FXOCourse

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FX Options Course Notes

Autumn 2010

Week 1

0 : Preliminaries, FX Spot

In Foreign Exchange (FX) all currencies have equal status as assets that can be bought and sold. Everycurrency transaction involves buying (going long) one currency and selling (shorting) another.

For each pair of currencies in the world we can quote a spot (immediate delivery, as opposed to a forward,which is for some future delivery) exchange rate. For example, the “EURUSD” exchange rate is the value of one EUR denominated in USD.

All currecy pair exchange rates are quoted the same way, i.e. the value of one unit of the right-hand (base)currency in units of the second (term, or quote) currency.

EURUSD = 1.3809 means one EUR is currently worth 1.3809 USD.

In practice the market quotes with a bid-offer spread. A more realistic EURUSD spot rate is 1.3808/1.3810.This means we can buy one EUR and pay 1.3810 USD (paying the market offer), or sell one EUR and receive1.3808 USD (giving the market bid).

Not all spot rates are independent of course, for example :

EURUSD = EURGBP × GBPUSD

The tradability of all crosses is unique to FX, and has important implications for FX Options markets inparticular.

Contrast all this with, e.g. the UK stock market, where all the assets are denominated in GBP : stocks arebought by paying GBP and sold to earn GBP; it is very rare to consider exchanging RBS shares for Shell shares,and the “cross” is never quoted.

Every currency pair has a conventional quote order, although it is permitted (if potentially confusing) to quotethem the other way around. Indeed you will probably have noticed that EURGBP is almost always quoted asGBPEUR in the media (much to every FX trader’s annoyance).

Convention Alert #1 : it is very common to leave “USD” implicit when discussing deals, for example when FXand traders talk about a “CHF” deal, they will be talking about USDCHF. Similarly FX Options traders mighttalk about “1month EUR options” in which case they mean EURUSD.

1 : Spot Ladders and Delta

We will spend a lot of time studying spot ladders for the various derivatives we’ll cover. These are simply agraph of some property of the derivative as a function of the spot exchange rate. For example, the chartbelow shows a spot ladder of PnL as a function of EURUSD spot for a simple long EURUSD spot position in1million EUR at 1.3800. If EURUSD spot goes up to 1.3900 we can now sell the 1million EUR for 1.39 million

USD, so the original trade has a Mark-to-Market of USD 10,000 at 1.3900. We can plot this MTM as a functionof EURUSD spot and it looks like this :

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PnL vs Spot

-20,000

-15,000

-10,000

-5,000

0

5,000

10,000

15,000

20,000

1.36000 1.36500 1.37000 1.37500 1.38000 1.38500 1.39000 1.39500 1.40000

2 : Options – a first look

As is well-known, a plain option gives the owner the right, but not the obligation, to buy one currency (the callcurrency) and sell another (the put currency) at a specific exchange rate (the strike) on a specific date (the

expiry; actually at a specific time on this date, we will return to this).

Options only have value in a world where the value of spot exchange rates in the future is uncertain. If weknew exactly where (say) EURUSD spot would be in one week’s time, there would be no point in doing anyoptions at all.

Let’s study a EUR call, USD put option (we did so in NMOSS) struck at 1.3800, in EUR100mio face amount (thePnL is USD) :

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Option PnL vs spot

0

1,000,000

2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

8,000,000

1.30000 1.32000 1.34000 1.36000 1.38000 1.40000 1.42000 1.44000 1.46000

1W

2D

0D

Focus on the red line (0D) first : this is the PnL profile for the option at expiry.

Below the strike the option is worthless (as we would not wish to buy EUR at 1.3800 if we can do better in themarket). Above the strike the option has the same PnL profile as a spot deal in EUR100mio, since this is

exactly what it is – we will certainly exercise the option to buy EUR at 1.3800 if the prevailing market rate ishigher; even if we don’t actually want the Euros we can sell them out for a profit.

Now consider the blue (1W) line : this is the PnL profile for the option with one week to go before expiry, i.e.there remains considerable uncertainty about where EURUSD spot will end up. This means that the blue line isabove the red line everywhere (even if only by a tiny amount). The option has a limited downside (zero) andunlimited upside – this asymmetry means that uncertainty always generates positive value for the owner.

Far below the strike, to the left of the picture the option is still close to worthless, as we are virtually certainthat EURUSD can’t move far enough to result in our option expiring in the money (we say the option is deeplyout of the money).

Far to the right of the picture the option looks just like the option at expiry (the red line), and in turn this isidentical with the value of a spot deal in the full face amount. This is because we are virtually certain thatEURUSD can’t move far enough to result in the option expiring out of the money (i.e. not being exercised).

The interesting region is in the middle near the strike where we can see the biggest effect of the uncertainty.For example with spot at 1.36, although the option is out of the money, there is enough time and uncertaintyfor there to be a probability of a move to the right, generating value. Similarly with spot at 1.40 there is achance of a move further to the right, generating more positive value, but for a move to the left there is a capon how much money we can lose as we pass through the strike.

So you can see how an option’s value comes about from uncertainty in the world – the more certain webecome of where EURUSD will end up, the more the option’s value will decay toward the red line. Oneimportant way in which we can become more certain is as time passes. The green (2D) line shows the value of the option with only two days to go before expiry, and we see that it is everywhere between the blue and redlines (but has the same features).

There are two important features to note about each of these three pictures :

• The value of a call option is always increasing with increasing EURUSD spot – it is LONG delta

• The value of a call option is always curving upwards, i.e. the slope of the curves increases from

left to right (this will mean it is LONG gamma, see below).

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Returning to the blue curve, we observe that the slope on the far left is zero (zero delta), and on the far rightthe slope is simply the same as a spot deal, i.e. 100 delta (this is percent of face amount, which we’ve also setto 100 for convenience).

It is important to realise that in the middle of the picture the option has a slope (a delta) somewhere between

these two extremes, indeed roughly near the strike the slope is the same as 50 EUR of a spot deal. When spotis right at the strike the option is behaving like a spot deal in half the face amount – this is intuitively sensibleas we observe there is approximately a 50% probability of spot going up (and the option turning into a 100EUR spot deal after we exercise it) and a 50% probability of spot going down (and the option expiringworthless, i.e turning into a zero EUR spot deal).

3 : Delta Hedging

Returning to our 1wk option struck at current spot (1.3800) we observe that we can flatten the PnL profile byselling 50 EUR of spot :

PnL profile with delta hedge

0

500,000

1,000,000

1,500,000

2,000,000

2,500,000

3,000,000

3,500,000

4,000,000

1.30000 1.32000 1.34000 1.36000 1.38000 1.40000 1.42000 1.44000 1.46000

1wk

0D

So now the PnL profile is locally flat (zero delta) in the middle of the picture (on the blue line). For a smallrange of spots we don’t care what happens to spot (our PnL doesn’t change).

As spot moves away from where we started (1.3800) two things happen :

• Our PnL goes up (in either direction)• Our delta changes – we get shorter as spot goes down (the slope is downwards) and longer as spot goes up(slope upwards)

This last feature is great – if we nod off at the desk, or go for lunch and come back to find spot has moved,then we always get to make favourable spot transactions : we can sell after spot has gone up, and buy back if it goes back down, and so on. This is called “trading Gamma”.

Note that it doesn’t come for free – as time goes by the value of the deal will decay down toward the red line.If the market is efficient (which it isn’t), the money you make trading gamma will offset this decay exactly.

The picture can still be understood intuitively :

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• Far to the left our option expires worthless and we’re left with our short 50 EUR spot hedge• Far to the right our option is exercised, generating a long 100 EUR spot position, offset by our short 50 hedge

As an exercise you should repeat the above analysis for a EUR put option. After you’ve hedged the delta youshould end up with the same profile as above.

In the class we sketched the Delta and Gamma profiles – we’ll return to these in more detail later.

Week 2

4 : Variance and Standard Deviation

Options’ value stems from uncertainty in the world, but we need to be a little more scientific about this to priceoptions properly.

We measure the uncertainty we’re interested using a statistical measure called “Variance.”

Imagine for a moment you are a company which makes 50mm screws. Your manufacturing processes are notperfect, so screws will randomly vary slightly in their length; we say that the screw’s length is a random

variable; here are the lengths in mm of ten screws drawn randomly from your supply line :

49, 49.5, 49.5, 50, 50, 50, 50, 50, 51, 51

We want a statistic which tells us how much we expect the average screw to deviate from the desired length of 50mm – the smaller this number, the better our manufacturing processes, and the less uncertainty there isover the length of a given screw.

The mean of these numbers is 50mm (otherwise we couldn’t be claiming to make 50mm screws). Thedeviations from this mean are (in order) :

-1, -0.5, -0.5, 0, 0, 0, 0, 0, 1, 1

You might immediately think we can just average these deviations, but of course the average deviation fromthe mean is zero (this defines the mean). Let us consider instead the square deviations from the mean; inorder these are :

1, 0.25, 0.25, 0, 0, 0, 0, 0, 1, 1

The mean of these numbers is 0.35, which we define to be the Variance (for any mathematicians out there,I’ve computed the population variance, not the sample variance). Note that the screw which is 1mm shortcontributes as much to this measure as the screws which are 1mm too long.

We can go further and study many more screws and build up a picture of the relative likelihood of differentscrew sizes from our manufacturing process – we call this the distribution of screw lengths. Thesedistributions can take many forms, but the most common and most important is a so-called normaldistribution. I quickly set up a model of our screw-manufacturing process, assuming a normal distribution of screw lengths with a mean of 50mm and a variance of 0.35. I got this :

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0

5

10

15

20

25

30

48 48.5 49 49.5 50 50.5 51 51.5 52

Frequency

Expected

The red dots show the number of screws observed with the given length ±0.1mm, from a sample of 200. Theblue line shows the expected number from a perfect normal distribution.

The blue line represents the classic “bell shaped curve” so beloved of scientists. There are a number of thingsto note :

• The distribution is symmetric, and peaked at the mean.

• Even though I generated the red data points from a normal distribution, they are themselves scatteredaround the blue line – as I make more and more observations the variations between the red and blue lineswould disappear.

• The width of the blue curve is related to the variance – in fact at half-height (in this case roughly 12.5) thewidth is twice the square root of the variance (!). The square root of the variance is called the standard deviation (which is approx 0.62 here), because it is the “characteristic” width of the curve.

• The chances of observing a screw with a deviation of more than 1.5mm from the mean are very small. Infact we can make fairly precise mathematical statements about the likelihood of a given screw being more thana certain number of standard deviations from the mean. For a true normal distribution 68% of samples drawnwill lie within one standard deviation of the mean, 95% within two standard deviations, and 99% within three

standard deviations.

• If we were to refine our machining processes and reduce the variance of our screw lengths, the curve wouldbecome narrower and taller (and vice versa).

One other interesting feature here – if we break up our set of 200 screws into 10 sets of 20, and measure theirmeans and variances we get something like this :

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Sample Mean Variance

1 50.01 0.58

2 50.28 0.39

3 50.15 0.31

4 49.82 0.25

5 49.91 0.316 50.08 0.27

7 50.03 0.32

8 50.01 0.32

9 50.01 0.27

10 49.86 0.24

Mean 50.02 0.33

Note how the means and the variances of each of the samples are themselves scattered around the true meanand variance, but the average of all the samples is a good measure of the true mean and variance.

There are two useful things to take away from this :

• Any statistical measure (mean, variance, etc.) from a subset of all the available data will only be an estimate– different samples will give different estimates.

• The more data you can pack into your sample (in this case, the more nails you measure), the better yourestimate of the mean, variance, etc. will be.

We must now turn our attention from screw manufacturing back to foreign exchange, and in particular the(percentage) changes in fx spot rates. If we chart any sensible exchange rate (or other asset price) we’ll seethat the daily changes (or hourly changes, tick data etc.) are apparently random. We can study theirdistribution and hence statistics like the variance and standard deviation.

5 : Volatility

FX exchange rates are random variables, so we can ask questions about the distribution of returns just like wedid for the distribution of the length of our screws. We consider percentage returns :

Percentage return = ( [New Spot] – [Old Spot] ) / [Old Spot]

What we find is that the percentage returns are themselves roughly normally distributed, with a mean of approximately zero, and a variance which we can measure.

The variance of spot returns is governed by two key factors :

• The horizon over which we look – one minute, one hour, one day, one year. The variance of returns (i.e. ouruncertainty about where spot will be) grows as the timescale gets longer.

• Some intrinsic “noisiness” of the currency pair – in an extreme example USDAED is actually pegged and haszero intrinsic noisiness, compared with some very noisy currency pairs like USDBRL.

We therefore find it useful to strip out the timescale component and write the following :

Variance = (Volatility)² × Timescale

Where the timescale is measured in years. So the (square of the) volatility is the variance rate, and is simplythe “intrinsic” noisiness of a currency pair. A currency pair with a higher volatility has a much broaderdistribution of returns (like a very rough-and-ready manufacturing process where the screws are of veryunreliable length). In contrast we’re virtually certain where an exchange rate with a very low volatility will bein the future – the distribution of returns is very narrow.

Volatility is right at the core of the FX options market, it is used in analysis, pricing, and is deeply rooted in thelanguage we use to discuss options.

6 : Black–Scholes

The Black–Scholes model is of vital importance to options markets. It models spot as a so-called log-normal

process with a fixed volatility. In essence this means that the percentage returns (strictly speaking, thelogarithm of th returns, but these are very close) of spot are distributed normally.

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In this model it is possible to price vanilla options precisely. Besides the obvious contract details (Currencypair, Maturity, Strike, whether it is a call or a put) the formula takes a few market data inputs :

• Spot – the exchange rate now• Interest rates – these are basically used to discount future cashflows• Volatility – the intrinsic noisiness of the currency pair

The contract details given to us, and the interest rates and spot rate are essentially external inputs too – sothe only parameter we’re free to move around is the volatility.

The volatility completely determines the price of the option (given the other market data).

Indeed, we generally quote options in terms of volatility itself (rather than premium). If you ask a premiumprice for an option (which does happen) it MUST be done with a fixed spot reference (and the delta hedgeshould be executed at that rate or else the price is wrong) – in the first week we learned how the spot ratechanges the premium price of an option quite quickly.

Instead we observe that for small changes in spot and interest rates (and even sometimes for small tweaks inthe maturity, strike, etc.) we don’t expect the intrinsic noisiness, i.e. the volatility, of a currency pair tochange. So we quote this volatility and then when the trade is executed, we use the prevailing market spotand interest rates, either to hedge between ourselves and the counterparty, or else where we can hedge in themarket, and figure out the premium from those.

It is an extremely convenient way to quote option prices; it allows us to hold our prices for several minuteswhile our counterparty decides what to do. Moreover, and this is important, it is a level playing field tocompare different options.

If I tell you that a 6mth EURCHF 1.29 CHF call option costs 1.2% eur, how can we compare this to a 2mth86.80 JPY put in USDJPY, which costs 0.0.25% usd?

But if I tell you they are respectively 11.5 and 12.25 volatility prices, this makes them much easier tocompare. Indeed I can make statements about the relative value of different currency pair volatilities,abstracted from any particular options contract : we can talk about EURCHF options trading at a volatility of 10%, and immediately compare with EURUSD which trades at nearly 14% - we can conclude somethingmeaningful about the level of uncertainty in these two pairs straight away, and in a way that is opaque if wewere simply to quote premium prices for a bunch of specific options.

Since we know that options become more valuable as the level of uncertainty increases, we know for sure that

options become more valuable if we increase the volatility we’re using to price them. This is encoded in theGreek called Vega.

7 : Vega

The most important Greek of all (which isn’t actually named after a Greek letter, but in fact a very bright starin the summer triangle) is called Vega. It is the sensitivity of the option price to an increase in the inputimplied volatility (the units are quoted per 1% in vol, e.g. from 10% to 11%).

Vega is vitally important – it is what we trade, as we try to buy Vega in options markets where volatility lookscheap, and sell options (i.e. Vega) in markets where volatility looks expensive. We’ll come back to exactlywhat this means.

Vega is also important for understanding how the premium price of your option moves. If you’ve been quoteda EURUSD option price of, say, 10%, and the option has a Vega of EUR75k, then if you wish to take EUR 7.5k

of margin, you need to raise the vol you show your customer to 10.1%. This avoids tedious iteration in thepricing machine, and (crucially) this spread will be independent of spot and rates etc. (for small-ish moves).

The Vega as a function of spot for a vanilla option (this is a EURUSD 1mth 1.3850) looks like this :

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Vega (EUR)

0

20,000

40,000

60,000

80,000

100,000

120,000

140,000

1.20000 1.25000 1.30000 1.35000 1.40000 1.45000 1.50000 1.55000 1.60000

Important notes :

• Vega is positive everywhere for a vanilla option – more volatility is always good for the owner of an option.• Vega is peaked around the strike – we’re most sensitive to changes in the volatility when it can have themost impact, i.e. when the option is really an option and not just a delta position (see section 2).• Vega decays to zero more or less symmetrically as we either go deep in the money or deep out of the money– there’s only so much an increase in implied volatility can do if we’re many standard deviations away from thestrike.

Week 3

8 : How do we calculate the “correct” volatility?

Volatility is simply a measure of the expected variance in future spot returns, so one thing we can do ismeasure what that variance actually was historically. For example we can take daily (hourly, 5-second tick,whatever) spot returns for the past three months, and compute their variance, and hence their volatility. Thiscan all be done in Excel, although there are various tools to compute these numbers for you : X-trader,Bloomberg, RBSM, Plato, etc.

It is important to note this is an estimate of the realised (or “historic”) volatility – as we observed with oursample of nails, as you take different samples (different times of day for daily returns, different sets of tickdata, and so on), you will get a different estimate of the realised volatility. Moreover if you have very few data

points then your estimate of the volatility is likely to be very inaccurate – if you measure the historic volatilityusing daily returns over the last month, for example, then you will have approximately 20 data points, whichas we observed in our example with nails, doesn’t tend give a very good picture of reality.

In practice, for most liquid currency pairs there is a fairly well-established marketplace for implied volatility,and we only have to resort to guesswork and historics etc. when we are faced with a very illiquid currency pairfor which there’s no interbank market.

However we still use historic volatility measures as a guide for extracting value from the options market – wecan attempt to buy volatility if it looks cheap (compared to recent historics, for example), and sell if it looksexpensive.

Note that many pairs have a “risk premium” built into them – meaning the implied volatility which trades in themarket can be very far from simple measures of historic volatility. A good example is USDCNY, where if younaively measure the historic volatility of spot you will get a number very close to zero, because the CNY is

pegged to the USD. However options trade somewhat above zero because there is a risk that the peg isreleased, in which case CNY will move a fair way. Something similar is true in many emerging markets – HUFand CZK have had periods when they’ve been very heavily managed by their respective central banks (keeping

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realised volatility very low) but implied volatility is much higher reflecting the relatively remote possibility of alarge move.

9 : Relationship Between Volatility and Maturity

An option’s value, and hence all its Greeks, are driven by the expected variance (or uncertainty) in where acurrency pair wil be in the future. Variance has two components, as we’ve noted before :

Variance = Volatility² × Maturity

So we should expect, and will find, that increasing the volatility in a currency pair, and increasing the time tomaturity, have much the same effect on the Greeks for an option.

9.1 : Value

Consider our familiar EUR call, USD put (strike 1.3845 below) :

Value for different Maturity/Volatility

0

500,000

1,000,000

1,500,000

2,000,000

2,500,000

3,000,000

3,500,000

4,000,000

4,500,000

1.3300 1.3400 1.3500 1.3600 1.3700 1.3800 1.3900 1.4000 1.4100 1.4200 1.4300 1.4400

Zero

Low

High

This chart should look familiar – if there is zero time to Maturity then the option value profile is the familiar redline, which is simply zero when out of the money, and the same value as a forward in the total notional (here100mio EUR) when in the money. Total certainty gives no additional value. The same is true if the volatility is

zero, as again there’s total certainty about where EURUSD will be in the future.

As we increase the time to maturity (from red to green to blue) the value of the option increases everywhere,but by the most in the middle near the strike. Exactly the same is true as we increase the volatility – moreuncertainty leads to more probability of the option ending up in the money, and hence a higher value (NB thisis simply the same as saying an option is long Vega).

It is useful to think of the volatility as setting the scale for the x-axis. In a low volatility environment a movefrom 1.3600 to 1.3700 is much larger (in terns of its likelihood of happening, or, more precisely, in terms of the number of standard deviations this represents) than the same move in a high volatility environment.Increasing the volatility has the same effect as stretching the scale on the x-axis.

9.2 Delta

Consider a similar chart for the Delta of the same option :

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Delta for different Maturity/Volatil ity

0

20,000,000

40,000,000

60,000,000

80,000,000

100,000,000

120,000,000

1.3300 1.3400 1.3500 1.3600 1.3700 1.3800 1.3900 1.4000 1.4100 1.4200 1.4300 1.4400

Zero

Low

High

At the moment of expiry, or in a zero volatility environment, the Delta is either exactly zero (out of the money)or exactly 100% (in the money) – there is no uncertainty left.

As we increase the time left to expiry (green line), or equivalently increase the volatility, the Delta acquires itsfamiliar S-shape – basically 50% Delta at the strike, and tending smoothly to 0/100 in the limits.

Focus for a moment on the 1.3700 spot on the graph above. Our 1.3845 EUR call is totally out of the money(zero Delta) in the zero volatility environment (red line), roughly 2% Delta in the low volatility environment(green line), and about 10% Delta in the higher volatility environment.

Delta is very useful as a measure of distance to a strike (and is related, mathematically, to the number of standard deviations between us and the strike). A low Delta option is far away, a 50-delta option is at-the-money, and therefore very close by, and a high delta option is far away (but in the money).

What we observe is that the same 1.3845 EUR call, as viewed from 1.3700 spot, is closer to us (10 delta vs 2delta) in the high volatility environment compared with the low volatility world. By “closer” I mean “more at-the-money” or more likely to be in-the-money at expiry, and therefore exercised.

The volatility again sets the scale for the x-axis in the graph above –increasing the volatility is the same asstretching the x-axis. In a ludicrously high-volatility limit then basically all options are ATM, and the Delta

profile would be a flat horizontal line at 50-delta.

9.3 Gamma

Consider the same picture for Gamma :

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Vega for Different Volatility Levels

0

10,000

20,000

30,000

40,000

50,000

60,000

1.30000 1.32000 1.34000 1.36000 1.38000 1.40000 1.42000 1.44000 1.46000 1.48000

Low Vol

High Vol

This is easy to interpret – increasing the implied volatility is like stretching the x-axis, so the Vega profile getsbroader – if we look at the option from 1.3600 spot (say) then off a higher vol base the option has come closer(it will be a higher delta), it has become more like an ATM option, and therefore has more Vega. The ATMoption itself can’t get any more ATM than it already is, and its vega doesn’t change. In a world wherevolatilities are infinitely high, basically all options are ATM, and all have the same Vega.

Note that the vega profile extends further out (in spot) than the Gamma profile.

Now we turn our attention to how Vega changes for options of different maturities :

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Vega for Different Maturities

0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

80,000

90,000

1.30000 1.32000 1.34000 1.36000 1.38000 1.40000 1.42000 1.44000 1.46000 1.48000

Short Date

Longer Date

The blue line is a 1week 1.3845 EUR call, the green line is a 2wk 1.3845 EUR call.

Now we have two effects – first the Vega profile gets stretched along the x-axis : this is just the familiar effectwhere with more uncertainty, all options become more ATM than they were before.

But now the peak is also higher. Since variance is the product of volatility (squared) and maturity, for a longermaturity any increase in volatility has a longer time to work its magic, so has a relatively larger effect. If Iincrease volatility from 10% to 11% for a week, this will have a smaller effect than the same increase for twoweeks, so a two week option has more Vega than a one week option.

Indeed (geek alert) because Variance equals Volatility² × Maturity, the Vega of options scales as the squareroot of the Maturity. So a two week ATM option has √2 (approximately 1.4) times as much Vega as a oneweek ATM option. If you remember this, and remember that a 1Y ATM option has 0.4% of face Vega (so100mio EUR of 1Y EURUSD ATM option has €400k of Vega), then you can figure out the Vega of any ATMoption (e.g. a 3Mth ATM option has 0.2% Vega, as √(0.25)=0.5).

Week 4

10 : Cut Times

Vanilla options expire at a specific time on their day of expiry, called the “cut”. The vast majority of optionsexpire at “New York Cut”, which is 10am NY time (EST).

Some options (especially those dealt in Asia) expire at “Tokyo Cut,” which is 3pm Tokyo time. Emergingmarket pairs often have their own cut (RUB expires at Moscow cut, PLN at Warsaw, etc.) Each pair has adefault cut, and if you don’t (or your customer doesn’t) qualify the price request, you will get a price for thedefault cut.

This is something to be especially wary of in Asia time where, even for G10, both NY and TOK cut options traderegularly.

Why is it important? Well if the option expires earlier in the day, then there is less time to expiry (lessvariance) so TOK cut options (with the same strike) are always cheaper than their NY equivalent.

Indeed we observed that the difference in price between 1week EURUSD NY and TOK cut ATM options wasnearly 1vol.

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So be careful when you ask, and don’t be offended if a nearby options trader tries to double-check which cuttime you want (and don’t be afraid to double-check with your customer in turn).

11 : ATM Options, a More Precise Definition

We have repeatedly referred to “ATM” options, loosely, as options whose strikes are nearby in some sense. Inaddition to this rough meaning of ATM there are some precise definitions of some specific “nearby” strikes :

Delta-Neutral-Straddle (DNS, almost always called, simply, “ATM”). This is the (single) strike at which a calland a put have the same (offsetting) Delta. Hence if you buy both the call and the put with the same strike (astructure called a straddle, see section 13), you are automatically Delta-neutral.

ATMF (at-the-money-forward) : this strike is precisely at the outright forward (i.e. spot+swap points). The “ATM” part of the name is fairly redundant here.

ATMS (at-the-money-spot) : this strike is set at current spot. Here the “ATM” part of the name is actuallymisleading rather than redundant – there are many markets where the ATMS is actually a long way from At-the-Money in the rough sense we’re used to (e.g. a 1y USDBRL ATMS option is actually a 30-Delta low-strike!)

Of these, by far the most common and useful is the DNS; if you ask for “1month ATM in EURUSD” you will getthis strike. Some emerging markets (e.g. BRL) and very long-dated G10 options (post 10y expiry) default toATMF for the standard (i.e. interbank-quoted) “at-the-money” option.

Make sure you are clear about which strike you want – it is horribly common to have disagreements aboutstrikes; customers used to other asset classes are often ignorant of the DNS (rates traders will insist that ATMFis the “at-the-money”, and equities traders have a mysterious fondness for ATMS). Do not be afraid to clarifywith your customer, and do not be upset if a trader queries exactly what strike you want.

12 : A Subtlety about Delta

Delta is a ubiquitous measure in the FX options market. When we say an option is 25-delta, this is for thepackage of the option and the premium we pay (or receive) for it. This means that the premium currencyaffects the delta of an option, or equivalently affects precisely what strike we’re talking about when we requesta specific delta option.

For example, at the time of writing, a 1Y EURUSD 1.2700 EUR put is worth about 3% EUR, or equivalently 410USD pips (i.e. for €100m face, either €3,000,000 or $4,100,000, depending on which currency we pay the

premium). Note that the two premia have the same value today, so the option’s value isn’t affected, but theDelta of the package is.

Let us imagine for a moment we account in USD. If we pay for this option in USD, then note that the premiumbears no FX risk at all for us (its value to us does not depend on EURUSD spot). For us this option is a 30-delta option (as it happens).

Now consider if we were to pay for the same option in EUR instead. Now we have borrowed €3mio fromsomewhere to pay for this option. The value to us (in USD) of this liability is therefore variable with EURUSDspot – indeed it has an FX delta of €3mio!

So if we pay a EUR premium for this option then we are short an additional 3 EUR of delta (per 100 face)compared with someone who has paid in USD.

And, indeed, if we price up this option we will see it is a 33-delta EUR put - in a sense the option contributes

short 30 (remember this is a EUR put!), and the premium an extra 3.

So note an imporant point – the exact DNS strike (see section 11) is dependent on which premium currencyyou use, because the definition is in terms of delta.

In practice each currency pair has a default premium currency, which everyone agrees to use. So notecarefully if a customer tries to switch to an unusual premium currency after dealing with you – it will changethe delta you have to cover, and possibly change the details of the whole trade!

The default premium currency is always USD if it is part of the pair. Otherwise it is normally the basecurrency. So EURUSD has default premium currency USD, EURGBP is EUR, etc.

13 : Simple Option Structures

13.1 : Straddles

A straddle is a Call and a Put at the same strike (in the same amounts). A straddle is indistinguishable (interms of Delta, Gamma, Vega) from a delta-hedged Call in the full amount, or a delta-hedged Put in the full

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amount. The Vega profile is therefore exactly what we’ve already discussed for a simple vanilla option (seesection 7).

The most common option contract is the ATM Straddle. It is delta neutral by construction because the ATMstrike (DNS) is defined that way.

13.2 : Risk Reversals

A Risk Reversal (RR) is a spread of a Call and a Put. E.g. one might buy a put and sell a call to obtain a payoff profile like this (this is for a buyer of the low-strike and seller of the high-strike) :

RR Profile

-4,000,000

-3,000,000

-2,000,000

-1,000,000

0

1,000,000

2,000,000

3,000,000

4,000,000

1.28000 1.30000 1.32000 1.34000 1.36000 1.38000 1.40000 1.42000 1.44000 1.46000

Note that it is normal to quote a specific delta RR, e.g. a 25-delta RR, in which case both strikes will be 25-delta by themselves. Note that the deltas of the two options reinforce each other, and for the 25-delta RR, thedelta of the structure is 50 (for a 15-delta RR it would be 30, and so on).

NB Risk Reversals are sometimes called “collars” and “cylinders”.

Let’s look at the vega profile we get from the RR above :

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RR Vega Profile

-80,000

-60,000

-40,000

-20,000

0

20,000

40,000

60,000

80,000

1.28000 1.30000 1.32000 1.34000 1.36000 1.38000 1.40000 1.42000 1.44000 1.46000

Low K

High K

RR

The blue line is the vega profile of the low strike by itself, the green line is the vega profile of the (short) highstrike by itself; the red line is the vega profile of the sum of the two, i.e. of the RR. Note that at current spot(in the middle) the RR is vega-neutral!

13.3 : Strangles

A Strangle is a structure where we buy (or sell) both a call and a put, but at different strikes. Again it is mostcommon to quote strangles where the two options have the same delta, e.g. 10-delta, in which case it is a 10-delta strangle. Note that the two option deltas cancel out, so in fact most strangles are approximately delta-neutral. A Strangle’s payout profile looks like this (similar to how you might hold your hands if you were tostrangle someone!) :

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STR Profile

0

500,000

1,000,000

1,500,000

2,000,000

2,500,000

3,000,000

3,500,000

1.26000 1.28000 1.30000 1.32000 1.34000 1.36000 1.38000 1.40000 1.42000 1.44000 1.46000

STR Profile

Note that a strangle will be cheaper than an ATM straddle, because the options we’re buying (or selling) arelower-delta and hence lower premium.

Let us look at the vega profile from the strangle :

STR Vega Profile

0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

1.26000 1.28000 1.30000 1.32000 1.34000 1.36000 1.38000 1.40000 1.42000 1.44000 1.46000

Low K

High K

STR

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The blue line is the contribution from the low-strike, the green is the contribution from the high-strike, and thered line is the sum, i.e. the vega profile of the strangle. A strangle will have a wider vega-profile than a simplestraddle, but note the twin vega peaks which are generally at lower-delta (i.e. not ATM, as would be the casefor an ATM straddle).

14 : The Volatility Smile

Up to now we have used the Black–Scholes model, which says that the percentage returns of spot exchangerates are normally distributed, and gives us a very neat and simple formula to price vanilla options in terms of a parameter called the volatility and some other market data inputs (interest rates and spot). In the pureBlack–Scholes model the volatility is simply related to the width of the distribution of spot returns, and is aconstant.

In practice, volatilities are not constant, they change as the world becomes more or less certain. Also, inpractice, we find that spot returns are not quite normally distributed, they exhibit extreme moves more oftenthan a normal distribution predicts.

The Black–Scholes model is just an approximation, but is an extremely useful starting point, and if we bend itsrules slightly it is still very powerful. The volatilities we quote for pricing options are just a code for the actualmarket prices – of course in the end we will only ever pay real money when we buy options (i.e. not vols!) sothe volatilities we quote are simply implied volatilities, i.e. a number we must pass into a coding machinecalled the Black–Scholes formula in order to generate the market prices of options.

This would be an insane thing to do except for the fact that the coding machine (the Black–Scholes model) isclose enough to reality to give us some intuition, and in particular we have a strong intuition about what theinput parameter (volatility) means.

So how do we go about bending Black–Scholes to fit the market? We find it is necessary to use a differentimplied volatility for each strike (and maturity). This generates what is called the volatility smile :

A Volatility Smile

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.9 1.1 1.3 1.5 1.7 1.9 2.1

Vol Smile

ATM

This is a graph of the volatility one must put into the Black–Scholes formula to match the market price of different strikes. Note that the x-axis above is the option strike, not spot. It exhibits two interesting features :

• Curvature – the smile curves upward, higher thatn the ATM vol, for both high- and low-strikes. This simplysays the market is willing to pay more for these options (or demands to earn more to sell them) than a simpleBlack–Scholes model predicts. This isn’t a huge surprise – we have already mentioned that in reality spot

markets exhibt extreme returns much more often than the simple normal model on which Black–Scholes isbased, so these distant (low-delta) options are more likely to end up in-the-money than Black–Scholes predicts(i.e. options struck there must be more expensive).

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• Skew – the smile above shows a slight skew : the 1.3 strikes trade at a volatility higher than the ATM strikes,while options a similar distance above the ATM (1.7 or so) are at roughly the ATM vol. This is an extremelycommon feature of FX volatility smiles; there are many reasons why it can develop. The simplest is simplydemand/supply issues : as spot goes down (for example) there is a natural demand for options in the directionspot is moving (either as protection from the move, or as a bet that it continues). Similarly people who havebet successfully on the move may well be taking profit on (in-the-money) options which are now high-strikes

(and maybe rolling them into new low-strike positions). This generates a natural demand for options in thedirection spot is moving to (especially if it is doing so in a noisy fashion), and a supply of options where spothas come from.

Note there are plenty of markets which exhibit a structural skew irrespective of which direction spot is moving– this is very common in Emerging Markets. Again this simply arises from imbalanced supply/demand : youwill no doubt have heard lots of talk in 2010 about buying Asian emerging market currencies against the USD,a classic example of the “carry trade.” Carry trades have a habit of appreciating slowly and then going wrongin a blaze of (volatile) glory – the massive structural long positions in Asian currencies vs short USD generate anatural skew to the option smile in favour of Asian puts / USD calls (as protection from the carry tradeunwind).

The above volatility smile is for one particular maturity – we can imagine lots of them together for everydifferent possible maturity, leading to what we call a volatility surface.

15 : Changes to the Volatility Surface

Some traders will use options to express fairly simple spot views, but there is much more to the optionsmarket than this, and to really be able to trade properly (and to advise our clients on how to best-execute theirdesired trades, we need to think beyond simple spot views.

The volatility surface will change shape with time and changing markets. Understanding options markets islargely about understanding what these changes are, how they have come about, and understanding the jargon that options traders use to describe them.

It is useful to consider three idealized volatility smile changes; in practice real vol surfaces experiencedistortions which are some combination of these changes,

Let’s begin with the simplest change an options surface can undergo, namely a near-parallel shift, i.e. a changein the overall level of volatility in a currency pair, without the shape of the surface changing (here as we gofrom Smile 1 to Smile 2, the world has become more uncertain, with volatility increasing globally) :

Mode 1 : Increase in Overall Volatility

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.9 1.1 1.3 1.5 1.7 1.9 2.1

Smile 1

Smile 2

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Owners of options of any strike will make money as we go from Smile 1 to Smile 2 (because all vanilla optionsare long vega!). Owners of ATM-options will make the most money (because they have the most vega). Sothe best option strategy to consider if you think the overall volatility of the market is mispriced (or about tochange) is an ATM straddle.

Traders with Risk Reversal positions will make money on one half of their strategy, and lose money on theother side, so are relatively neutral on the move above. Owners of strangles will also make money, but not as

much as owners of ATM options (as they have less vega).

The second key mode to consider is a change in the skew of the smile :

Mode 2 : Increase in Skew (Risk Reversal)

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.9 1.1 1.3 1.5 1.7 1.9 2.1

Smile 1

Smile 2

In the evolution from Smile 1 to Smile 2 above the skew of the smile has increased : low-strikes have becomemore valuable (more bid) and high strikes have become less valuable (more offered). Note that the overalllevel of volatility hasn’t really changed. Owners of ATM straddles will experience no real change in the value of their positions, whereas people who are long low-strikes will make money, and people who are long highstrikes will lose money.

The strategy which benefits most from changes in the skew (as the title of the graph suggests) is the Risk

Reversal. Here the owner of the strategy shown in section 13.2 (long the low-strike and short the high-strike) will benefit from this change to the surface (on both legs – their low-strike has become more valuableat the same time as the high-strike they’re short has gone down in value).

In the above change from Smile 1 to Smile 2 we would say the Risk Reversal has become more bid for low-strikes.

Note that owners of strangles will also experience no real change in the value of their position as the change invalue of one option will largely offset the other.

Finally consider this mode, a change in the overall curvature of the smile :

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Mode 3 : Increase in Curvature (Fly)

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.9 1.1 1.3 1.5 1.7 1.9 2.1

Smile 1

Smile 2

Here note that owners of ATM options will experience no real change in their position (the overall level of volatility has not changed), and traders with Risk Reversals will have offsetting profit and loss from each half of their trade. Owners of strangles will benefit most from the change above (from Smile 1 to Smile 2).

Note that owners of strangles have a mixed exposure – they will make some money from both modes 1 and 3;indeed for a pure change in the curvature of the smile, owners of strangles have some wasted exposure (theirnet vega). We can do better than a simple strangle, with what is called a (vega-neutral) butterfly (or simply,fly).

16 : The Butterfly

A butterfly is an option strategy designed to be insensitive to modes 1 and 2 in section 15, and benefitmaximally from a change in the overall curvature of the smile (mode 3). It is a spread of a strangle and anATM straddle, organized to be vega-neutral at inception :

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Fly Vega Profile

-60,000

-40,000

-20,000

0

20,000

40,000

60,000

80,000

1.26000 1.28000 1.30000 1.32000 1.34000 1.36000 1.38000 1.40000 1.42000 1.44000 1.46000

Strangle

DNS

Fly

The blue line is the vega profile of the strangle (see section 13.3), the green line is a short position in an ATMstraddle, in an amount designed to precisely vega-neutralize the straddle at current spot. The red line is thesum of the two, and therefore the vega profile of the butterfly.

17 : Summary of Volatility Smile Changes vs Strategies

Let us summarize the fundamental modes of change in volatility smiles, and their effect on the value (PnL) of the option strategies we’ve discussed :

PnL effect ATM straddle Risk Reversal Butterfly

Change in overall vol Large Zero Zero

Change in skew Zero Large Zero

Change in curvature Zero Zero Large

It will now be a lot easier for us to analyse options markets and decide on what sort of strategy will benefit usmost, given views or other information about the volatility surface.

Example :

Imagine we think that the skew of the volatility surface is low (either historically or from some other analysis)and simultaneously think that volatilities are high. We are considering a combination of modes 1 (in reverse)and mode 2 from section 15. Our view is that the smile will evolve from Smile 1 to Smile 2 as below :

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Example Smile Evolution

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.9 1.1 1.3 1.5 1.7 1.9 2.1

Smile 1

Smile 2

ATM

The optimal strategy is to be short a high-strike, as this is the part of the smile which is changing most. Notethat we will have modest gains if we are either short an ATM option (alone) or simply long a Risk Reversal(long the low-strike and short the high-strike), but neither of these is optimized to our view.

Week 5

18 : The Volatility Surface in RBS’s Systems

We have observed that we have three structures (ATM Straddle, RR, Fly) which are each sensitive to preciselyone of the main shape changes the vol surface can undergo. It is therefore natural to use the price of theseinstruments to parameterize the volatility surface itself. Here is a screenshot of the EURUSD surface out to 5Yfrom XTrader :

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Note the “Cut” and “Prem” default boxes – respectively “NYK” (New York Cut) and USD-premium as discussedpreviously.

For each Tenor (standard expiry date) we see the following :

ATM – the price (in vols) of the ATM straddle (the little flag “a” or “f” tells us whether this is a DNS or an ATMFrespectively)

RR – the price (in vols) of the 15-delta (this is what the flag “15s/15f” tells us) Risk Reversal. In RBS systemsthe number quoted for the Risk Reversal price is always the vol for the low strike minus the vol for the highstrike, so in EURUSD above the smile is “bid for low strikes” i.e. skewed much like the pictures above.Something like USDZAR will be skewed the other way (bid for high strikes), and the RR numbers will benegative (and red).

Fly – the price (in vols) of the 15-delta Butterfly. This is the average price (in vols) of the 15-delta stranglestrikes minus the price of the ATM (recall the fly is a spread of a strangle vs a straddle).

So for example the 2Mth row says the “central” vol of the smile is 13.75 (for the DNS), and the 15-delta RiskReversal and Fly are 2.65 vols and 0.85 vols respectively.

With a little bit of maths this tells us the vol prices of three strikes in the market (the ATM and the two 15-delta strikes), and our model then fills in all the gaps with a smooth curve, ending up with a picture much likethe example smiles we’ve been looking at up to now.

A note on the flag which says “15s/15f” : the little s/f flag simply tells us whether the delta is a forward deltaor a spot delta. We have a choice when we delta hedge an option whether we use a spot deal or an outrightforward. Because of discounting effects an outright forward in a notional amount of (e.g.) 25 will have an FXdelta of slightly less than 25 (where a spot deal would be exactly 25).

Up to now when we have defined a 25-delta strike we have meant a strike which is precisely delta-hedged by25% face of a spot deal. It is also possible to define a 25-delta strike to mean a strike which is precisely deltahedged by 25% face of an outright forward to the expiry date. This latter is a slightly different strike, and is

called a 25-forward -delta strike (and would get a little “f” flag in the vol form above).

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NB I attempted to demonstrate this effect but was destroyed by the very low USD interest rates and NMOSS’srounding – I will demonstrate again in AUDUSD where the effect is noticeable (or try it yourself).

19 : Terminology for Trading Risk Reversals

Traders have an unfortunate habit of ambiguous statements like “buying the Risk Reversal” – you should be

aware that this means different things in different markets.

When the RR is clearly bid for low-strikes (a positive number in RBS’s systems, e.g. AUDJPY, EURUSD, etc. attime of writing), then “buying RR” means buying the low strike, and selling the high strike. The trader isbuying sensitivity to the RR becoming more skewed in the direction it already points.

When the RR is clearly bid for high-strikes (a negative number in RBS’s systems, e.g. USDZAR, USDTRY), then “buying the RR” means buying the high-strike and selling the low-strike. Again, note that the trader will makemoney if the RR becomes more skewed than it already is – this is the sense in which the trader means she is “buying” the RR.

Note that this breaks down entirely when the RR is nearly zero and it is no longer obvious what “buying theRR” means. At this point it is essential that traders be clear with each other and an example price might thenbe : “0.2-0.4 favouring [or bid for] low-strikes”.

If you’re unsure what a price means : ask. This costs less than dealing the wrong way around!

20 : Live Pricing vs Delta Exchange

As a trading desk we assume that any deal done by sales with a client will arrive in our book delta-hedged.This has many benefits, not least that it gives you a bit of breathing space between executing the deal andbooking it!

There are two main ways that this delta-hedge can be executed, but note that it is always the responsibility of the person executing the option deal to sort out the delta (so, when dealing with clients, it is the sales person’sresponsibility). Which situation you find yourself in is largely dependent on the behaviour of your customer(they will tend to prefer one to the other) but in principle anyone can ask for either.

20.1 : Delta Exchange

This is the simplest case : we agree to exchange the requried delta as part of the contract we’re pricing. The

interbank market (both through the brokers and direct) operate this way. So, for example, if I buy 100 of 1y30-delta USD call, JPY put with some bank, I will also sell them 30 USDJPY of spot (or an outright forward).

If your customer asks for a price and gives you a spot reference then it is understood they will exchange deltawith you at that spot rate (otherwise, you are right to ask, why have they mentioned it?).

So for example if they ask for 100 of 1y 30-delta USD call, JPY put with a spot reference of 83.90 then thecontract you are pricing will include 30 USDJPY at 83.90 (which obviously has no cost), so everyone can focuson the option price without worrying too much about where spot is.

Watch out for two things :

• Check the spot reference they’ve given you is not miles off market

• Customers who ask for a price with a spot reference, but then try to deal live (see next section) at the last

minute – they’re almost certainly trying to pick you off!

Note that a trader may well give you a price with a spot reference even if you have to turn this into a live pricefor your customer. I’ll tell you how to do this in week 6.

20.2 : Live Pricing

In this case the customer doesn’t want the delta hedge, and is only interested in the option. This means youhave to execute the required delta hedge in the market at the time of trading.

We’ll return to how to do this for exotics after we’ve studied a few of them, but for a vanilla you will obtain avolatility price from a trader (or from the system). The system can turn this into an appropriate premium pricefor you given the prevailing spot rate. Crucially you have to tell the system what is the prevailing spot rate(bid and offer). Enter the vol price you are given in the “Spot Prem Vol” box below (if it differs from theautomatic system bid/offer).

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You can enter the correct spot rate in the “Spot” box near the top – and use the cursor keys to make the ratetick up/down. It may be wise to give yourself a bit of slippage, so price the option using a wider spot rate thanyou see in the market so that you can hold the price for a little (and maybe take some AV if you can executeyour hedge better). Note that the premium price shown by the pricer above includes the effect of the spotwidth (so the offer for this option is priced using the bid of spot, and the bid is generated from the offer of spot– I set up a USD put which has a negative delta, which is why the bids/offers are crossed).

If the customer deals this option with you the instruction is to sell 13 USD (vs JPY) at 83.77 (the spot bidyou’ve entered) – this should be the FIRST thing you do as soon as the customer says the deal is done.

Obviously if you’ve priced the option off a spot bid of 83.77, but manage to sell your 13 USD higher, then thatmoney is extra AV. However you should not leave deltas unhedged in the hopes of punting a bit and making

extra money.

If a customer tries to switch from pricing off a spot reference (implication : they will exchange delta) to a liveprice, then you should essentialy consider this a new option price, and you need to double-check where youcan hedge the delta etc.

Week 6

21 : Pricing Spreads of Options

A spread of options in when you deal a collection of options, buying some and selling others. A perfectexample is a Risk Reversal, which is a low strike spread against a high strike, i.e. one is bought and one issold.

You should be familiar with how vol prices are quoted for spreads : it is common to “choice” one leg and

spread the other (or if there are several options, choice all but one of them, and spread the last).

A “choice” price is one price on which you (i.e. your customer) can either buy or sell (at his or her choice). Itdoesn’t happen very often on individual options (generally only when someone is showing off) but it is a crucialpart of making a spread. For example, consider this spread :

Request : 2Y NZDUSD 0.60 vs 0.85 on 100 NZD.Price : 18.75 ch the 0.60 vs 14.60/15.10

This means the client can buy the 0.85 at 15.10 vols, and sell the 0.60 at 18.75 OR sell the 0.85 strike at14.60 and buy the 0.60 at 18.75.

It is a lot easier to quote the price this way rather than spreading both options : it is easier to see the bid-offerspread, and it is less easy to make a mistake and deal both options the same way around.

Note that the net bid-offer on spreads of options such as that above should be less than on the pair of optionsindividually (since the individual prices carry the option to buy both, which obviously has more net risk thanthe spread, where there is at least some offset).

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Don’t get caught out by clients trying to deal on the choice leg but doing nothing on the other side – this mightseem obvious but I’ve seen it happen!

22 : Expiries

At some point the options you deal will expire, and you should know what to do and how to communicate what

is happening with the trading desk.

To understand the most convenient way to communicate expiries, let’s consider a typical expiry. RBS is long100m EUR of a 1.3100 EUR call, USD put, expiring in a minute or so. Spot is very close to 1.3100. Look atthe delta position we have from this option :

Spot above 1.31 : +100m EUR of delta (the option is in the money and will be exercised, generating €100)Spot below 1.31 : zero delta (the option is out of the money)

A trading desk would consider this a very unbalanced position, and would sell €50m to balance it :

Spot above 1.31 : +€50m of delta (+100 from the option, -50 from the spot deal)Spot below 1.31 : -€50m of delta (0 from the option, -50 from the spot deal)

There is nothing new here – this is simply an ATM option, so we delta hedge it with half the notional – byselling €50m – in practice of course we will have been risk managing this deal through its life, so the short 50EUR will be the aggregate of possibly hundreds of spot trades over the life of the trade.

Now before we move on, let us consider what our delta position would be if the option were a EUR put thatRBS was long instead of a EUR call. First with the option alone :

Spot above 1.31 : zero delta (the option is out of the money)Spot below 1.31 : -€100m (the option is in the money and will be exercised, generating -€100m)

Obviously this will have been delta hedged, so in practice the delta risk for the trading desk will look like this :

Spot above 1.31 : +€50m (from the spot hedge)Spot below 1.31 : -€50m (-100 from the option, +50 from the hedge)

Note that this is precisely the same as for the EUR call. This point has been made before : after delta-hedgingit is virtually impossible to distinguish between calls and puts. As a trading desk we are therefore only

interested in the net amount of strike (i.e. the amount of options) we have expiring, and not whether they arecalls or puts.

If we are long a strike, our delta will go from being short to long as we go through the strike (from below toabove), and the amount of the jump is the size of the option. If we are long a strike our delta will jump theother way, from long to short (maybe just repeat the exercise above to convince yourself of this).

In either of the scenarios above, if spot were to finish above the strike (our call ITM, or our put OTM), wewould have €50m to sell to rebalance our risk. If spot finishes below the strike (our call OTM, or our put ITM),we have €50m to buy to rebalance our risk.

So as a trading desk we don’t care whether options are calls or puts, we simply wish to hear a very clear call of “above” or “below”. And more pertinently perhaps, this is what you will get from the trading desk.

If you have a close expiry, make sure you think for a minute about what you’re going to say at the time.

Speaking the right language at expiry time will go a long way to building a good professional relationship withthe trading desk. Moreover, yelling vague statements like “I exercise!” across the trading floor will quitequickly wreck your relationship with the desk!

23 : Call-Put Parity

Consider this structure in EURUSD :

Long €100m of 1Mth 1.3100 EUR callShort €100m of 1Mth 1.3100 EUR put

At expiry, one of two things can happen :

• Spot above 1.31 : EUR call is ITM (we buy €100, sell $131), EUR put is OTM (we do nothing)• Spot below 1.31 : EUR call is OTM (we do nothing), EUR put is ITM (we buy €100, sell $131)

So either way we will buy €100, and sell $131. So there is NO optionality – only an obligation. Indeed thiscontract is precisely a long EURUSD forward.

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This underlines the fact that calls and puts are indistinguishable up to a net delta (i.e. the forward). Indeed wehave this incredibly useful relationship :

Call – Put = Forward (the call-put parity relationship)

Where it is understood the strikes and maturities all match. Note of course that a forward has no vega, further

proof that the optionality has vanished from this structure.

24 : Call Spreads and Put Spreads

Let us turn our attention to a particularly important new structure, the Call Spread (or Put Spread).

In this structure the buyer of the call spread buys an OTM call, and sells a more OTM call. It is most commonto deal them in equal amounts (a “1x1 call spread ”), but occasionally they are dealt in other ratios, e.g. 1x2call spread and so on.

Let’s look at a few spot ladders.

21.1 : Payout profile

Start with the payout profile of a 1.32/1.34 1x1 call spread in EURUSD :

Call Spread

0

500,000

1,000,000

1,500,000

2,000,000

2,500,000

1.30000 1.31000 1.32000 1.33000 1.34000 1.35000 1.36000

If spot expires below both strikes, both are OTM and expire worthless. If we expire anywhere to above the1.32 strike then the call spread is worth some money up to a maximum payout (in this case $2mio) when thesecond strike is also ITM.

The second strike eats away all the potential (infinite) upside of the first vanilla, so this call spread is MUCHcheaper than the 1.32 vanilla by itself, though obviously it has a fixed potential upside.

24.2 : Delta

Let’s look at the delta profile. Consider the 1.32/1.34 call spread with 1wk to go, with spot at 1.31. The 1wk1.32 is approximately a 35-delta option (as I price it now), whereas the 1.34 is approximately 15-delta.

So it is clear that the 1x1 call spread will have approximately 20 EUR of delta here now (35-15 = 20), i.e. lessdelta than the 1.32 call alone. This makes sense because although we still make money as spot goes up, we

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will not make as much money (because the payout is capped) and not as quickly. The call spread is a muchmore modest strategy than the outright EUR call – it has less delta and will cost a lot less.

The delta profile looks like this : the blue line is the delta profile of the 1.32 by itself, the green is the 1.34 byitself (NB we’ve sold it so we’re short delta from it), and the red line is the sum of the two lines, i.e. that of thecall spread.

-150,000,000

-100,000,000

-50,000,000

0

50,000,000

100,000,000

150,000,000

1.25000 1.27000 1.29000 1.31000 1.33000 1.35000 1.37000 1.39000

Call Spread

1.32 Call1.34 Call

Note that the owner of the call spread is always long delta – we always make money as spot goes up, but if spot goes far enough we stop caring where spot goes and the delta comes back to zero (we’re miles throughboth strikes and our PnL is $2mio no matter what.

As advertised, the delta of the call spread at 1.31 spot is approximately 20.

24.3 : Vega (and Gamma)

We can just build the vega profile as we know what the vega profiles of the constituent options look like. Theblue line is the vega profile of the 1.32 call in isolation, the green curve is the vega profile of the (short) 1.34 –note, of course, it is just the same as the blue curve shifted to the right; the red line is the sum of the two, i.e.that of the call spread.

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-60,000

-40,000

-20,000

0

20,000

40,000

60,000

1.25000 1.27000 1.29000 1.31000 1.33000 1.35000 1.37000 1.39000

Call spread

1.32 Call

1.34 Call

There are two important things to take away from this :

• The red line is makes intuitive sense : off to the left volatility is good for us as we’re more likely to end up inthe money, but have no downside if spot randomly moves much lower. To the right of both strikes we areshort vega because we want spot to stop moving around – we’ve achieved our maximum upside and only bad

things can happen if spot moves around more.• The red vega profile is exactly the same profile as that of a Risk Reversal .

This last point should not be a surprise – a Risk Reversal is a spread of a put and a call – the put only differsfrom a call by a forward, which doesn’t affect the vega profile at all. Indeed if you look at the payout picturefor the call spread above (in section 24.1), and rotate it by 45º clockwise, the picture looks identical to thepayout of a Risk Reversal. (Adding a delta hedge is exactly the same as rotating the PnL graph by someangle).

The only difference between a call spread and a risk reversal, therefore, is where spot is when you put it on.

• Risk reversal : spot is between the strikes at inception, so you end up roughly vega-neutral• Call spread : spot is to the left of both strikes at inception, so you end up small long vega• Put spread : spot is to the right of both strikes at inception, so you end up small long vega

This isn’t academic rambling : the buyer of a call spread has a more modest vega position than the buyer of avanilla outright, and moreover they have short vega as well as long vega, i.e. they have a Risk Reversal position (in the sense that the value of their position is impacted significantly by where the market is pricingRisk Reversals, i.e. to the skew of the volatility smile).

What this means is that when we are pitching options products to clients who wish to take directional bets (orexecute favourable hedges), our decisions about which products to pitch should be informed about views (oursor those of the customer) on implied volatilities, risk reversals, flies, etc.

In this example the buyer of the EUR call spread is buying Risk Reversal (i.e. their position becomes morevaluable if the smile twists more in favour of EUR puts vs EUR calls), so at the very least we should not berecommending this product at the same time as telling our clients “gosh, don’t EURUSD Risk Reversals lookincredibly high, they are sure to come off.” We wouldn’t simutaneously send research recommending bothlong and short EURUSD spot positions!

NB the Gamma profiles are the same shape as Vega.

Week 7

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25 : European Digitals

A European digital is in many ways the simplest derivative. The owner simply receives a rebate if, at themoment of expiry (often using some published fixing) spot is above some strike, and receives nothing below(for a European digital call; the reverse is true for a European digital put).

The term “European” reminds us that all the action happens at expiry only (no path dependence). This is incontrast to an American digital (or one-touch) which we’ll cover next.

Let’s turn our attention to the Greeks : the blue line below is the payout at expiry for 1mio USD rebate of a1.31 European digital call; the red line is the value with some time to go (a week in the example below) :

European Digital PV at expiry (blue) and before (red); 1mio USD rebate, valued in USD

0

200,000

400,000

600,000

800,000

1,000,000

1,200,000

1.28000 1.29000 1.30000 1.31000 1.32000 1.33000 1.34000

This graph may look oddly familiar – it looks exactly like the picture we would draw for the Delta of a vanillaEUR call option. This isn’t a coincidence, and we will find that all the Greeks for the European digital are oneout of step with those for a vanilla (so the Delta will look like the vanilla’s Gamma, and so on).

The above is for a USD-rebate digital, valued in USD. Let us quickly look at a EUR-rebate digital, also valuedin USD :

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European Digital PV at expiry (blue) and before (red); 1mio EUR rebate, valued in USD

0

200,000

400,000

600,000

800,000

1,000,000

1,200,000

1,400,000

1,600,000

1.28000 1.29000 1.30000 1.31000 1.32000 1.33000 1.34000

The picture looks very similar, but there are two key differences :

• The size of the PnL jump is now USD 1,310,000, which is of course just the value of €1mio at 1.31 spot.• The payout (blue line) has developed a very slight slope to the right of the picture. This is because thepayout is fixed in EUR, which is obviously worth more USD on a higher spot (e.g. USD 1,330,000 at 1.33 spot).

So the payout has a small intrinsic delta.

Let’s have a look at Delta and Gamma (we’ve gone back to the 1m USD rebate 1.31 European digital call):

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-30,000,000

-20,000,000

-10,000,000

0

10,000,000

20,000,000

30,000,000

40,000,000

1.26000 1.27000 1.28000 1.29000 1.30000 1.31000 1.32000 1.33000 1.34000 1.35000 1.36000

Delta

Gamma

Note, as promised, that the delta profile looks the same as the gamma profile of a vanilla. It is positiveeverywhere (the owner always wants spot to go up), but once the digital is in the money there is diminishingreturn in spot continuing to increase, so the delta decays back to zero.

The gamma profile is similarly simple to understand : to the left of the strike we have no downside and plenty

(1mio USD) of upside, so spot movement is good for us (it only has upside). The reverse is true when thedigital is in the money – random spot movement can only have negative consequences, as the digital can’t getany more in the money, but could easily end up out of the money. Hence we get the classic gamma profileshown in red above.

Note that the gamma profile is essentially the same as that of a call spread, and hence of a Risk Reversal, sothe full market price of a European digital is dependent on the shape of the smile (i.e. on where the RiskReversal trades).

The European digital is like an extremely tight vanilla call spread, so the greek profiles shown above look verysimilar to those we’ve studied already. (And, behind the scenes this is the mathematical reason why thegreeks end up very closely related to vanilla option greeks.)

Note that when spot is at the strike, the European digital is worth 50% right up to the moment of expiry. Thisisn’t surprising as we have a 50-50 chance of moving either way and ending up with a digital ultimately worth

100%, or zero.

Note that the delta has nothing to do with probability of being in the money (this is ONLY true (and onlyapproximately) for vanillas). Actually the probability of being in the money for a European digital is simply itsprice!

Finally, observe that we can build a vanilla option from a pair of European digitals. Consider the following :

• Long €100mio of 1.31 European digital call• Short $131mio of 1.31 European digital call

If spot at expiry is below 1.31, nothing happens (with either digital). If spot finishes above 1.31, then thestructure above is identical to an obligation to buy €100mio and sell $131mio, i.e. to buy 100 EURUSD at 1.31,which is precisely the same as the economics of a vanilla EUR call option struck at 1.31.

NB to set up a European Digital in NMOSS use the codes EC and EP (for calls and puts respectively). If youdon’t explicitly request otherwise, trading will assume that the rebate currency is the base currency, and giveyou a price in percentage of base.

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26 : American Digitals (One-Touches)

Here is our first proper exotic. In contrast to the European digital, these American digitals are path-dependent, and depend on the whole history of spot from inception of the deal to expiry.

So an American-style digital (normally called a One-Touch) will pay the owner a rebate if at any time duringthe life of the trade, the barrier level is touched.

For the same barrier level this will clearly be more expensive than a European digital (in fact the value of anAmerican-style digital is more or less double that of a European, as we shall see).

We will collect some thoughts on how we go about monitoring barriers after we’ve looked at a few moreproducts.

Setting up one-touches in NMOSS requires a bit of decoding. We use a four-letter code built as follows :

A {U or D} {I or O} {M or H}

The A stands for “American”, the {U,D} are for up or down barriers, the {I,O} denote knock-in or knock-outbarriers, and the {M,H} determine whether any possible rebate is payable at Maturity or at barrier Hit.

So for example a one-touch, which pays a rebate at maturity if EURUSD touches 1.34 (with spot at 1.31)would be considered an AUIM. 1.34 is an Upper barrier, the owner gets the rebate if the barrier touches (sothe product knocks In to the rebate), and any rebate is due at Maturity (which is the overwhelming default).

For each one-touch there’s an equivalent no-touch (AUOM for the example above) which pays a rebate unless

the barrier triggers.

Let’s look at the Greeks we get from a one-touch :

NPV

0

200,000

400,000

600,000

800,000

1,000,000

1,200,000

1.26000 1.27000 1.28000 1.29000 1.30000 1.31000 1.32000 1.33000 1.34000 1.35000 1.36000

With spot far away from the barrier it is worth virtually nothing, and as spot gets closer to the barrier the valuegradually increases until the barrier is touched (at which point it is permanently worht the rebate amount,subject to some discounting).

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Note that with spot above the barrier, the one-touch is simply cash, and will therefore have no interestingGreeks of any kind (your bank statement doesn’t include Gamma or Vega!)

You might observe that this NPV profile looks very similar to the profile of the European digital (until thebarrier is hit).

Delta and Gamma look like this :

0

5,000,000

10,000,000

15,000,000

20,000,000

25,000,000

30,000,000

35,000,000

40,000,000

45,000,000

50,000,000

1.26000 1.27000 1.28000 1.29000 1.30000 1.31000 1.32000 1.33000 1.34000 1.35000 1.36000

Delta

Gamma

As promised the delta and gamma both vanish above the barrier. Note however that the delta jumps to zerodiscontinuously (you can see this in the NPV graph because there’s a kink in the curve at the barrier). Thisdelta jump has important consequences, and occurs for many barrier exotics.

Consider what happens when the trading desk is risk-managing the above position. With spot at 1.33 we willhave (net) sold approximately 42 EURUSD to be delta-neutral. As spot ticks up toward 1.34 this will increaseto a short position of 45 EURUSD. However as spot touches the barrier the underlying OT essentially vanishes,and we’re left with the naked hedge which has to be covered. In this case the desk will have to buy 45EURUSD as spot goes through 1.34 (a stop-loss). In practice we will pass this order to the spot desk to watchfor us, and you will hear barrier-related stop-loss orders being executed all the time (and may observe theirimpact on the spot market if the orders are large enough).

Note that both the delta and gamma graphs look like the left-hand half of the same pictures for a Europeandigital! We can use this intuition to predict what we expect the profiles to look like if we raise the vol base (orincrease the time to maturity) – as normal this is simply a change of scale on the x-axis, but this time thegraphs have a fixed point – the barrier – through which nothing changes :

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NPV (blue) vs NPV off higher vol (red)

0

200,000

400,000

600,000

800,000

1,000,000

1,200,000

1.26000 1.27000 1.28000 1.29000 1.30000 1.31000 1.32000 1.33000 1.34000 1.35000 1.36000

So from the various connexions we have drawn this looks simply like the picture we drew a long time ago forthe vanilla delta on two different vol bases (the left-hand half of it at least). The one-touch must worth moreeverywhere if we raise the volatility or the time to maturity – in both cases we increase the probability of hitting the barrier before expiry (indeed this is just the statement that the OT is long vega).

The delta change on a higher vol will also look familiar from our understanding of how the vanilla gammachanges (from tall narrow peak to a lower, broader one) :

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0

5,000,000

10,000,000

15,000,000

20,000,000

25,000,000

30,000,000

35,000,000

40,000,000

45,000,000

50,000,000

1.2600

0

1.2700

0

1.2800

0

1.2900

0

1.3000

0

1.3100

0

1.3200

0

1.3300

0

1.3400

0

1.3500

0

1.3600

0

Delta

Gamma

Delta (higher vol)

Gamma (higher vol)

Finally let us make our connexion between vanillas, European digitals and one-touches more explicit. The priceof a European digital is essentially precisely the probability of it ending up in the money; this role is played bythe delta for a vanilla (approximately). Also, we observe that at any time during the life of a European digital,it is worth essentially 50% of its rebate. Given that, trivially, a one-touch is worth 100% at its barrier, we findthat the one-touch behaves very much like double the amount of a European digital (check you follow this!)

So, approximately (assuming everything has the same maturity, strikes etc.) :

Price of One-Touch = 2 × Price of European Digital

Price of European Digital = Delta of Vanilla

So, e.g. a one-touch with a barrier at the strike of a 20-delta vanilla will have a price of approximately 40%.

These are an incredibly useful couple of rules – for a vanilla trader the delta of vanilla contracts determine thescale (i.e. tell us how “far away” options are); for an exotics trader this role is played by the prices of Europeanand American digitals. If you sit anywhere near exotics traders while they are pricing anything, you willregularly hear questions like “what is the TV of the equivalent one-touch?” (TV is just the Black–Scholestheoretical value, i.e. the price in a particularly simple model).

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