Implied volatility is thus the volatility of the underlying instrument, which, when included in the theoretical pricing model, yields a theoretical value identical to the market value. Because of put-call parity, implied volatility should be identical for call and put options.

While historical volatility is a measure of the past, implied volatility reflects market expectations for the future of the underlying’s price fluctuations over the remaining life of the option.

If we analyze historical data, we find that the implied volatility is usually smaller and less volatile than the historical volatility, although both follow a mean-reverting process. The implied volatility of short-term options tends to be further from the mean than long-term options.

In the long run, historical volatility is found to have the strongest influence on implied volatility. In the short run, near-future events such as OPEC energy announcements, earnings news, or takeovers are likely to increase implied volatility.

Depending on the type of option (i.e., vanilla options, exotic options) and its return characteristics, different option pricing models may be used to derive implied volatility. The Black and Scholes (1973) formula is one of the most famous option pricing models.

It is often used for basic financial instruments with approximately lognormal prices, such as European options with no dividends. This method is analytically advantageous.

However, there are many modifications to the Black-Scholes model, along with alternative pricing methods such as binomial methods, analytical methods, and approximation methods that account for different option characteristics. For a comprehensive overview see Haug (2006).

For example, if we use the Black-Scholes formula to derive the theoretical value of a call option C, the input variables would include

- the volatility of the underlying or the expected future volatility,
- the expiry date of option,
- the strike price of option,
- the price of the underlying,
- the prevailing interest rate r, and, in certain circumstances,
- dividends paid by the underlying. The single nondirectly observable value would be the future realized volatility.

Since the Black-Scholes option pricing function is strictly monotonically increasing in the volatility, in other words, if all other input variables are equal, there is only one single value for the theoretical option price for a certain volatility, it is possible to obtain the inverse function in the volatility of the pricing function, so that observed option market price implies a volatility which is referred to as implied volatility. If the call option’s price is higher than null and lower than the price of the underlying, there is a one-to-one solution.

Option pricing models are generally rather complex. Thus there is often no closed-form solution for the implied volatility. However, a root-finding technique, such as the NewtonRaphson method, can be used to obtain a solution for it.

Because of the rather high volatility of prices, it is important to use an efficient algorithm to obtain a solution for the implied volatility. If the pricing function, like the Black-Scholes formula, is well behaved, and there is a closed-form solution for vega, the Newton-Raphson method can be an extremely efficient method that can converge quadratically.

However, if there are multiple local extrema, and vega must be estimated, other numerical methods (such as Brent’s method) or approximations (such as the Brenner–Subrahmanyam formula) may be more efficient. For more details see Antia (2002) and Hallenbach (2004).

The observed implied volatility is usually not constant. It varies with different underlyings, strike prices, and expiry dates. This illustrates the limitations of the BlackScholes formula, which states that there is only one implied volatility, independent of the strike price.

If all other input variables are equal, the graph of the implied volatility will have a characteristic shape, which is persistent over time and is characteristic of the respective option and market. Before the crash of 1987, the shape of the implied volatility relative to the expiry date was U-shaped, or what we call a smile.

However, since that time, the smile has transformed into a downward slope, which is referred to as skew (usually equity call options), or a smile/smirk (usually equity put options). Figure 1 shows the smile of an equity put option.

If a smile is observed, there is a premium for options with very high or very low strike prices that is not captured by Black-Scholes, which implies that options are no perfect substitutes for each other.

One explanation for this phenomenon may be found in behavioral finance, which states that investors are willing to pay a premium to hedge their portfolio against extreme losses.

The illiquidity of out-of-themoney options or stochastic volatility may also explain these approaches to the premium. However, in any event, theory does not capture what makes one option more desirable than another.

By considering the expiry as a second input variable, we can derive a static implied volatility surface that illustrates term structure and volatility smile at the same time. In practice, a volatility surface can provide insight into whether there are any irregularities to be exploited.

For the sake of interpretability, the input variables are usually standardized; thus, instead of the strike price, the spot moneyness is used. The high implied volatility of close-to-expiry, deep-in-the-money, and out-of-the-money options here is a result of the high bid-ask spread and is thus an illiquidity of the options. Furthermore, close-to-expiry options are not continuous, but exhibit discrete jumps.

In addition, the volatility surface is far from constant—it changes over time (the evolution of the implied volatility surface). Several rules and models exist to explain and predict the development of the surface, that is, the stick-strike rule, the sticky-delta rule, and the sticky-implied-tree model. Several options exchanges offer (implied) volatility indexes like the Chicago Board Options Exchange Volatility Index (VIX) as a proxy for expected volatility.

Implied Volatility |