Explanation
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).

Explanation
From the put-call parity, we know that Call - Put = Stock - Bank deposit. Since the bank deposit has a zero Gamma, and the Gamma of the Stock itself is also 0, we get the relationship Gamma of Call - Gamma of Put =
0. Therefore, if the Gamma of a call option is 0.08, the Gamma of the corresponding put option is also 0.08.

Explanation
The delivery price of a futures contract is given by Se^(rt), just as in the case of a forward contract. However, a key difference is that a forward is settled at maturity whereas a futures contract pays out the P&L daily. So if the spot price increases from S to S, the holder of a futures contract immediately receives the change in the delivery price without any discounting to the present. That is, the holder of the futures contract receives (S + S)e^(rt) - Se^(rt) = Se^(rt) right away. Therefore the delta of a futures contract is e^rt, which given positive non-zero values of r and t can only be greater than zero.
Therefore Choice 'b' is the correct answer. Note the difference from a forward contract where this difference is not received till the delivery date, therefore making the delta of the forward contract to be equal to 1.

The return on an asset is a function of the covariance of the asset's return to the returns of the market portfolio.
They do not depend upon the standard deviation of the asset itself. Therefore Choice 'b' is correct and Choice
'c' is incorrect.
The expected returns on an asset are equal to the risk free rate plus the beta times the market risk premium, therefore Choice 'a' is correct.
Portfolios on the efficient frontier will all have a different Sharpe ratio, which is the ratio of excess returns to portfolio volatility.
Choice 'c' is the correct answer (note that the question is asking for an identification of the INCORRECT statement).

Explanation
The transformation line represents the combination of a 'risky bundle' and the risk free asset. Investors can choose different combinations of these assets depending upon their risk appetite. The transformation line meets the y-axis (portfolio returns) at the point equal to the risk free rate. Choice 'b' is the correct answer and the rest are incorrect.The highest possible transformation line, ie the transformation line with the maximum slope, is the transformation line joining the risk free rate on the y-axis and the portfolio with the maximum Sharpe ratio on the efficient frontier. This line is called the 'capital markets line'. Investors can pick any point on this line according to their risk appetite, and doing so would maximize the return they can obtain for their desired level of risk. The capital markets line is tangential to the efficient frontier. The Sharpe ratio stays constant throughout the CML.