Why Volatility Clusters: A Walk Through GARCH

Pull up a long-dated chart of NASDAQ returns — say, the last thirty years — and squint at it. Don’t look at the price. Look at the jitter. Look at how violently the daily moves expand and contract over time.

What you’ll notice is that the chart doesn’t behave like a steady drumbeat. It looks more like weather. There are long stretches where the market seems to barely breathe — narrow daily moves, gentle drift, the kind of placid tape that lulls portfolio managers into thinking risk has been solved. And then, almost without warning, the lid comes off. The same index that was meandering by 0.3% a day starts swinging 3%, 4%, 5%. The storm settles in for months, sometimes years, before the calm returns.

Two things should jump out:

First, volatility is not constant. It changes with time. In statistical language, it is stochastic — itself a random variable, not a fixed parameter you can look up in a textbook. The simple assumption every introductory finance course leans on — that returns are drawn from a normal distribution with a stable standard deviation — is a useful fiction. In the real world, that standard deviation is alive and moving.

Second, and more interestingly, today’s volatility looks an awful lot like yesterday’s. Quiet begets quiet. The mid-1990s (roughly 1992 through 1998) were a long, gentle hum. Storms beget storms. The dot-com unwind from 1998 through 2002 was a sustained period of elevated turbulence where one wild day was almost always followed by another. In 2008 it happened again. In March 2020 it happened again.

This is the property that statisticians call autocorrelation in volatility — the regression of a variable on a lagged version of itself. Less formally, it’s the reason traders talk about volatility clustering. Volatility doesn’t sprinkle itself uniformly across the calendar like raisins in a scone. It bunches up.

The puzzle this creates

Once you accept that volatility clusters, you have a problem. The classical toolbox of finance — mean, variance, the bell curve — assumes the world is well-behaved in a particular way. It assumes that the size of tomorrow’s surprise is drawn from the same distribution as the size of last Tuesday’s surprise, and last June’s, and the surprise from three years ago. Statisticians call this assumption homoskedasticity: equal scatter, equal variance, across time.

But the NASDAQ chart is screaming the opposite. The scatter is unequal across time. There are calm regimes and turbulent regimes, and they persist. The data is heteroskedastic — different scatter in different periods — and crucially, the heteroskedasticity is predictable, because volatility today carries information about volatility tomorrow.

For decades, this was treated as an annoying empirical wart on the elegant theoretical models. Then, in 1982, Robert Engle proposed a way to actually model it.

ARCH: yesterday’s surprise predicts today’s variance

Engle’s idea, which would eventually win him the 2003 Nobel Prize in Economics, was disarmingly simple. What if today’s variance isn’t a fixed number, but a function of recent shocks?

He called it ARCH — Autoregressive Conditional Heteroskedasticity. The name is a mouthful, but the components are doing intuitive work:

• Autoregressive: today depends on yesterday.

• Conditional: variance is conditional on what just happened, not assumed in advance.

• Heteroskedasticity: variance changes over time.

Strip the jargon and ARCH says this: the bigger yesterday’s unexpected return was, the bigger we should expect today’s variance to be.

That’s it. If the market got slapped with a 4% surprise yesterday, ARCH says the distribution from which today’s return is drawn has a fatter standard deviation than it would have had after a sleepy 0.2% day. The model is essentially a feedback loop where shocks beget more potential for shocks.

It worked. Empirically, ARCH captured the clustering in equity returns, exchange rates, commodities — markets across the board.

GARCH: variance has memory of itself

Four years later, in 1986, Tim Bollerslev proposed a generalization. Engle’s ARCH said today’s variance depends on yesterday’s surprise. Bollerslev said: that’s true, but it’s also true that today’s variance depends on yesterday’s variance itself.

This is the GARCH model — Generalized ARCH. It adds one more ingredient: variance has its own memory, separate from the memory of individual shocks. Today’s variance is a weighted blend of three things:

1. A long-run baseline level of variance (the regime average, where things eventually revert to).

2. Yesterday’s surprise (the ARCH part — sharp shocks lift expected variance).

3. Yesterday’s variance itself (the new GARCH part — if vol was already elevated, it stays sticky).

That third ingredient is what makes GARCH so good at capturing real markets. It explains persistence. A single 4% day doesn’t just elevate today’s expected vol and then snap back to normal tomorrow; it elevates a state variable — variance itself — that decays slowly. Storms have inertia. So do calms.

The math is more involved than ARCH, but the intuition is what matters: variance is sticky, and GARCH is the simplest equation that takes that stickiness seriously.

What traders actually do with this

Here is the part that I think is underappreciated by people learning this material from textbooks. Most discretionary options traders have never written down a GARCH equation in their lives. And yet they use it constantly.

Nassim Taleb makes this point in Dynamic Hedging — that experienced options traders are, in his phrase, “calculating GARCH in their head.” They look at the time series of an asset’s prices and they can see the regimes. They feel where vol is clustered high and where it’s clustered low. They notice when a sleepy market just delivered a fat-tailed day and adjust their pricing of optionality upward, not because a model told them to, but because their pattern recognition is doing the same job the model is doing — weighting recent shocks, weighting the prevailing regime, expecting persistence.

This is, I think, the right way to hold the relationship between the model and the practice. GARCH didn’t invent volatility clustering. Bollerslev didn’t teach the market that storms persist. The model is a mathematical formalization of something traders have known in their bones for as long as there have been markets — that risk has a memory, and the recent past is the best guide to the near future.

What Engle and Bollerslev did was give us a way to write that intuition down, estimate it from data, and bolt it onto the rest of finance — option pricing, risk management, capital allocation — so that those frameworks could finally start treating volatility the way the tape has always behaved: alive, moving, and clustered.

We know that:

Expected profit = Summation of [probability of profit* Payoff ]

Before we head into calculating probability of profit of an option, let’s quickly delve into understanding cumulative distribution function (cdf)

A CDF is a tool used in statistics to show the probability that a random variable will take on a value less than or equal to a certain amount. In simpler terms, it tells us how likely it is that something will happen up to a certain point.

When we use the market’s option prices to back-calculate this CDF, we are looking at the Risk-Neutral Probability. It’s not necessarily a "prediction" of the future, but a map of what the market is currently charging for those outcomes.

For instance, following is the cdf of a dice roll.

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Silver had a massive deleveraging event on January 29 2026, resulting in a 35% move lower in the spot price in 1 day, which is really unheard of except in the most extreme market conditions. This came from news that the LME had raised the margin requirements for holding a futures position in the metal.

1 month at-the-money implied volatility finally settled at a sky high 98% after the move, which reflects a more than healthy risk premium embedded in the market and also expectations of further aftershocks in the spot price action. The volatility skew also reached elevated levels given the speed and ferocity of the downside move.

This resulted in an opportune time to sell implied volatility via option structures that are immune to the high frequency realized volatility of the underlying asset which market makers can monetize via delta hedging gamma positions. One such structure is simply to sell a 1 month 25 delta put on silver after the event with a partial delta hedge, as it seemed the brunt of the move had occurred already. This would have ended up being a profitable trade as the silver price pretty much saw it’s low on that day for the next month and also implied volatility eased from these lofty levels fairly soon.

Any questions, feel free to let us know in the comments. Until next time!

1. Implied vol of the stock goes down by 5 days consecutively.

2. Percentage of days that 30 min historical volatility is lower than implied vol.

3. Implied vs Realized vol metric is at a low percentile versus historical.

From here we can determine the cheapest vols to buy from a list of stocks.

Note, the reader still needs to want to go long or short the underlying stock via options to utilize the screener, the screener is simply there to help identify cheap options from an implied volatility basis.

Daily frequency measures of realized volatility are calculated by taking the standard deviation of daily log returns of a financial instrument's price over a specified period. This method is straightforward to compute and has been widely used in academic research and practical applications, including options pricing models such as the Black-Scholes model. However, daily frequency measures may not capture sudden or abrupt changes in volatility that occur within shorter time intervals.

Intraday frequency measures of realized volatility, on the other hand, are calculated by taking the standard deviation of intraday log returns over a specified period. This method can capture more frequent and rapid changes in volatility, which may be important for short-term trading strategies and risk management purposes. Intraday volatility measures can be computed using various approaches, such as high-frequency data, tick-by-tick data, or low-latency trading data.

The application of daily frequency measures of realized volatility is commonly used in options trading to estimate the implied volatility, which is a critical input for pricing and hedging options contracts. Implied volatility represents the market's expectation of future volatility and is often used as a proxy for risk assessment. By using daily frequency measures, traders can estimate the average level of volatility over a specified period, allowing them to set more accurate option prices and manage their risk exposure.

In contrast, intraday frequency measures of realized volatility are particularly useful in options trading for two primary reasons. Firstly, they allow traders to assess changes in volatility at shorter time intervals, which can be vital for understanding market dynamics and detecting potential opportunities or threats. Secondly, intraday volatility measures can help traders identify the optimal time to enter or exit a position, as rapid changes in volatility can significantly impact option prices.

In conclusion, daily frequency and intraday frequency measures of realized volatility offer unique insights into the behavior of financial instruments' prices and are valuable tools for options trading. Daily frequency measures provide a broader view of average volatility over extended periods, while intraday frequency measures capture more frequent and rapid changes in volatility. By utilizing both approaches, traders can make better-informed decisions regarding option pricing, hedging, and risk management, ultimately leading to more profitable trading outcomes.

Call Option A - 1 month volatility = 22% ( 30 days )

Call Option B - 2 month volatility = 25% ( 60 days)

You want to buy one of these options but you are not sure which one is cheap and which one is expensive.

Interesting exercise. Let’s read on!

To understand which is cheap or expensive, one must understand the concept of forward volatility.

Forward volatility is a measure of implied volatility of an option over a period of time in the future.

In the above example , let’s assume both options have the same strike price.

Remember that variance is additive, and that variance is equal to volatility ^2.

Annualized variance of option A = 0.22^2

Daily variance of option A = 0.22^2/365

30 day variance of option A = 0.22^2/365*30=0.0039

Similarly, 60 day variance of option B = 0.25^2/365*60 = 0.0102

So the 30 day forward variance = 60 day variance of option B - 30 day variance of option A = 0.0062

Now we need to convert this back into annualized volatility. First, we calculate the annualized variance which is 0.0062/30*365 = 0.0766

Therefore, the annualized implied forward volatility = sqrt(0.0766) = 27.67%

Let’s summarize:

Option A volatility = 22% ( 1 month )

Option B volatility = 25 % ( 2 months )

Forward vol from 1 month to 2 month = 27.67%

Now here is an important concept:

If there is no significant news expected between the 1st month and 2nd month, forward volatility from month 1 to month 2 should be same as the implied volatility of option A (1 month).

In light of the above, forward volatility seems high (assuming all else equal i.e. no significant volatility premium for news).

Therefore, as a trade you could short option B and long option A (recall that they have the same strike prices).

Have a look at the following volatility curve for the FX pair USDINR as an example:

Notice how different tenors have different implied volatilities. A series of forward volatilities can be bootstrapped from the curve.

In the next piece of this series, I will teach you how to bootstrap a given volatility curve from basic parameters and introduce the concept of vol carry. Stay tuned, Options Nerds out!

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In options trading, volatility is a critical factor that can make or break your trades. As traders, we often look at implied volatility (IV) to gauge market expectations of future price movements. However, to truly understand the context of current volatility levels, it's essential to delve into the concept of volatility percentile. In this article, we’ll explore the volatility percentile, why it matters, and how you can use it to make more informed trading decisions.

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What is Volatility Percentile?

Volatility percentile is a statistical measure that compares implied volatility to its historical levels over a specified period. It tells you the percentage of time that implied volatility has been lower than its current level. Essentially, it provides a context for whether the current IV is high or low relative to its historical range.

For example, if the IV percentile of a stock is 80%, it means that the current implied volatility is higher than 80% of the historical IV readings for that stock over the selected period. Conversely, if the IV percentile is 20%, the current implied volatility is lower than 80% of historical readings.

Why Volatility Percentile Matters

1. Relative Context: Volatility percentile offers a relative measure of volatility, helping traders understand if the current IV is high or low compared to past levels. This is crucial because an IV of 30% might be high for one stock but low for another.

2. Strategic Decisions: Knowing the volatility percentile can help traders decide which strategies to employ. High volatility percentiles might indicate a good time for strategies like selling options (e.g., covered calls or iron condors), as premiums are richer. Low volatility percentiles could suggest buying options (e.g., long calls or puts) as premiums are cheaper.

3. Risk Management: Understanding where the current volatility stands in its historical context can help traders manage risk more effectively. High volatility percentiles can signal potential market turbulence, prompting traders to be more cautious or hedge their positions.

Calculating Volatility Percentile

To calculate the volatility percentile, follow these steps:

1. Gather Historical IV Data: Collect implied volatility data for your chosen asset over a specific period (e.g., 1 year).

2. Rank the Data: Sort the historical IV data from lowest to highest.

3. Find the Current IV: Determine the current implied volatility level.

4. Calculate the Percentile: Identify the position of the current IV within the ranked data set and calculate its percentile rank.

Here’s a simple formula to determine the percentile:

Percentile=Number of data points below current IV/Total number of data points×100

Practical Example

Let’s say you’re looking at the implied volatility of XYZ stock over the past year. You have 252 daily IV readings (one for each trading day).

1. Sort these readings from lowest to highest.

2. If the current IV is 25%, and there are 201 readings below 25%, the percentile would be:

Percentile=201/252×100≈79.76%

This means the current IV is higher than approximately 80% of the past year’s readings, indicating a relatively high volatility environment.

Using Volatility Percentile in Your Trading

High Volatility Percentile (>70%)

• Strategies: Consider selling options to capture high premiums. Strategies like iron condors, covered calls, and vertical spreads can benefit from high IV environments.

• Caution: Be aware of potential market events or earnings reports that might be causing high volatility. Adjust your risk management accordingly.

Low Volatility Percentile (<30%)

• Strategies: Look into buying options, as premiums are lower. Long calls, puts, and straddles can be more cost-effective.

• Opportunity: Low volatility can provide opportunities for entering positions with less risk of significant price swings.

Conclusion

Volatility percentile is a powerful tool that provides context to current implied volatility levels, helping traders make more informed decisions. By incorporating volatility percentiles into your analysis, you can better gauge market conditions, select appropriate trading strategies, and manage risk more effectively.

Stay tuned for more insights and strategies in our upcoming newsletters. As always, happy trading!

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In today’s post, the options nerds will describe how to screen for cheap options for those of you out there who are interested in buying options in your favorite stocks. We know that the most important element of driving your P/L in stock options is moves in the stock price (delta), but it is extremely helpful if the implied volatility will reprice higher as well after you purchase the option.

How can we predict that the implied volatility will reprice higher after our buy? Well, that depends on our ability to identify when the implied volatility is low. In this series of posts, we will introduce a series of metrics that can help us identify when implied volatility may be due for a rebound. Let’s dig in!

1. Implied Volatility Waning Momentum

   We introduce this concept based on the idea that the implied volatility surface of an asset (stock, FX, crypto, etc) can only sell off for a limited number of trading days before either 1) stabilizing 2) rebounding higher.

We can define the number of days where the ATM vol has sold off (adjusted for daycount convention) and use this as a criteria for screening for cheap call options, as on first order the call skew should sell off proportionately.

For example, we can set the number of days to 5, and screen for stocks whose ATM vol has sold off for 5 trading days in a row.

At the end of this series of posts we will also start publishing a curated list of cheap stock options from a volatility perspective. Hope you’re all excited about this! Until next time, TopOptionsNerds signing out!

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In today's post, we would like to dive into more detail on the impact of the equity dividend yield on option pricing and how this can materially impact the P/L of your trading portfolio with some examples of spectacular losses at Wall Street trading floors in the past.

We will also go into detail on the concept of Wrong Way Risk, and how exposure to Dividend Risk can be compounded into a wrong way risk, especially in markets where there is a high degree of structured product issuance. From this we can see the importance of keeping abreast of the scale and growth of activity in the structured products markets as an important metric as we think about second order risk in our portfolios, as well as for providing potential trade ideas, skills which any experienced options trader will possess. Let's dive in!

For complete beginners to the impact of equity dividends on the P/L of the option holder, take a look at our previous Level 1 post about option pricing here.

The risk associated with dividends in an option portfolio tends to increase in proportion to both the average maturity of the options held and the average moneyness of the portfolio. This implies that as the average maturity of the options portfolio extends or as they become more in-the-money, the portfolio risk due to potential dividend payouts becomes greater. Many option traders mitigate their exposure to delta risk by hedging with spot positions (for example, an FX options trader will hedge with cash FX and an equity options trader will hedge with the underlying stock), however, this leaves the forward risk unhedged (which incorporates dividend and interest rate risk). The reason they do this is because of greed.

The options trader wants to retain as much of the client markup on the deal as possible, and it is much cheaper (lower bid offer spreads) to delta hedge with spot than with the forward, especially for longer-dated and emerging market currencies (liquidity is worse for both longer-dated forwards and forwards on EM pairs). The options trader is usually mandated to hedge the forward risk with the bank’s own forwards desk in order to internalize flows, and the forwards trader, when learning that the options trader has just won a juicy structured product deal, and is a captive client (has to deal with the internal forwards desk), will often charge an egregious bid offer spread to hedge the forward. Ultimately what ends up happening is that the options trader will hedge with spot and leave the dividend risk unhedged.

Even in cases where forward risk is hedged, it is generally more challenging to hedge against dividend risk compared to interest rate risk due to the greater liquidity and availability of interest rate derivatives in the market. As a result, market participants often find it easier to manage and hedge interest rate risk, which is more readily tradable, while dividend risk presents a greater challenge due to the limited availability of dividend derivatives and the complexities associated with pricing and trading them effectively.

In many equity option markets, clients who do not engage in delta hedging often take the position of outright buyers of calls or sellers of puts. As a result, market making desks are left with a similar risk exposure, namely, being long dividends. However, market makers have a limited capacity to mitigate this risk by engaging with hedge funds through instruments like dividend swaps. These swaps provide a means for market makers to transfer the risk associated with dividends to hedge funds, thereby managing their overall risk exposure more effectively. However, they tend to be only available on liquid equity indices such as the S&P 500 and not single stocks, so there is still a lot of dividend risk unhedged.

In part 2 of this series, we will go over the concept of Wrong Way Risk, and how this manifests in the structured products market in relation to dividend risk on equity options and FX forward risk in the context of FX structured products! Hope to see you there!

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There are three main ways of measuring Skew.

a) Strike skew

= Difference in vol between 2 strikes

= 90% - 100% of ATM ( for example)

Remember not to divide the strike skew by ATM volatility (this is a common mistake). You could argue that a stock with a 10% difference in a low-volatility stock is more significant than a 10% difference in a high-volatility stock, so the temptation to divide by ATM is understandable. However, this is wrong because it doesn’t account for the fact that the Difference in width of 2 strikes also increases with the vol.

b) Delta skew

This is very similar to strike skew with a very high R-square.

= Difference between constant delta put and call

= 25 delta put vol - 25 delta call vol (for example)

c) CBOE skew index ( third moment)

The CBOE has created a skew index on the S&P500 which is strike-independent.

Skew = 100-10*3rd moment.

I suggest checking here if you want a deep dive into how the third moment is evaluated.

We will move on to other essential features of volatility skew.

Skew and other option greeks

Implied Skew, in general, represents non-symmetry. In theory, it is measured by calculating:

Vanna, which is d vega/d spot (rate of change of vega with respect to spot level)

or,

d delta/d vol (rate of delta change with respect to implied volatility level).

or,

Vanna = d^2 PV/ d(spot)d(vol)

Why is skew negative?

Skew for equities is normally negative; i.e. volatility of puts is higher than calls. This makes sense because big market jumps tend to be more down than up (of course not always). Here is the keyword - ‘jump’. Markets could be ‘trending’ up but the direction of ‘jump’ ( or potential expectation of future jump) could be down to make the Skew negative.

Volatility is a measure of risk and leverage increases when equities decline. If we assume no change in a company’s number of shares outstanding or debt, a simple reduction in stock price increases company leverage (debt/equity ratio).

Investors typically invest in equities and want protection against price declines, thus are willing to pay a price (premium) for puts.

When the actual correlation of stock prices and volatility is negative, ‘realized skew’ is negative.

Implied Skew is almost always negative.

Index skew vs single stock skew

Imagine an index where all constituent stocks have a flat (or zero) skew i.e. Volatility of different strikes of the same maturity is the same. What will be the index skew? Flat as well—-right?

No, wrong.

The index skew will be negative. Low strike index volatility will be almost equal to single stock volatility as the implied correlation is close to 100%. However, ATM index volatility will be less than this value because the implied correlation for ATM strikes is less than 100% ( diversification is most effective when spot = strike).

Remember , σIndex² = implied correlation × average(σSingle stock)².

Therefore even if single stock constituents have zero Skew, the index can have a skew as low strike index implied vol > index ATM implied vol.

Furthermore, the magnitude of the index skew will be greater than the average single stock skew.

You can somewhat think of it this way.

Index skew = Single stock skew + implied correlation skew

A less diverse index will have a lower implied correlation skew and hence, a lower index skew. A more diverse index will have a higher implied correlation skew (ATM vol is lower but low strike vol is higher) and hence, a higher index skew.

Relationship of Skew with time

Consider buying a 1 year put option (long skew position). When it becomes a 3-month put option, its implied vol will have risen, assuming the same ATM vol and spot because

“skew increases with the passage of time”

In other words,

vol of 1y 99% put - vol of 1y 100% put < vol of 99% put - vol of 100% put

How to trade Skew

Remember, you can have a view on Skew based on expected market correlations or panics or trends, but your real PnL or delta will be determined by the volatility of the volatility surface that reflects the true regime and the model that maps the regime best.

What do I mean by this?

There are 4 different ways to model a vol surface.

1. STICKY delta

2. STICKY strike

3. STICKY local vol

4. JUMPY vol

STICKY delta

This model has the same vol for options of the same strike as the percentage of spot (or same delta). Sticky delta literally means if the delta of the option is the same, the vol is the same, irrespective of the spot move.

For instance, 90% put strike with SPX at 4800 is the same vol as 90% put strike with SPX at 4500.

Think about this.

Well, it’s sometimes possible when everyone is expecting the market to go lower and has already bought a lot of puts, and when the market actually goes lower it’s not surprising anymore and vol of lower strikes remains the same.

In such cases, if you are long Skew, you lose money.

Repricing PnL from long Skew = negative

Skew theta = Always negative. ( skew theta is always the price you pay for owning Skew)

You can define skew theta as →

theta per unit gamma of all strikes of a given maturity - theta per unit of gamma of ATM strike.

Net PnL = negative.

Spot-vol correlation = positive.

STICKY strike

As the name suggests, the volatility of a particular strike remains the same even if the spot market has moved.

For instance, consider SPX is at 4800 and 90% put strike has vol of 15.

Now, when SPX actually moves to the 90% put strike, the strike will turn into an ATM option.

This ATM vol in this case is now expected to be 15 (which was earlier predicted by Skew)

If the market keeps dropping, this ‘new’ ATM call volatility still remains at 15.

According to the sticky strike model, vols do reprice exactly as Skew suggests.

In this model, if you are long Skew,

Repricing PnL = 0

Skew theta = negative

Net PnL = negative.

Spot-vol correlation = 0

Constant Local Vol Model

If the local volatility surface stays constant, the amount volatility surfaces move for a change in spot is equal to the Skew (i.e., ATM volatility moves by twice the Skew, once for moving up the Skew and another by the movement of the volatility surface itself)

In this case,

Repricing PnL = positive

Skew theta = negative

Net PnL = 0

Spot vol correlation = negative.

You can argue this is the ‘only’ model that prices skew correctly.

Jump Vol Model

This is the only model where a long skew position actually makes money where repricing repricing PnL > skew theta. Best long skew profitable trades happen when market moves in panicky fashion in the direction the Skew is priced for ( for instance - a large rise in implied vol following a decline in spot)

Spot-vol correlation = Very negative

Repricing PnL = positive

Skew theta = negative

Net PnL = Positive

Ok, that is it for today. In a subsequent section, we will discuss practical examples of how to monetize skew.

Thanks for reading and feel free to leave questions in the comments!

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Now, if you’ve been having a look at options market commentary, you might see something similar to the below (which is an example from the cryptocurrency options market):

The Put Call Ratio and Max Pain Point are two things that options traders like to look at and are also useful for directional traders. Why is this the case?

Put Call Ratio

First of all, the Put Call Ratio is mathematically defined as:

The news item above, it is describing that of the options expiring on Nov 24th, the Put Call Ratio is 0.83 for BTCUSD.

This gives an indicator that option buyers have more positions expiring on Nov 24th expecting the BTCUSD price to be higher than lower. Usually the sellers of options are the option market-makers who are continuously delta-hedging their portfolios, so if the Put Call Ratio for options expiring today is skewed towards puts, the market-maker delta-hedging activity will likely result in the spot drifting down, and vice versa for calls.

The Put Call Ratio can also be used to look at the trading activity on any given day. This gives a sense of which side of the market (up or down) option buyers have been more excited about on any given trading day. A typical put-call ratio for equities that is considered a good basis for evaluating market sentiment would be 0.7 — that is, a ratio below 0.7 would mean that options buyers are adding to bearish positions on the day and above 0.7 would mean that options sellers are adding to bullish positions on the day. The reason that 0.7 is the typical neutral put-call ratio (in the equity markets) is that normally there are more investors buying calls than buying puts, so 0.7 is a typical put-call ratio for evaluating sentiment.

Max Pain

Max pain, or the max pain price, is the strike price with the most open options contracts which causes the most losses for option buyers at expiration. It is called max pain due to the idea of maximum pain theory, which assumes that option buyers will buy and hold the option (without delta hedging) until expiration, while option sellers are market-makers who will be continuously delta hedging their option portfolios. In this case, the max pain price is the price at which the maximum amount of financial losses would occur for these option buyers. This is because they are not delta hedging and thus their payoff is solely based on how far the underlying is at expiry from the strikes of the options that they purchased. Note, that this theory is controversial in the industry because of the assumption that only option sellers are continuously delta-hedging their option portfolios.

Calculating the Max Pain Price

To calculate the max pain price, one needs to sum up the difference between the current spot price and the strike price for each in-the-money strike price, multiply by the open interest of that strike, add together the dollar value for calls and puts, and repeat for each strike price. Finally, we need to determine which strike price has the highest dollar value, and this is the Max Pain Price.

In formulas, for each strike, we can define the Pain Value as:

and then find the strike that maximizes the Pain Value across all strikes with open interest.

Important point: The OptionNerds believe that an improved Max Pain Price metric should be scaled by the time to maturity to account for the higher gamma of shorter-dated options which would have a larger influence on the potential path of the underlying towards the Max Pain Price. We therefore define a Modified Max Pain Price based on a Modified Pain Value:

Note: If the Max Pain Price is being calculated for a given expiry (i.e. options expiring today, options expiring tomorrow), then the Modified Pain Value is not needed as the Gamma will be directly proportional to the Open Interest.

Let’s dive in to a numerical example of calculating the Max Pain Price!

Let’s say the Stock Price is $100, and the open interest is 10 lots of $110 calls, 20 lots of $115 calls, and 5 lots of $85 puts. The options open interest graph looks like this:

The $110 strike has a Pain Value of (($110-$100) * 10), the $115 strike has a Pain Value of (($115-$100) * 20), and the $85 strike has a Pain Value of (($100-$85) * 5). When we compare these values, it is clear that the Max Pain Price is $115.

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Well hold on, let’s look into:

1. what is the VIX futures curve,

2. the shape of the curve on July 7th when we recommended the trade,

3. whether simply buying a 3-month VIX future at trade inception would have been profitable,

4. whether the futures curve trade has been profitable so far.

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What are VIX futures?

VIX futures are like any other listed financial futures contract, in that they are traded on an exchange with daily margin requirements, and allow speculators and hedgers to have exposure to the VIX index on that future date (known as the settlement date). The most actively traded futures contracts are the quarterly contracts that settle/expire in March, June, September, and December (and have codes H, M, U, Z). Since these are financial contracts (not physical commodities futures contracts), they simply represent the traders in the market’s expectation of where the VIX index (otherwise known as spot VIX) will be on the settlement date.

What was the shape of the VIX futures curve when we recommended the trade?

By plotting the prices of the futures contracts (price on the y-axis, settlement date on the x-axis), and then drawing a smooth line through the points, we can see the market’s pricing of the VIX futures curve. If we recall the chart in the previous post, the VIX futures curve was upward-sloping in July, which is usually the case when spot VIX is abnormally low as volatility is a mean-reverting process1.

Would simply buying a 3-month VIX future at trade inception (July 7th) have been profitable?

Although October 7th (3 months out from trade inception) is not an actively traded futures settlement date, the Chicago Board Options Exchange helpfully publishes the interpolated (remember that smooth line..) VIX for standard tenors i.e. 9-day, 3-month, 6-month, and 1-year on each trading day, which can be pulled up in my favorite graphing tool, TradingView.

On July 7th, the VIX3M was ~17.25 vol, while spot VIX today (about 3 months later) is 18.6 vol:

Also, interestingly the implied correlation that we mentioned in our inaugural post
has picked up (gone higher) from 20% to 27.7% (using another handy ticker on TradingView, COR3M).

So yes, buying a 3-month VIX future at trade inception would have been profitable, to the tune of 1.35 vols.

Would the futures curve trade that we recommended have been profitable so far?

We recommended buying 2x the 5-month future versus selling 1x the 3-month future, so let’s see if this trade would have done better than simply buying the 3-month VIX future!

As described above, being long (a.k.a. buying) the 3-month future would have resulted in a 1.35 vol profit. So selling 1x the 3-month future would have resulted in a 1.35 vol loss. What about the P/L on buying 2x the 5-month future (which would have 2 months to settlement today)?

Unfortunately, the CBOE does not post a VIX2M or VIX5M, so let’s use VIX3M and VIX6M and do a little bit of arithmetic to approximate the answer!

On July 7th, VIX3M was 17.25, and VIX6M was 19.3. Using a linear interpolation, VIX5M should be:

To find the P/L we need to find VIX2M today. Again from TradingView/CBOE we know that spot VIX is 18.6 and VIX3M is 19.5. Using the same linear interpolation technique, VIX2M should be:

Therefore, we would have made 19.2-18.6 = 0.6 vols, which on 2x the position would mean we made 1.2 vols on the long 5-month future, but lost 1.25 vols on the short 3-month future. So in fact the total P/L on the position is -0.05 vols.

Why did this trade not work out as expected? In fact, the VIX futures curve is higher but also flattened out. This means that traders expect this current rise in volatility to be fairly well contained, or that the mean reverting nature of volatility will start kicking in soon.

We hope this article was enlightening as both an intro to the VIX futures curve and an analysis of how the previous trade recommendation went. Happy to answer any questions, and until next time!

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The Black-Scholes formula states that the price of a Call option ( C ) is:

Where N is the standard normal cumulative distribution function:

Wow! This sounds way too complicated.

Not to worry. Let’s attack this from first principles.

According to the definition of the call option payoff:

which is effectively the same as saying:

Can we split the above statement into two parts? Yes! It would look like:

and

In other words,

or

Strike * Probability that spot at expiry is greater than strike

or K*N(d2) where N(d2) is probability that Spot at expiry is greater than strike.

(Hold your horses, I will come back to explaining N(d2) shortly)

Moving on to the second term in the decomposition:

or:

The conditional expected value of Spot on expiry given St>K * Probability that spot on expiry is greater than strike

Lets summarize what we have learned so far:

Now we need to convert a future payoff to present. Taking present values (multiplying by the discount factor exp(-r*T)) :

Assume the stock price today is $10 and strike price is $10.

The picture below can be viewed as the:

“truncated average of stock price truncated above the strike price”

See the below chart:

And this one:

Picture credit: quantpie

The above 2 snapshots essentially decompose the Black-Scholes equation for you.

Therefore, the price of the option above is: $5.80 - $4.21= $1.59.

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If you recall from the earlier post on the assumptions behind Black-Scholes, the model for pricing and hedging options assumes that the option trader can trade the underlying asset continuously, with zero transaction costs, without interruption. In reality, asset markets are open only at certain times. For example, Foreign Exchange markets are open 24 hours a day from Monday, 7am in New Zealand to Friday, 5pm in New York. The New York Stock Exchange is open from 9:30am to 4pm Monday to Friday, with the exception of Exchange holidays. In fact, the cryptocurrency markets are most likely the market that has the closest structure to what is assumed in Black-Scholes, in that cryptocurrency exchanges are open for trading 24 hours a day, 7 days a week, 365 days a year.

These market mechanics play into two major considerations when it comes to our realized volatility calculation. The first is the annualization factor, while the second has to do with modifications for accounting for the ‘overnight return’. Let’s dive in!

Annualization Factor

Recall that the realized volatility is defined as the annualized standard deviation of the log price returns. The reason that the figure is annualized is so that traders can compare realized volatility across different asset classes where markets trade a different number of days per year. Common factors for daily frequency volatility calculations include sqrt(252) for equities, sqrt(260) for FX, and sqrt(365) for cryptocurrencies. When sampling realized vol at higher frequencies, we need to take that into account by multiplying the daily annualization factors by the square root of the number of observations per day.

Equity Dividend Adjustment

As you may well know, the price of a stock will gap lower on the ex-dividend date as the fair value adjusts for the fact that the owner of the stock no longer receives the dividend past the ex-dividend date. This gap would cause what seems to be realized volatility but in reality, needs to be removed in order to get an accurate estimate of the true realized volatility.

The recommended way to make this adjustment is to multiply all the stock prices before the ex-dividend date by an adjustment factor defined as:

This will serve to make the price series smoother in terms of removing the price gap seen on the ex-dividend date.

Overnight Return

The overnight return is particularly relevant for equity markets and is the change in the price from yesterday’s market close to today’s market open. There are a few major methods that traders use to deal with this issue:

1. Ignore the overnight return entirely, and only calculate realized volatility based on the price returns during market hours. This is similar to ignoring the price jump after an earnings announcement. While it is simple, it would most likely result in a realized volatility calculation that is ‘too low’ as there are real price returns that occur during the overnight period due to market-moving news and pent-up supply/demand that builds up overnight. Therefore, we do not recommend this method!

2. Treat the overnight return in the same way as the other price returns in the historical time series for the purposes of calculating realized volatility. For example, if we are sampling the price of a stock on the NYSE every 30 minutes, every day we would have 14 price observations from 9:30am to 4pm, plus an additional price observation at 9:30am the following day. Therefore, we can calculate the realized volatility using 15 price observations or 14 log returns every trading day.
   

   Now, you may say that it does not feel particularly natural to compare a price return over a 30-minute period with one that is over a 17.5-hour period — that is, it does not feel like an apples-to-apples comparison. In which case, we can look at the 3rd method!

3. This is an interesting method that involves some additional calculations and some parameters that can be tweaked for the aspiring options nerds out there.
   
   The goal is to estimate the percentage of the total price variance which occurs during the trading day. One popular way to do this is to compare the realized volatility estimate based on a daily close-to-close calculation with the higher frequency intraday (i.e. every 30 minutes) calculation over a given lookback window. Be sure to use the correct annualization factor in these calculations!
   
   Once this percentage is estimated, we can simply take the realized volatility estimate we calculated in method 1 and divide it by this percentage in order to adjust the estimate (higher) to incorporate the additional volatility that occurs during the overnight period.
   
   
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We will now run through a quick numerical example of both the equity dividend adjustment and overnight return adjustments for calculating realized volatility on equities using IBM stock (IBM).

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Scenario: You are on your first week at the job on a Wall Street trading floor, and the head of sales’ favorite client lifts your offer for $100mio notional of a 2-year, at-the-money call option on the S&P 500 (traded on an over-the-counter basis). It is 5 minutes to the close of trading on the NYSE and CBOE (Chicago Board Options Exchange), so you cannot wait for the salesperson to enter all the trade details into your risk system (we know boomers type slowly!) before doing some vega hedging of this trade.

Question: How much of 1-year ATM call options on S&P 500 should you buy to have a vega-hedged portfolio? Let’s assume that 1-year is the longest maturity options that have reasonable exchange liquidity.

Recall that Black Scholes vega is defined as:

where:

r is the risk-free rate of the numeraire

T is time to maturity in years

S is the spot price of underlying.

For an at-the-money option,

and in particular,

for both the 1-year and 2-year ATM options, and for small levels of r,

therefore, the Vega of a 2y ATM call / Vega of a 1y ATM call is equal to:

So, you need to buy 1.4*100 = 140mio USD of 1-year at-the-money call options before the end of the day to vega hedge the portfolio before market close. Don’t mess it up, and don’t forget the salesperson’s sales credits!

On a more serious note, the vega multiplier as a square root of the time to maturity is something that every option trader knows by heart, and we highly recommend committing some of these common multipliers to memory as it will help you to quickly calculate the vega risk of your options portfolio if it contains options with different maturities. See the chart below!

Hope this was fun and hope you’re excited about the next edition of this series! Until next time,

The OptionsNerds

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Delta Hedging

As mentioned in the post about the Black Scholes model, modern options pricing is based on the fundamental assumption that the price of an option is based on the expected costs of replicating the option by trading the underlying asset.

For example, an options market maker who sells a 1-month at-the-money (50% delta) call on TSLA will buy shares of TSLA equal to 50% of the option notional in order to achieve delta equivalence, as previously described. This is also known as the 'initial delta hedge' as it is the first delta hedge that the option seller will make in her attempt to replicate the option via dynamic hedging.1

Due to the positive gamma of a long call option (which represents negative gamma for the option seller), in order to keep delta equivalence the market maker should continuously adjust the delta hedge amount every time the underlying asset moves. In the Black Scholes model's world, the market maker is trading the underlying asset in continuous time (i.e. every millisecond) with zero transaction costs. In reality, though due to the existence of the bid-offer spread on the underlying asset, the option market maker will delta hedge at certain moments in time, usually based on either:

1. a predetermined time-frequency (i.e. every 5 minutes while the market is open, once a day at market close, etc) or

2. once the underlying asset reaches certain levels away from the current level.

Note that due to the negative gamma of being short the option, in order to delta hedge, the call option seller will need to buy the underlying asset when it moves up and sell the underlying asset when it moves down (otherwise known as stop loss trades (S/L)). If realized volatility is high and the underlying price moves up and down rapidly, the market maker will incur delta hedging losses by buying high and selling low! If the option is fairly priced, then we can expect that the option premium the market maker receives for selling the option is perfectly offset by the delta hedging losses. Intuitively, this would be the case if the realized volatility during the lifetime of the option is equal to the implied volatility that the market maker sold the option at (the implied volatility can be computed using the Black Scholes model based on the option premium at the time of the initial option trade).

Why should I do it?

Although the majority of option traders are looking for a low-cost way to speculate on the direction of the underlying, delta hedging an option can be profitable if the option trader believes/forecasts that the realized volatility of the option through its lifetime is different from the implied volatility that the option is trading at in the market.

Hope this makes sense from a big-picture perspective. Now, we will go into more advanced concepts as to why even if the realized volatility calculated during the lifetime of the option is equal to the implied volatility at trade inception the market maker can end up making a loss at the end of the delta hedging process!

The reason that the market maker can end up with a loss (or a gain!) is due to the concept of path dependency.

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Also, for the newbies, just a reminder that this is our launchpad for Newbies to Options Trading!

Until next time,

The OptionsNerds

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If 1-year volatility is 8% (annualized), the option premium of at-the-money straddle = 4.2 * 10% * sqrt ( 365) = 8%

Ok, you get the idea. We won’t delve into why the calculations for straddle premium are simplified in this post but we do love simplicity in estimates as they offer great insights into trade ideas. A lot of traders use volatility and ATR-based (Average True Range) stop losses and profit targets and a glance at the volatility curve (and basic high school math) can give them an approximate idea of straddle breakevens and premia.

There are 2 crucial caveats in this:

1. Firstly, interest rates have been ignored (set to zero) and so has the effect of forward carry.

2. Secondly, if the trader thinks that spot will move by more than the breakeven, it does not necessarily mean that the trader should purchase the option, perform delta hedging, and expect to monetize the position 100% of the time. When delta hedging is performed, there are more important considerations such as expected realized volatility which we can discuss in a later edition.
   
   Ok, let’s move on to another concept.

What is the probability that the option will move more than the breakeven price?

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