Q4. Delta and Gamma with Uniform Distribution (20 points)

Current underlying price at 100, and you expect price at expiration follows uniform distribution with mean absolute deviation of 20.

You short 10 PUTs with strike at 90.

Q4a. What is the TOTAL delta of your 10 short PUT position? (2 points)

Q4b. How many shares do you need in order to offset PUT delta from Q4a? Do you long or short the underlying shares (2 points)?

Q4c. What is the total gamma value of your hedged PUT positions from Q4a and Q4b? (2 points)

Q4d. For the hedged put position (short 10 puts, hedged with shares), what is the PnL from (starting) delta, and from gamma when underlying moves from 100 to 80, respectively? (4 points)

Q4e. Calculate option price when stock moves $80. How much of the total PnL of Q4d is from option position, and how much is from the stock position? (2 points)

Q4f. If underlying moves down from 100 to 80, what is the new delta at stock price of 80 for your overall position based on gamma? If you need to re-hedge to flatten delta with underlying shares, what trade do you need to do in the shares to flatten the delta? (2 points)

Q4g. What is the PnL from the delta and gamma when underlying moves from 100 to 120, respectively? (2 points)

Q4h. For Q4g, how much of the total PnL is from option position, and how much is stocks, when underlying moves from 100 to 120? (2 points)

Q4i. If underlying moves up from 100 to 120, If you need to re-hedge to flatten delta with underlying shares, what trade do you need to do in the shares? (2 points)