Welcome to the Volatility Handbook, here is where I will aim to introduce you to volatility, concepts surrounding volatility, how volatility can be traded, and how this differs to directional trading in outright contracts.

Let's begin!

Fair warning first, there is some maths involved here.

But nothing that can't be sorted with some Googling or Investopedia.

### What is Volatility?

There are a multitude of definitions and interpretations for the word 'volatility'.

Some may refer to it in the qualitative sense as the choppiness of an instrument, whilst others, including me, describe volatility as a quantitative measure of dispersion.

Qualitative: "This asset moves around a f*ck-tonne"
Quantitative:   "This asset has a volatility of 85%, that's a f*ck-tonne"

The typical (read: standard) definition of asset volatility is the standard deviation of logarithmic returns.

Where the standard deviation is the square root of variance.

Where variance is the sum of the squared distances of data-points from the mean. In mathematical notation:

To introduce the reader to the concept of standard deviation and variance, let's look at various IQs of a quasi-fictional population, Macrodesiacistan:

Let's call the average IQ of a Macrodesiac 100, we will call this the mean. Let's also say that, on average, 68% of the population's IQs will be within 20 points of the mean. We will call this +-20 point range, the standard deviation. To summarise:

The population of Macrodesiacistan is 300, the mean IQ is 100, and the standard deviation is 20. Let's look at the distribution of Macrodesiacistan's IQ:

From here it is quite clear to see that the majority of the IQs are centered around the mean, 100, and quickly fall away, with some notable outliers - an unlucky chap with an IQ of 29 (a -3.55 standard deviation value) , and a very fortunate fellow with an IQ of 158 (a 2.9 standard deviation value).

As we can see, there are 3 IQs that exceed an absolute 3 standard deviations over the mean, as a proportion of the data, that's 1%.

If we were to compare this to the density of a perfectly normal distribution we would see that this is above the expected proportion.

However, if the population were to increase to, say, 1,000,000 (I'm sure DB wouldn't mind that); we would see the proportion of excess results drop.

This is a concept known as the central limit theorem, which states that, in the limit (a very large sample size), a sample distribution will converge to a normal distribution.

With population 1,000,000:

The proportion of IQs greater than 3 absolute standard deviations now falls to a more reasonable 0.2726%.

The key point to drive home is that the mean and standard deviation of a dataset are measures of expectation. We expect 68.268% (we received 68.2%) of the data to be within 1 standard deviation of the mean**, and we expect the probability of a value larger than this occurring to decay exponentially!

** for data that is, presently, accurately described by a normal distribution.

## Asset Prices and Volatility

Now that we have discussed the use case for the standard deviation as a descriptor of dispersion, we can take that knowledge and apply it to financial markets.

A crucial exercise for volatility trading is to understand the standard deviation from the logarithmic returns between daily closing prices.

This is simply known as close-close volatility. Let's take  EURUSD as an example.

We will plot out the price plot with daily closing prices, and from these, take the daily log returns. As per our IQ example, I'll also show the standard deviation bands.

Let's address the elephant in the graph first.

Clearly, these returns are not normally distributed, though arguably they are now in recent years. 2008 and 2009 clearly go against the grain.

Did anything happen back then?

🤷

In any case, let's go to our density plot:

Perhaps it is easier to illustrate the magnitude of daily movement in terms of the number of standard deviations away from the mean.

This normalisation process is  called Z-Scoring.

Here is the corresponding plot of Z-Scored daily returns, and the accompanying density plot:

In the density plot, a normal distribution (gaussian curve) has been fitted to act as a reference point.

These moves ought not to happen.

It is pointless to give the usual '1 in x years' statistic, but their appearance shows that we should be wary of relying on assumptions of normality when trading.

Large moves tend to be created by shock catalysts, so on the rare occasion this happens:

So keep a tight stop, or use defined risk option structures - one of which I cover later on.

## Annualised Volatility

When dealing with volatility, it is convention for it to be quoted as an annualised figure. This allows for the normalisation of differing volatilities, so that an 'apples to apples' comparison can be made.

Let's first look at the annualisation calculation.

Starting with a key assumption of a random walk:

Variance grows in linear proportion to time.

This means that total variance = Variance * Time

Recall that standard deviation is the square root of variance.

If we have annual variance:

Periodised Variance * 1 year/time = Annual Variance

(Periodised Variance*1 year/time) = Annualised Standard Deviation

Knowing that the square root of the product == the product of the square roots

Standard Deviation * (1 year/time) = Annualised Standard Deviation

So consider the following:

An asset has a daily volatility of 2.5%. There are 252 business days in a year. What is the annualised volatility?

2.5% * (252/1) = 39.7%

Let's apply this to our EURUSD example:

Standard Deviation of daily log returns = 0.75%
0.75% * √(252) = 11.9%

This implies that, in a year's time, we expect EURUSD to be between +-11.9% of the current price with a likelihood of ~68%.

Charted, here's what the +-1SD range looks like:

Further from this, we can apply a fairly basic rule of thumb to transform an annualised standard deviation to an expected daily move:

Annual SD/20

Ex: 80% Annual SD = 80%/20 = +-4% daily expected move

Simple! From our annual standard deviation we have chopped down to a daily expected 'mean absolute deviation'. A handy rule of thumb, especially when we look at option trading.

## Rolling Volatility

It is important to note that sampled realised volatility that a trader receives is not necessarily the same as the population volatility.

Volatility is time variant.

Assets go through different 'regimes' of volatility. Simply, we can break these down to low, medium, and high.

Sticking with EURUSD, lets take the rolling annualised weekly volatility, and see what comes out...

This rolling feature also allows us to compare the price returns vs the change in volatility, and establish a correlation.

Before we cross the bridge to discuss volatility derivatives. This final table shows the weekly annualised volatilities for a number of asset classes:

## Options: A Volatility Derivative

As has been discussed in my beginner's guide:

Options, as per the Black-Scholes-Merton world, are first and foremost volatility derivatives.

The ability to transact in the underlying market so that the trader eliminates their exposure to that market (delta-hedging) can be a funny concept to wrap your head around. But it is that activity that provides the trader with a pure (or dirty, if you're a vol/var swap kinda trader) exposure to volatility.

Just a refresher, the P/L of an option position (with 0 carry consideration) is given by:

dΠ = (dS * Δ) + (0.5 * (dS)² * Γ) + (dσimplied * ν) + (dt * Θ ) + ...

d stands for change

Π is the option price

S is the underlying price

Δ is the trader's delta, their local sensitivity to underlying price movement (where local means for small price changes)

Γ is the trader's gamma, their delta's local sensitivity to underlying price movement

σimplied is the implied volatility

ν is the trader's vega, their local sensitivity to implied volatility movement

t is the time increment

Θ is the trader's theta, their local sensitivity to the passage of time

... are the ignored higher order terms.

For simplicity, we are ignoring higher order terms as these have little effect for small changes in the inputs.

You should note that gamma is the only second order term.

For the more mathematically inclined reader, they may consider the option's P/L to be described by a Taylor expansion, generally, up to the third order (cross derivatives included!).

Returning to the P/L...

dΠ = (dS * Δ) + (0.5 * (dS)² * Γ) + (dσimplied * ν) + (dt * Θ ) + ...

The change in option price is given by the change in underlying * delta, + half of the change in underlying price squared * gamma, + any change in implied volatility * vega, + the passage of time * theta + negligible higher order terms.

The bold part, (dS * Δ), describes the immediate sensitivity of the trader's position to movements in the underlying.

Delta (Δ) is commonly referred to as the hedge ratio, and describes how much of the option's underlying must be bought or sold to remain locally insensitive to price movement.

For the above example, we can short Δ units of underlying, and introduce the following term to the P/L: (dS * -Δ). Astute readers will realise that this now leaves the overall P/L explain with:

dΠ = (0.5 * (dS)² * Γ) + (dσimplied * ν) + (dt * Θ ) + ...

And if we drop any higher order terms and assume a Black Scholes Model world of unchanging volatilities, where dσimplied is always 0 :

dΠ = (0.5 * (dS)² * Γ) + (dt * Θ)

Where in a fair market with no obvious edge:

(0.5 * (dS)² * Γ) + (dt * Θ) = 0

and so...

(dt * Θ) = (-0.5 * (dS)² * Γ)

Where dS can be rewritten as (dS/S)²  * S².

Realising that, in expectation, (dS/S)² is equal to the variance of returns (with mean return 0) means we can rewrite and rearrange to produce the famous BSM equation:

In English: Time decay P/L is equal to 0.5 * -gamma * realised variance.

It is from here that we also see that delta hedging an option provides exposure to realised variance - not volatility.

This becomes important when replicating variance swaps,  and explains why they are easier to replicate than a volatility swap.

The presence of 'S²' sneakily represents that these exposures are also slaves to the price of the underlying; ruining our dream of a pure, constant exposure to these greeks. The development of the 'log contract' aims to overcome this.

One final point: the breakeven daily move in the underlying (dS) for delta hedges to offset time decay, can be solved through the following rearrangement:

Θ*dt = 0.5 * -Γ *(dS)²

2Θ*dt = -Γ *(dS)²

2Θ*dt/Γ = (dS)²

√(2Θ*dt/Γ) = dS

So as an example, let's take a straddle struck at 25% IV against an asset trading at $100: Where daily Θ = 0.01355 Where Γ = 0.015833 √((2*0.01355)/0.015833) =$1.31

$1.31/$100 = 1.31% move per day

You may also note that when you change the IV from an annualised figure to daily figure, you hit somewhat of a jackpot:

(25%/√365) = 0.0131 = 1.31%

A nuance here is that because the assumed mean return of the underlying is 0, the mean absolute deviation == 1 standard deviation. Just another thing to keep in mind.

Anyhow, that's enough math for now.

The key points to take are:

Constantly delta hedging an option removes directional exposure, and creates exposure to realised variance. The purity of your variance exposure is determined by hedging frequency. ∞ hedges = pure exposure but infinite cost. Therefore the trader must optimise between variance reduction and trading costs.
Time decay (theta, Θ) is the cost of convexity (gamma, Γ).
For the same σimplied, where there is more gamma, there will be more theta paid, usually when the strike is ATM.
With regard to magnitude of greek exposures, there is always a dependence on spot price in relation to the strike.

Let us turn to 'implied volatility', what this means, the effect that it has on option prices, and how it breaks the above equation.

## Implied volatility

Implied volatility, or σimplied as I will be referring to from here on out, is the volatility needed to be input to the BSM formula to output the current market price.

In the BSM world, σimplied is supposed to be a constant, drawn from the underlying's realised volatility.

Here is how that would look on a surface; where the surface is the visual representation of all strikes, shown as  σimplied, across spot and time for 1 underlying:

But in reality σimplied surfaces are very different, they are non-uniform through both strike price and time to maturity.

Here is how a surface can look for, say, a dollar denominated currency:

So, appreciating that volatility is non-constant, we must respect the '(implied * ν)' term of the P/L. But also the effect that changing σimplied has on delta (Δ), gamma (Γ), and theta (Θ).

Proper PITA.

Suddenly measures of spot/vol correlation and covariance matter a lot more, and this has an impact on the shape of the σimplied surface , and the spot/vol cross-section of the surface, called skew.

Let's examine a few skews (data from QuikStrike) by delta (taking the option strike that has delta 'x', and displaying it).

Clearly we see that, in 2 out of 3 cases, σimplied either side of the At The Money (ATM) strike is higher.

This means that market participants place a relative premium on those strikes, out of respect that large moves can happen in those directions.

Seller's are compensated for this risk, and Nassim Taleb keeps them in business!

No offence to the deadlifting squid ink maestro, of course.

Generally we can make the distinction that skew (the difference between equivalent points on call wings and put wings) is generated by spot/vol correlation, i.e the skew of the distribution; and that the premium on the wings over the ATM is created by the level of 'kurtosis' in the distribution, i.e how fat the tails are.

Note that this does not mean an option with a higher σimplied is more expensive than an option with lower σimplied, the σimplied is merely an 'accounting' term to create like for like comparisons.

Here is how option prices may look for an equity style skew vs flat vol. For the skewed puts, each strike lower commands an σimplied 0.5% higher than the last, for the calls, each strike higher is 0.2% lower than the last.

Here's a simplistic/wrong summary: Buy a straddle and you will profit from a move in any direction. Sell a straddle and you will lose from a move in any direction.

Here is the payoff from a long straddle, struck at a 25% σimplied , with a time to maturity of 1 year:

At face value, this seems like the perfect trade. If the underlying moves in any direction, the straddle buyer will profit.

However this is a trap that many a beginner option trader may fall into. See, the underlying will not just jolt in one direction constantly (looking at you tho BTC 😉).

Prices tend to exhibit some level of mean reversion after large moves. Perhaps a simulation would be easier to visualise this:

Let's take our long straddle, struck at 25% σimplied, against an underlying with an annualised volatility of 25%. For simplicity we will assume an unchanging σimplied. We can value our straddle at each point along each price path. The result looks like this:

One more, exceptionally useful, thing we can now look at is a scatter plot of P/L as a function of spot price and time.

We can see our P/L at various points in time, including after 1 week, 6 months, and at maturity:

But what if we were to buy the straddle at 35% σimplied, with the underlying realising 25%?

Whilst there are some exceptional results, we find that the majority of ending underlying prices are not far enough to either side of \$100 to turn a profit.

This particular trade is a loser 70% of the time, and the average overall P/L is negative too, meaning those outsized winners didn't make it big enough to drag the average P/L into the green.

The increased σimplied meant that we paid more in Θ for every moment of time passed, and we received less Γ (just how it works! **).

Unfortunately this combination is a recipe for losses. There's a reason you buy low and sell high.

** zomma (dΓ/dσimplied), if you're curious.

Some quick maths again (2+2 = 4. -1, that's 3. Right??). When delta hedging a long option to expiration, the P/L can also be written as:

ν * (σrealised -  σimplied)

Or, vega multiplied by the spread between realised volatility and implied volatility. Option buyers want σrealised > σimplied, and vice versa for option sellers. As a rule, the 'volatility risk premium' means that options are generally priced at volatilities above realised volatility.

So again, some key points to take away regarding implied volatility (σimplied):

σimplied is an accounting term, and allows a trader to compare options of different strikes and maturities in an apples-to-apples fashion.
σimplied is a useful barometer to see how much time decay (Θ) will be paid, and how far the underlying must run to create a profit (unless delta hedging).
When delta hedging to expiry, the spread between σimplied and σrealised is all* that matters.

*ish. The time at which the moves occur which contribute to realised volatility also play a role because gamma is higher closer to expiration, this can create a funk with non-continuous (read: realistic) delta hedging.

## Other Volatility Derivatives

The VIX!

Everybody's favourite 'fear' index and 'measure of expected volatility'.

Well, technically wrong.

Not completely wrong.

The VIX is a non-tradeable derivative quote that represents the fair strike of a constant maturity 1 month variance swap on the S&P 500, quoted in terms of an annualised volatility.

Due to the nature of variance (hello, squared term) compared to volatility, the VIX will run hotter than ATM 1 month volatility. Generally around the 30Δ put level.

It's a bit of a niche distinction, but important nonetheless if you want to know what you're talking about.

I can't really talk much more about variance or volatility swaps. Not least because they're OTC products (and not for peasants), but because there is much more nuance to them than I can do justice. Some great papers on these swaps include:

## The End

When tasked to write this handbook, two of the points I was asked to answer were 'is volatility good or bad?' and 'is volatility important?'.

Two difficult questions to crack!

In terms of good or bad, I suppose it depends which side of the fence you're sitting on. A tail-risk manager ought to thrive in volatile environments, a portfolio manager for a book of high yield credit may not want such volatile conditions.

One thing I've seen, on Twitter, time and time again prior to this year is that volatility is too low.

Then, we saw Q1-Q2 2020:

Is there a sweetspot between high and low volatility? Perhaps. But the same complaint comes whether vol is high, or if vol is low:

"I can't make money"

Maybe it's just their strategy.

Talking about volatility both in and out of financial assets:

In the policy planner's perfect world, volatility would not exist, every future budget and forecast could be made with complete confidence, and assets would grow at the riskless rate; and volatility itself would have no volatility.

In the anarchist's perfect world, volatility would remain consistently high. So perhaps nobody want's high vol of vol (except pesky volga traders).

I believe that the 'goldilocks' level of volatility is ideal. Not too high so that there is no confidence in the system, but not too low so that the great unwashed are never cleansed - so to speak.

Volatility keeps everybody on their toes, it forces the creation of contingency planning, it lights a fire under the arse of everybody who wants to improve, and to not be left in the dust (or is that drift - who knows?).

## Is Volatility Important?

Yes. Arguably, volatility is the most important thing. It's what distinguishes between a 'riskless' government bond yielding less than a penny on the pound annually, and the digital 'currency' that bounds around with reckless abandon (and indeed, everything in between). Without volatility, there can be no 'skill', or 'alpha', or 'excess return'.

Whether or not you and your strategy is robust/simple/sophisticated enough to survive and thrive off varying levels of volatility is an entirely different question. One that, if you are incorporating systematic, quantifiable elements, is available for rigorous review from backtesting and stress-testing. If you prefer a qualitative approach, this is a more difficult question to answer.

In any case, I hope that this brief look into volatility has proved useful, and that you've managed to take something from it. There is, as I'm sure you can imagine, so so much more that can be explored, in this area. So I'll leave you with some great books:

Osseiran & Odel: Unperturbed By Volatility: A Practitioner's Guide To Risk

Blow your mind with mathematics with the most in-depth look at volatility trading and risk management I've ever read (and always come back to).

An indepth look at volatility, derivatives, and nuances - without the heavy jargon.

Sheldon Natenberg: Option Volatility and Pricing

What else can be said? It's the OG.

Best,

Jamie Glancy

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