8. A put option on the S&P 500 has an exercise price of 500 and a time to maturity of one year. The risk free rate is 7% and the dividend yield on the index is 3%. The volatility of the index is 20% per annum and the current level of the index is 500. A financial institution has a short position in the option.

a) Calculate the delta, gamma, and vega of the position. Explain how they can be interpreted.

b) How can the position be made delta neutral?

c) Suppose that one week later the index has increased to 515. How can delta neutrality be preserved?

The answers are given. Please provide the detailed process! Thanks!

a) 0.371, -0.0038, -1.85, b) Sell 0.371 of index for each option sold, c) Delta changes to 0.317 so 0.054 of index must be bought back