Basic Points

• Convexity is a term used to describe the nonlinear payout structure of option strategies as compared to hard deltas, which retain a fixed directional exposure regardless of outcomes.
• Long options, especially without a spread and even more so when out of the money (starting with low deltas), make dramatically more in percentage gains from big moves than would comparable moves in the underlying security; hard deltas from the underlying always have a linear payout. 
• In general, convexity is a reference to the curvature which can be observed in an option’s risk chart. 
• Options have the most gamma convexity when they are shorter in duration.

Expert: Volga, the Other Convexity

There is also a type of convexity which comes from IV (implied volatility), which is caused from the tendency of volatility bidders tend to pile on while IV is increasing. This is known as volga, which is a change in vega in relation to a change in IV; a positive volga value means that vega increases when IV increases, which means a larger slice of the option’s price is determined by IV. High volga levels set the stage for potentially explosive price increases in an option if there is a spike in IV (the expected percentage range over the next year with 68.3% confidence based on options pricing). 

Expert: Strategy

One way to manage convexity as traders is to use gamma scalping, which is an offensive style of delta-hedging (rebalancing directional exposure) after sharp moves which partially lock in profits, while still maintaining some exposure if there’s a continuation. Some advantages of this are that there becomes less risk on the table after gamma-scalping, some profits are secured, and more money can be made if large moves continue.

An alternative strategy is rolling to lower deltas, which also reduces risk, secures cash profit, and allows for more opportunity if the underlying security keeps moving ahead of the expected move or if IV keeps expanding. Rolling also maintains strong convexity, and if the debits get small enough then adding more quantity to the contracts held can be justified. This is how thousands of percent (as compared to the original debit) can be made from managing convexity on extreme moves. But with gamma scalping however, each action to delta-hedge is going to dull the convexity somewhat, which is something to consider.