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As I understand it, IV is defined as an annualized 1 standard deviation range for an underlying. So what is the meaning of implied volatility for a strike?

  1. The IV on each strike is calculated by inputting the market price of the option into the Black-Scholes model. If the pricing model calculates IV per strike, how do you get a value for the underlying as a whole? For example Tastyworks says that for January expiration SPY has an IV of 21.7%. Is this an average or other aggregation type formula that combines IV from all the strikes? Is there such a standard formula or do other trading platforms it differently?

  2. For example the 250 put on SPY is 20 points out of the money and has a 23.76% IV. Does this mean that someone buying the 250 put for the current market price is valuing the volatility for this cycle at 23.76% and say someone else buying the at the money put (270 with 18.7% IV) for the current market price is valuing the volatility for this cycle at a lesser value?

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Historical Volatility is the standard deviation of daily price changes, not Implied Volatility.. It's the most commonly used determination of HV but there are other methods as well.

I have never seen a calculation of the Implied Volatility for a strike. There can be several options at the same strike or as many as 3 dozen (the heavily traded SPY). Each option has its own implied volatility and it is calculated as you wrote in paragraph 1).

Different web sites calculate the Average Implied Volatility in different ways. Some just average the different IVs. Some do more complex calculations (IVolatility which uses a proprietary weighting of the delta and vega of 4 ATM options per expiration. If you really want a headache, read about the calculation of the CBOE Volatility Index (VIX).

Some brokers will provide an IV per option, an average IV per expiration, and an average IV for the stock (all options).

Options of the same expiration can have very different values. A volatility smile is when the IV of each option is higher as the option gets further out-of-the money. There are also Option Smirks (IV has either a Forward or Reverse Skew).

  • I got the IV definition from tastytrade which says "[IV is] a percentage that indicates the annualized expected one standard deviation range for the stock based on the option prices". Maybe that's a simplification. – oceanus Dec 11 '18 at 4:37
  • Sorry about phrasing it as IV for a strike. What I meant to say was IV for an option. Can you clarify what you meant by "There can be several options at the same strike or as many as 3 dozen"? I know for a strike level there are calls and puts, what are the dozens of options? – oceanus Dec 11 '18 at 4:40
  • I understand that the IV increases for options as they get further out of the money along the volatility smile. What I'm having an issue with is the definition of IV related to a strike level. If the 250 out of the money put has 23.76% IV and the 270 at the money put has 18.7% IV, what do those numbers tell me? – oceanus Dec 11 '18 at 4:43
  • HV is the one year standard deviation of past data. Some calculate 20, 50, 100 day HV, etc. IV is forward looking, assisting in gauging volatility sentiment of a stock, ETF or the market. The key word in the tasty trade definition is "expected" which means in the future or "forward looking". What I meant was that the SPY has 30+ expirations so there will be up to 30 different options trading at active strikes (closer to ATM). An illiquid stock with monthly options might have on 4-4 expirations. – Bob Baerker Dec 11 '18 at 13:31
  • I don't think that in isolation that the 23.76% IV versus 18.7% IV tells you much of anything. Other factors might be at play. Bad data? IV calculation of last trade which is not concurrent with current price rather than current B/A? If the IV numbers are correct, what's the big picture? Smile, Smirk, Frown? And even with that info, I doubt that there's much to be gained for a retail trader. Maybe if you're delta neutral trading and pennies in size mean something but otherwise, not much. – Bob Baerker Dec 11 '18 at 13:31
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As I understand it, IV is defined as an annualized 1 standard deviation range for an underlying.

That's incorrect. That definition is for Historical Volatility, which is not the same as Implied Volatility (or IV).

what is the meaning of implied volatility for a strike?

Implied Volatility is one of the parameters of many option pricing models. Simplifying:

call price = f(underlying price, strike, time to expiration, dividends, risk-free rate, implied volatility).

For a given option (say, call) contract, you know strike and time to expiration by the contract definition. You know call price by looking at the market. You know dividends and underlying price by looking up information about the underlying stock. You know the risk-free rate by tracking treasuries rates. The only unknown is implied volatility. So you can solve the equation and find the IV.

That is the definition of IV.

  1. how do you get a value for the underlying as a whole?

There's no standard definition of "the IV of the underlying", nor "the IV of the underlying for month X". You'd have to ask your broker to know exactly which definition they are using. I don't know what Tastyworks use.

It's typical to follow the definition of the VIX, so you can look that up on the Cboe website.

  1. Does this mean that someone buying the 250 put for the current market price is valuing the volatility for this cycle at 23.76%

Not exactly - like mentioned before, there's no such thing as "the volatility for this cycle"; at least not as a standard definition. What's happening is that the market is pricing that 250 Put at 23.76% implied volatility -- in other words, 23.76% is the "magic number" that is found by solving the equation defined by the option pricing model you are using.

The difference in IVs that you mention is very common in the marketplace - it's commonly referred to as volatility smile. There's no single explanation for the phenomenon.

One common explanation is the fact that market returns are not normally distributed - there are fat tails - while option pricing models typically assume normal or log-normal returns.

The fat tails mean that prices jump really far more often than predicted by normal distributions. If the IV of OTM options was the same as ATM options, those OTM options would be underpriced - that is, the buyer of those options would have a positive expected return. But the marketplace prices contracts in a fair way, otherwise arbitrage opportunities would exist. That's why IV is higher for OTM contracts.

  • I got the IV definition from tastytrade which says "[IV is] a percentage that indicates the annualized expected one standard deviation range for the stock based on the option prices". You're right I left out the word "expected" in my question. – oceanus Dec 26 '18 at 19:59

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