's a great hedge.

If in 12 months the option actually is exercised.

So if in 12 months the option of the underlying trades at 120, let's say then you know, everything worked out fine because we bought the stocks at 100 we borrowed the money at 5%, we need 105 from our client or our counterparty to repay the borrowed money, and that's the strike of the option.

So that works. However, there's another outcome, right? And that is what if the spot price has declined or is trading below 105, then other than in case of the forward, our counterparty will not go through as the transaction. They're not gonna pay us $105 anymore because they just walk away from the option.

What we have now is a bunch of shares and we have to repay $105 because we borrowed a hundred per share, right? But the market price is not at 105 because remember, the reason for the client to walk away here from this trade is that the underlying trades at a price below the strike.

So at 100, for example, maybe at 90.

So now we have a problem, we have a hundred shares each share, let's say it's worth $90.

Yes, we can sell them and we get $90, but what we have to repay is 105 because we borrowed a hundred to buy the stocks in the first place, we borrowed it at 5%.

So this is an obviously not going to work.

So other than in case of a forward option, hedging cannot simply be static.

We cannot just buy 100 stocks borrowed money for 12 months.

We have to be more dynamic.

And that means as a general rule or fairly simplistic view on things here you need to think about what is the probability that the option is exercised and the more likely it is that the option is exercised, the more or the higher the proportion of stocks you should buy.

So let's just assume for simplicity here, we know and you know, just let's, let's just assume this, we know this or we calculate the probability of the option to be exercised as 50% at the moment.

Now we have sold 100 options and we have a 50% chance that the option is exercised.

So we need, we have a 50% chance that we have to deliver a hundred shares into the option.

Does it maybe intuitively make sense that we should buy 50 stocks, right? And then if now the market starts moving and let's say the market price, the spot price goes up, it goes from where we are a hundred in the scenario at the beginning, let's say it goes all the way up to 110, what does that do to the probability of the option to be exercised? Hopefully it's insured if to say, yeah, that goes up.

Because if I have the right to buy an asset at 105 and that has a 50% chance of exercise when the spot price is 100, now the spot price has gone up quite significantly, that makes the option more valuable.

That makes, means the option is more likely to be exercised.

So probably we then should buy more shares, right? And it's great we can do this, right? The problem is now that life is not a one way street.

So the share price has gone up within buying a couple of extra shares, but we're buying them at higher prices.

Then afterwards the share price goes down and we see have to readjust our probability of option to be exercise. We wanna downsize our hedge, we're selling some shares, but now at lower prices.

And that means from a hedges point of view, you know, we cannot just sort of, you know, buy all the assets and then forget about it.

We have to take a dynamic approach and, and for reasons we've just intuitively explain, we will always buy at higher prices and sell at lower prices.

And that means that sort of movement in prices, volatility and uncertainty, basically, this is what it all comes down to is, um, really then our enemy if you wish, right? So if you're selling optionality, what ideally you don't wanna see is channeling a price to move volatility to move.

And that's then basically what the time value it's all about.

It's really about this uncertainty.

And the higher the uncertainty you have as an option seller, the more time value you are going to charge, right? So what drives uncertainty then? Ultimately two things that hopefully will be immediately, um, intuitive, and that is of course the time to expiry.

Because the longer the time horizon, the more uncertainty there is, right? It's probably relatively or I don't wanna say easy, but it's probably easier to forecast within a certain bandwidth of accuracy, the S&P 500 closing level today than it is to forecast the S&P 500 closing level within the same bandwidth you know, a year from now or 30 years from now, right? And a year from now, you know, you'll have a better shot still than, than probably 30 years from now.

So uncertainty increases with time to expiry of the option, which can also be referred to as an option 10.

So that means all else being equal, a three months option should be having a smaller time value than a other, the otherwise identical 12 months option.

For example. What also plays a role is then obviously the behavior of the underlying price, and that's the volatility.

Is this an asset that doesn't really move as a fairly well-behaved asset or is it an asset that tends to have a very volatile behavior? So it is having a significant price increases followed by significant corrections, et cetera, and that happens all the time.

That of course is increasing the uncertainty quite significantly.

And therefore there's this intuitive link between a positive correlation really between the time to expire of an option and the volatility of the underlying and the time value of an option. So these are two factors that I say would be immediately intuitive.

There's one more thing to say about the time value, and that is something that one needs to obviously be aware of and that is that time value approaches zero when the option goes to its expired date because when the option expires, but if definition, there's no uncertainty left, we know exactly the intrinsic value of the option at expire, we know exactly how much we will have to pay out as the option seller.

And that means there's no uncertainty. Varies then per definition. No time value.

Okay, so that's the general point. There's one third factor that Im impacts time value. And that's the one that I mentioned earlier when I said, you know, it's probably best not to think of the time value as the compensation for the risk that intrinsic value might go up.

So what we're doing on this slide here is we're having a look at the option premium of the same option, but at different points in its life.

So we're starting with a three months option here. That's a green line, right? It's a three months option at spot.

So at the time of trading this one we have changed the strike just to, you know, keep you on your toes. It's a hundred dollars. It's a three months call with 100 dollars strike call.

And we have calculated a price at the time and it was just below a $5 at a spot price of 100 dollars as well.

And now what we do is we just look at that, assuming nothing else has changed, just time has gone by.

So one month later the asset price is exactly where it was before.

So still at 100 dollars.

Volatility of the asset hasn't changed, just the time to expire has changed.

Where's the option premium? The lead line, is it two months to expiry option? And as you can see, it's a tiny bit below the green line.

What's the difference? That's basically just the shortening of the time to expire. This is no longer a three months option. It's a two months option. Now vol hasn't changed.

So the time value has decreased solely by time going by. That's a time decay of an option. The option has aged by a month, has now two months to expire. It is a shorter product, smaller time value.

All else was kept constant. And now we're going another month down the line. And that means there's a one month option and no surprises here.

The option premium has decreased.

So basically saying you bought that option here at $4 and something, then a month later, if the market hasn't done anything, it's worth less.

Another month later is worth less and now we go another month into the future.

And that means the option now is addicts expiry date.

We have a spot price of a hundred.

We have a strike price of 100 that's an only at the money option.

Time value now per definition is zero.

And the option only has intrinsic value, but because the strike equals the spot, the intrinsic value is zero. So as we travel through time, the time value decay to zero, that's fine. That's nothing new. That's, you know, time value driven by time to expire. Effectively, what we've seen on the previous slide, we kept vol stable.

So that's not really of any impact in this scenario.

But what I want to point out is sort of like these two parts of the chart, and that is when the option goes in the money, because this is a call, right? These are in the money options and this is out of the money options.

What happens when the option goes deep in the money and deep out of the money time value disappears because all these lines are converging. And that doesn't make sense if you think of time value as compensation for the intrinsic value to get worse.

Because even if the price price is at $120, it's not a, you cannot completely rule it out that the price will continue to rise and then the intrinsic value exceeds the $20 at that point, right? So that could, that might as well happen.

So why is time value going down here? Well that's basically then when you go back to what happens to the uncertainty, and I would invite you to think about real extreme case to make this even more visible than, than on the chart.

So let's say we have a call option and the strike is 100 dollars, right? And expiry is five in five minutes, right? And the spot price of the asset is $10,000, right? So we now have the right to buy the asset at 100 dollars and that right expires in five minutes and the spot price is 10,000. And yes, legally it's an option.

It says in the contract that we have the right to buy the asset.

Practically though, can we all agree that this is not an option? This is an obligation to exercise the option.

Because if you want, you would lose a significant amount of money.

And so everyone in the right mind would exercise this option because you've gotta buy the asset at a hundred if it trains at 10,000 in the market right now.

So if it's practically not an option, not the right, but the obligation, that's basically practically a forward contract and it should behave as such.

And the forward contract has no time value because there's no uncertainty.

What we know at this point, well you know, there's still five minutes left, so it's not 100 percent guaranteed, but you know, in absence of any sort of like extreme surprises here this option will be exercised in five minutes.

What we don't know to the last cent is the amount of money that we will generate as the option buyer, but that's the same for forward.

So effectively this has become a five minute full contract if you wish, and therefore shouldn't have time value because there is no uncertainty.

We know we're going to exercise this option and that's why the time value, when options move in the money is disappearing on this chart.

And the same for out of the money.

Now if you think we have a strike of 10,000 and the spot price now has declined to a hundred, it's fairly unlikely that over the next five minutes we're gonna see a rally from a hundred to 10,000.

That means there's also pretty much no uncertainty whether or not the option is exercised. This option is guar almost guaranteed not to be exercised. There's no uncertainty here for the option seller.

It shouldn't have time value and because it has no intrinsic value, either the option premium is zero.

And that ladies and gentlemen, is all I wanted to share with you here today.

I hope you found this beneficial. Thank you very much for your participation.

Have a great rest of your Friday, a super weekend ahead and I look forward to seeing you again soon on one of the other sessions.

Take care for now. Have a great weekend. Bye-bye.