LO 15.5: Describe the volatility smile for equity options and foreign currency

LO 15.5: Describe the volatility smile for equity options and foreign currency options and provide possible explanations for its shape.
The volatility pattern used by traders to price currency options generates implied volatilities that are higher for deep in-the-money and deep out-of-the-money options, as compared to the implied volatility for at-the-money options, as shown in Figure 1.
Figure 1: Volatility Smile for Foreign Currency Options Implied volatility
Strike price
The easiest way to see why implied volatilities for away-from-the-money options are greater than at-the-money options is to consider the following call and put examples. For calls, a currency option is going to pay off only if the actual exchange rate is above the strike rate. For puts, on the other hand, a currency option is going to pay off only if the actual exchange rate is below the strike rate. If the implied volatilities for actual currency options are greater for away-from-the-money than at-the-money options, currency traders must think there is a greater chance of extreme price movements than predicted by a lognormal distribution. Empirical evidence indicates that this is the case.
This tendency for exchange rate changes to be more extreme is a function of the fact that exchange rate volatility is not constant and frequently jumps from one level to another, which increases the likelihood of extreme currency rate levels. However, these two effects tend to be mitigated for long-dated options, which tend to exhibit less of a volatility smile pattern than shorter-dated options.
E q u i t y O p t i o n s
The equity option volatility smile is different from the currency option pattern. The smile is more of a smirk, or skew, that shows a higher implied volatility for low strike price options (in-the-money calls and out-of-the-money puts) than for high strike price options
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Topic 15 Cross Reference to GARP Assigned Reading – Hull, Chapter 20
(in-the-money puts and out-of-the-money calls). As shown in Figure 2, there is essentially an inverse relationship between implied volatility and the strike price of equity options.
Figure 2: Volatility Smile for Equities Implied volatility
Strike price
The volatility smirk (half-smile) exhibited by equity options translates into a left-skewed implied distribution of equity price changes. This left-skewed distribution indicates that equity traders believe the probability of large down movements in price is greater than large up movements in price, as compared with a lognormal distribution. Two reasons have been promoted as causing this increased implied volatilityleverage and crashophobia.

Leverage. When a firms equity value decreases, the amount of leverage increases, which essentially increases the riskiness, or volatility, of the underlying asset. When a firms equity increases in value, the amount of leverage decreases, which tends to decrease the riskiness of the firm. This lowers the volatility of the underlying asset. All else held constant, there is an inverse relationship between volatility and the underlying assets valuation.
Crashophobia. The second explanation, used since the 1987 stock market crash, was coined crashophobia by Mark Rubinstein. Market participants are simply afraid of another market crash, so they place a premium on the probability of stock prices falling precipitouslydeep out-of-the-money puts will exhibit high premiums since they provide protection against a substantial drop in equity prices. There is some support for Rubinsteins crashophobia hypothesis, because the volatility skew tends to increase when equity markets decline, but is not as noticeable when equity markets increase in value.
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