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# Dirfferent implied Volatilities on Differnt Strikes. Why?

I got overwhelmingly positive feedback about last week's column on implied volatility, so I decided to push a little further into the subject. As always, please drop me a line at dborun@zacks.com if you have any questions or comments.

The Black-Scholes option pricing model was revolutionary when it was introduced in 1973 and has been the gold standard for options pricing ever since. Robert Merton contributed real-world modifications to the basic equation and with Myron Scholes was awarded the 1997 Nobel Prize in Economic Sciences. Regrettably, Fisher Black had passed away before that particular recognition of his ideas and efforts.

Like most scientific and mathematical breakthroughs, the equation has been modified to more accurately reflect current market conditions, but the basic logic remains unchanged.

The original formula allowed for five basic inputs to determine the theoretical value of a European-style option:

Current Price of the Underlying

Time to Expiration

Strike Price of the Option

Risk-Free Rate of Interest

The Price Volatility of the Underlying – Expressed as  Standard Deviation of the Annual Return

You’ll notice that four of the five inputs are readily observable in the marketplace, while the fifth – volatility – requires the user to make a guess about how volatile the price movement of the underlying will be over the life of the option.

Since the introduction of the original formula, modifications have been added to incorporate dividends and the premium for American-style options which can be exercised before expiration, but volatility remains the wildcard variable.

The Black-Scholes model assumes that stock prices are distributed normally in a bell curve. (It actually assumes that returns are distributed lognormally – which effectively means that the price can’t go below zero – but the difference between the two is negligible at most reasonable prices for the stock.) A normal distribution looks like this:

Notice that it can be steep and narrow (low volatility) or flat and wide (high volatility) but that it is symmetrical.

Observed Price Movement

For years, traders used this normal distribution assumption and input a reasonable estimate of future price volatility for all options on a stock that had the same expiration date, until empirical evidence piled up to suggest that stock prices didn’t necessarily fit that pattern exactly.

Two alternatives became evident:

1)    Stocks tend to move down faster than they move up, so the left side – or “tail” - should be fatter that the right side. This is known as “Skewedness.”

2)    Extreme price movement in both directions (outliers) is more common than the normal distribution would suggest, so both tails should be fatter than a normal distribution would suggest. This is known as “Kurtosis.”

To accurately price options on stocks that move to extremes more than the normal distribution would suggest and which move faster to the downside than the upside, traders began using different volatilities on different strikes to more accurately represent the stock movement they expected. The result is commonly referred to as the volatility “skew” or “smile” because options far away from the current level of the underlying are priced using higher vols than at-the-money options with the result that they have higher theoretical values than a pure bell-shaped distribution would give them.

Market Forces

There is also a supply-and-demand component of the volatility skew. The average investor who owns equities generally wants to protect against downside price moves (buy puts) and also may be willing to give up some upside opportunity (sell calls) either to pay for the puts or to earn additional income on the position.

This applies to individual investors as well as professional, institutional managers who are responsible for billions of dollars worth of pensions, endowments and other funds. If you're trading options on the S&P 500, you're likey to encounter "1-lot" orders from relatively small retail accounts as well as orders for 1,000s of contracts from professionals - and evertuthing in between.

Because traders want to buy low and sell high, options market makers price the puts that investors want to buy at higher vols than they calls they typically sell. Market makers also adjust their vols based on the inventory of options they’ve accumulated in an attempt to sell options they own and buy back options they’re short.

Here is a graph of the implied volatilities for options on the S&P 500 at two different times:

Notice that whether vols overall are relatively low (2013) or high (2015), the vols at various strikes exhibit a "smile" shape, with puts priced higher than calls.

Practical Application

You’ll recall that in some of the trades we have previously constructed, I advised buying low volatility options and selling high volatility options. This doesn’t mean you should never buy a put or sell a call, because the market prices do exhibit both skewedness and kurtosis. It does mean however, that for any option trade you’re considering, it makes more sense not to buy the most expensive option or sell the cheapest one, especially if the vol smile is bumpy rather than smooth and you can trade a similar option at a mor attractive implied vol.

In the above graph, in 2013, a trader who wanted to sell a covered call on the S&P 500 ETF (SPY - Research Report) would have been better off selling the call that’s 10% above at the money (and priced at 13.1% vol), rather than the call that’s only 5% above at the money and priced at (9.7% vol.)

-Dave

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