Q1. You believe stock price by year end will have the following multinomial distribution (15 points):

Price Probability

60 -10% 80 20%

100 -40%

120 -20%

140 -10%

Q1a. What should be the stock price TODAY? (3 points)

Q1b. what is the prob that a 110 strike CALL will expire ITM? (3 points)

Q1c. what is the conditional average price of underlying stock when 110 strike CALL expires ITM? (3 points)

Q1d. what is the conditional average payment from the 110 strike CALL option when the CALL expires ITM? (3 points)

Q1e. based on Q1b-Q1d, how much should the 110 CALL be priced at? (3 points)

Underlying current trading at 200 with MAD of 40. Q2a. What is the probability of option expiring ITM for a 160 CALL? (3 points)

Q2b. What is the average underlying price when CALL expires ITM? (3 points)

Q2c. How much should the 160 CALL be priced at? (3 points)

Q2d. Out of the price in Q2c, how much of that is intrinsic value and how much is time value? (3 points) Hint: intrinsic value is the value of option if option expires NOW. Time value is the remaining.

Q3. Implied MAD with Uniform Distribution

Underlying price currently at 200, and follows a uniform distribution with mean of 200. You observed 80 strike PUT priced at $5.00. What is the implied MAD? (10 points)

A: put price equation

Which one to choose? Or need to keep one

A: need to double check to make sure strike price X stays inside the uniform form distribution boundaries. We throw away the smaller solution and keep the larger one.

Q4. 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)