By John Summa, CTA, PhD, Founder of OptionsNerd.com
As we've already learned, volatility comes in two forms: statistical (historical volatility) and implied (IV). We saw how past levels of IV can be used to determine how expensive options premiums are (relative to past levels). In this segment, the relationship of IV to historical volatility will be used to show how to evaluate whether an option is "overvalued" or "undervalued".
Recall that an option trades at prices that may not be the same as theoretical values. In other words, an option's price may not be in line with valuations from the perspective of option pricing models (i.e., BlackScholes). What should we make of this deviation, which is often referred to as mispricing? Academics make careers out of trying to recast options pricing in an attempt to find a better model  better in the sense that it does a more accurate job of explaining where option prices are at any point in time. This usually involves using different measures of historical volatility, but the issue goes well beyond the scope of this tutorial. (For related reading, see Understanding Option Pricing.)
IV Relative to Historical Volatility
When IV and historical volatility are derived using normalized methods relative to price, the two volatilities can be compared, in terms of both current levels and past levels, as seen in Figure 7. As the chart shows, the two levels of volatility can oscillate around each other at certain times, and can remain at different levels for extended periods before crossing again. The charts present a 20day exponential moving average of current volatility levels, which smooths (or dampens) daily noise.
The chart shows historical volatility and IV between June 2003 and August 2007 for Altria Group Inc. (NYSE:MO). Historical volatility represents the actual volatility of the underlying stock, while the IV plot represents the volatility that is implied by the market price of the options trading on MO. Historical volatility and IV can at times remain quite similar, even if they don't remain exactly the same. When they are the same or close to the same, this means that the actual options prices are implying a volatility level that is close to the same as the volatility of the underlying stock.
However, at times historical volatility and IV can significantly deviate from each other. This can be the basis for finding options that are significantly under or overvalued. Additionally, by looking at average levels of both historical volatility and IV, it is possible to determine when either of the two types of volatility have moved significantly away from their average or normal levels. Some argue that it is easier to predict the direction of volatility than the price of the stock itself. With the right options strategy, it might be possible, therefore, to trade a return of volatility to normal levels, which seems to happen at regular intervals. There are also seasonal factors at work that can be isolated for possible volatility trading.
IV and Historical Volatility Spikes
There are a number of conditions shown in Figure 7 that capture the above conditions. First, in 2003, a spike in IV can be seen (near 39%), which represents very high implied volatility. That is, option prices are very expensive. In September 2004, meanwhile, historical volatility goes below 10%, meaning the stock volatility has declined sharply. Obviously, just the reverse of high IV and low historical volatility can be seen at different times, which would represent the opposite option price valuation conditions.
When IV is greater than historical volatility, options are thought to be overvalued, and when IV is less than historical volatility, options are considered to be undervalued. In Figure 7, both conditions are indicated (IV < historical volatility and IV > historical volatility).
Generated by OptionsVue 5 Options Analysis Software 
Figure 7: Altria Group Inc. implied and historical volatility charts. Historical volatility (historical (statistical) volatility is marked by a brown line; implied volatility is shown in blue. 
IV is a measure that captures projected volatility, a value that depends on how expensive options are in the marketplace. Historical volatility, meanwhile, is a measure of how the volatility of the underlying stock or futures contract has been over the recent past. Because historical volatility is an input in pricing models, it indirectly affects theoretical price levels.
Options Mispricing
If actual prices for options are greater than historical volatility modeled theoretical prices, IV will pick up this mispricing. The mispricing can be above or below theoretical prices, which is perhaps the easiest way to understand overvalued options (above theoretical) and undervalued options (below theoretical) pricing. When the market prices are greater than theoretical prices, IV will be greater than historical volatility and when market prices are less than theoretical prices, IV will be less than historical volatility.
Figure 8 provides a summary of the ideal conditions for selling and buying options. The conditions of high or low IV and historical volatility refer to levels of both measures of volatility relative to past levels. As we saw in Figure 7, these levels can be evaluated using a lookback period over a number of years. The more time in the lookback period, the better.
  High Implied Volatility  Low Implied Volatility  High Historical Volatility  Low Historical Volatility 
Implied > Historical  Selling Bias  Mixed  Selling Bias  Mixed 
Implied < Historical  Mixed  Buying Bias  Mixed  Buying Bias 
Figure 8: Volatility conditions for selling and buying options 
When IV or historical volatility are high and IV > historical volatility, typically this is an ideal time for selling options. Here, both theoretical prices will be high due to high historical volatility (if historical volatility is high and IV > historical volatility, then IV must be high, too) and the actual highpriced options prices will be projecting even higher volatility, which adds additional overvalued premium in the marketplace. For buying, meanwhile, when IV or historical volatility are low and IV < historical volatility, the reverse valuation conditions exist. Options are cheap and undervalued. The remaining relationships in Figure 8 show mixed volatility conditions, which are not ideal for either selling or buying.
Conclusion
The concept of under and overvalued options is explained and illustrated with historical implied volatility relative to historical volatility levels. Additionally, high and low IV and historical volatility (not greater than or equal to) conditions were shown to indicate expensive and cheap options, respectively. The ideal conditions for selling and buying options were also presented. High IV combined with IV > historical volatility make for the best conditions for selling while buying options generally is statistically superior when IV is low and IV < historical volatility.
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