1.

Question 2 on next page, Excels spreadsheets in Blackboard

3.
ATM Call

OTM Call

a)	Assuming we sold one ATM options contract, the delta on the options position is (.616)*(-100)= -61.6, so you would go long 61.6 shares to make the portfolio delta neutral.The gamma of the portfolio  is 12.8 so to make the portfolio gamma neutral be take a long position in the OTM with gamma .053 of 12.8/.053=241.50 so 2 contracts. However we need to make this portfolio delta neutral again buy selling 92 shares. This makes the portfolio gamma and delta neutral.
b)	Going back to delta (non gamma neutral) neutral position in part a, we can make the portfolio by vega neutral by taking a long position of  vega of portfolio/ vega of OTM stock =(10.77)/.042 of 256.42 or 300 traded options. This would increase delta to 24 so we should need to sell 24 shares.

4.
a)
S&P 500 Calculations

Crude Oil Calculations

Using the above value we can calculate the 10 day Var of the Portfolio

b) Using historical simulation method

c) Back Tested Results

Return of portfolio on day i= (5/8)*Returns&p(i)+(3/8)*Returncrude oil(i)

5. Refer to excel for data

We can conclude from the volatility series that the EMWA has a smaller standard deviation making the volatility estimates smaller than the GARCH (1,1).  Since the standard deviation for the GARCH (1,1) is higher, giving it higher volatility estimates, ultimately leads to a more accurate prediction for volatility.  Both methods are relatively close, as seen above, however the GARCH (1,1) is a through and accurate tool to measure volatility.