Q2. You are interested in buying some put protection for SP500 ETF SPY.

SPY trading at 470. You are interested in a PUT option with 25 delta (ie delta per PUT is -0.25). Annual sigma is $125. DTE is 20 days.

Q2a. What is strike price that will generate that 25 delta for the put? (4 points) (hint: delta for put is connected to CDF function. Using delta, you can figure out z-score that generates that delta. You can find the z-score either using normal table or in EXEL with norm.s.inv() function. The input to that function is the cumulative probability and output is the z-score. Then you can convert the z-score to strike price).

Q2b. You bought 50 put contracts with 25 delta. Market moved downward by $5 the next day. You did not hedge your put position. What is the PnL from delta / gamma/ theta respectively? (4 points) (note: each contract corresponds to 100 shares).

Q2c. Consider a market marker who sold those 50 puts to you. He immediately hedged the position delta with underlying shares. How many shares does he need to sell immediate? (4 points)

Q2d. If annual sigma immediately jumped by $5 from 125 to 130 (still 20 DTE). How should market maker adjust hedging position? (4 points)

Q2e. Keeping IV and underlying price the same for 5 days. How should market maker adjusts hedging position after 5 days? (hint: recalculate option delta with new 15 DTE). (4 points)

Q2f. If market goes down by $5, how should market maker adjust his hedging position based on gamma effect to stay delta neutral? Note: there are still 20 days left for the option (4 points)