a. Consider the following two continuous time models. Model 1: dS = dt + odz Model 2: dS = Sdt + oSdZ where dZ is the instantaneous change in a Brownian motion and u and o are fixed parameters. Detail the key features of the models, explain how they differ and indicate which you think is most appropriate as a model for stock price dynamics. Also indicate any limitations of the particular model that you think is most appropriate. b. You currently hold a portfolio of options on the same underlying. Call this portfolio X. The delta, gamma and vega of the portfolio are as follows; AX= 75.0,X = 15.0, VX = 200 i. Explain the implications of these numbers for the risk exposures of the option portfolio. (5 marks) ii. If the price of the underlying was to move by $5, 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. You can also take positions in the underlying and two options Y and Z. Option Y has a delta of 0.6, a gamma of 0.02 and a Vega of 14. Option Z has a delta of -0.35, a gamma of 0.01 and a Vega of 10. You would like to adjust your portfolio so that it is delta and gamma neutral and has a Vega of +100. How would you do so through adding positions in Y, Z and the underlying to the portfolio X? Why might you want to hold such a portfolio?