B.3 a. Consider the delta of a European option. i. Explain concisely what the delta measures, the bounds on its value and how it is linked to the construction of the replicating portfolio for the option. Illustrate your answer by discussing the sign of the delta of an at-the-money put option. (10 marks) ii. Show how deltas of puts and calls (with the same maturity, strike price and underlying) are linked and what this implies for their respective signs. What are the implications of this for the man- ner in which one would use these options to delta hedge a long position in the underlying. (10 marks) b. You currently hold a portfolio of options on the same underlying. Call this portfolio X. The delta, and gamma of the portfolio are as follows; Ax -50.0, Fx=3.0 i. Explain the implications of these numbers for the risk exposures of the option portfolio. (7 marks) ii. If the price of the underlying was to fall by $2, by how much would you expect the value of the portfolio to change and by how much would you expect its delta to change? (5 marks) iii. If one wanted to delta hedge this portfolio using the underlying, what position in the underlying would one have to take. Explain your