The volatility smile or skew, is a phenomenon where strike prices of out of the money options have a higher or lower implied volatility then the at the money options.  When this type of curve structure occurs, a trader can take advantage of this situation in a number of ways. 

Directional speculation:

If a trader believes the market for an underlying asset is going up or down, he can place a call spread (for directional upward bets) or a put spread (for directional downward bets).  The call spread is where an investor buys an at the money call, and simultaneously sells a call with a strike that is higher (out of the money).  A put spread is where the investor purchases an at the money put, and simultaneously sells a put with a lower strike (that is out of the money).  If a smile exists, the implied volatility used for the out of the money calls or out of the money puts, will be greater than the implied volatility used for the at the money calls or puts.  This will reduce the overall theoretical premium that a trader pays for the call spread or put spread.

Delta Neutral Trades

In attempting to capture the skew or smile, a trader can create a structure that does not have any outright delta, in an effort to capture the premium associated with the smile.  This means that the trade does not have exposure initially to upward or downward movements in the underlying asset.  A structure in which trader purchases 1 at the money put and sells two out of the money puts can be used to take advantage of a high relative skew on out of the money options.  This can also be accomplished on the call side of an options structure.  Buy purchasing one put and selling two, the delta can be completely neutralized, and premium can be collected.  The structure will likely make money on the put side, unless the market falls below a specific point below the two out of the money options.  Out of the money, option can be staggered to make the strike level where a low is created very far away.  This type of structure mimics a miss option in the world of binary options.  When the smile is very high, it makes sense for a trader to purchase of miss option if the payout mimics the payout that would be receive from selling vanilla out of the money options.

Selling a strangle is also a way to benefit from the market created out of the money options with very large smiles.  A strangle is selling or buying out of the money options.  When a volatility smile is high, selling strangles can be very profitable.  A way to take advantage in a delta neutral profile, is to purchase a straddle (buying a put and a call on an at the money strike), and selling to strangles against it (selling out of the money puts and calls).  Similar to the structure described above, this strategy combines puts and calls and make the area that the market needs to reach to create a losing trade, very far from the at the money area.  The structure, which is also called a 2,1,1 provides significant protection if the market begins to move, and a trader can collect a significant premium from the out of the money volatility skew.

Smile arbitrage is a way for an investor to capture the high volatility if a particular strike moves outside the current linear calculation of the smile or skew.  This can happen if the demand for a strike is greater than the strikes around it.  For example, if the 60-dollar crude oil strike as a volatility of 50%, and the 59 strike has a volatility of 52% and the 58 strike has a volatility of 54%, and the 56 strike has a volatility of 58%, once would expect the volatility of the 57 strike to be at 56%.  If for some reason this strike had an implied volatility of 55%, a trader could purchase that strike and sell a different strike with the hope that the implied volatility difference would return to normal.

An investor can use numerous strategies to profit from the volatility smile.  The key to successful trading it to understand when the smile is rich relative to the at the money volatility or cheap relative to the at the money volatility.  A trader should examine different tools that are available to gauge the volatility smile, to enhance their potential advantage in trading this market.