Implied Volatility and Volatility Skew

Suppose you own a long call and calculated the value using the BSM model and the underlying's three-month volatility. Now—after you have initiated the contract—the open interest, or the number of calls written on the underlying at that exercise price, jumps dramatically after the option reaches a certain price, even though the underlying remains roughly unchanged. So it's clear that another investor has a different viewpoint than you. What might cause that?

That's not right. The option's time period is specified as part of the contract, so it's a given.

You got it!

The volatility calculation can vary between different investors because different time periods or different expectations can be used to estimate volatility. So the investor that initiated the large position could have a very different view of the volatility.

Suppose at-the-money puts had a higher implied volatility than at-the-money calls for the same underlying; how might you explain the difference?

Not really. This would suggest that the call options would have more value, but that alone doesn't explain implied volatility.

Exactly. This is commonly observed, and mainly for this reason. So it's most common to see a one-sided skew when graphing the implied volatility as a function of the strike price because of that asymmetrical fear of negative shocks.

Another common shape of this graph is often known as the **volatility smile**. What does that name suggest about implied volatility as you move further away from the strike price?

Yes!

No, it increases.

This isn't a rule, but a fairly common occurrence: 

What would be a clear reason for a worsening (larger) skew?

Suppose you saw a relatively large skew that you expected to correct back to a relatively flat shape soon. How could you make some money if you're right?

Not really. You'd need to take advantage of the temporary perceived mispricing between the two.

Since implied volatility can be calculated by using the BSM model, you can effectively trade volatility based upon the calculation, especially because implied volatility will vary over different exercise prices, time periods, and between calls and puts. You can base trading decisions on the term structure of volatility, which is based on the time to expiration. The implied volatility in regards to the exercise price is known as the volatility smile, or sometimes, the skew. Traders construct what's called a volatility surface, which combines the term structure of volatility with the volatility smile, so it's a three-dimensional plot that typically varies from the BSM model assumptions. Why would that be?

No, any mispricing will be corrected quickly. Recall that traders can synthetically create one option from another if there is an arbitrage profit to be made.

That makes sense. If long positions were being hedged, traders would be selling covered calls or buying protective puts, or both. This imbalance will increase the price of puts and lower the price of calls, adding to the skew. And of course, this imbalance can change over time.

Not really. Shorting the underlying shouldn't have a direct effect on options prices.

Not really; greater uncertainty overall should translate into higher implied volatility levels everywhere, shifting the entire graph upward.

That's right. The skew suggests lots of put buying relative to calls. If this risk differential reverses, then you can benefit by the rising call prices relative to put prices. This is called a **risk reversal**, conveniently enough.

That's not a good idea. If there's a large skew, plenty of investors have already piled into the puts.

That's right!

Not quite.

The BSM model assumes a flat volatility surface, so there is opportunity for trading based upon the term structure and volatility smile. There are also opportunities to trade volatility because it varies over time. As implied volatility increases, investors are communicating an increased level of risk, which also means that the price of hedging is also increasing.

After exploring all of this implied volatility, it might be nice to compare it to realized volatility—what actually happens. Go out and gather some stock prices. What's next?

Of course.

No, the volatility you want is of the returns themselves, so you need the returns first.

After calculating all of your daily returns for a month, you might have 21 trading day changes, and you can calculate the standard deviation of those returns.

How would you expect this to compare to the annualized standard deviation of returns?

Absolutely. There's rarely more volatility in a month than there is in a year. So the annualized standard deviation adjusts for this. You have 21 days of returns. A year often has 252. So you can annualize your standard deviation for the month as: > $$ \sigma_{annual} (\%) = \sigma_{monthly} (\%) \sqrt{\frac{252}{21}}$$. If traders become fearful, implied volatility will increase. But realized volatility is what it is—or more accurately, was. You can't get one from the other.

Probably not. There's no reason to expect the rest of the year to be much more calm than the past month.

That's not it. Recall the calculations for annualizing a periodic standard deviation of returns.

To summarize: [[summary]]

Time to maturity

Volatility calculation

Prices generally rise over time

Traders expect negative shocks more than positive shocks

Implied volatility increases

Implied volatility decreases

Sell calls and puts

Buy calls and sell puts

Buy puts and sell calls

There is greater uncertainty overall

There's a mispricing that won't be corrected until expiration of the options

Traders are hedging their long underlying positions

Many traders are shorting the underlying

The BSM model assumes a flat volatility surface

The BSM model assumes the correct term structure of volatility

Calculate returns

Calculate standard deviation

It's too low

It's too high

It's about the same

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