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What does it mean to long convexity of options?
Long convexity is achieved by owning long dated low delta options. When a significant move occurs in the underlying the volatility curve will move higher. Instead of a linear relationship between your long position and it's return, you receive a multiple of the linear return. For example: Share price $50 Long 1 (equals 100 shares) contract of a 2 year 100 call Assume this is a 5 delta option If the stock price rises to $70 the delta of the option will rise because it is now closer to the strike. Lets assume it is now a 20 delta option. Then  Expected return on a $20 price move higher, 100 shares($20)(.20-.05)=$300 However what happens is the entire volatility surface rises and causes the 20 delta option to be 30 delta option. Then The return on a $20 price move higher, 100 shares($20)(.30-.05)=$500 This $200 extra gain is due to convexity and explains why option traders are willing to pay above the theoretical price for these options.