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Which of the following statements are true:
I. For a delta neutral portfolio, gamma and theta carry opposite signs
II. The sum of the absolute value of gamma for a call and a put for the same option is 1 III. A large positive gamma is desirable in a delta neutral portfolio IV. A trader needs at least two separate tradeable options to simultaneously make a portfolio both gamma and vega neutral
A. I, III and IV
B. III and IV
C. II and IV
D. I and II
Answer: A
Statement I is true. Consider the Black Scholes PDE 202.23.e1
and substitute delta = 0 (as this is a delta neutral portfolio), and we get the result 202.23.e.
Since r is generally small, and and S are positive, theta will be negative when gamma is positive and vice versa.
Statement II is incorrect. The gamma of a call and a put are equal and do not add to 1. The relationship described applies to delta, and not gamma.
Statement III is correct because positive gamma means that the portfolio gains both from an increase and a decrease in the value of the underlying, given delta neutrality. (Generally, a positive gamma is a good thing, but like everything else, it does not come free. A trader can choose to keep their portfolio gamma positive, but in doing so he or she would be giving up premiums from options they would have sold to neutralize the gamma.) Statement IV is correct because the only way to hedge gamma and vega is through other options positions as only options have gamma and vega. If only one tradeable option is available, it would be possible to hedge either the gamma or the vega, but not both, as achieving neutrality in one will upset the neutrality of the other.
The only way to simultaneously hedge the two would be to use two different options on the underlying, and determine the number of options to be traded (using a system of simultaneous equations).

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