Underlying stock currently trades at 100 and is lognormally distributed as expiration. Riskfree rate of 0%. Sigma at 20% annually for log returns. 63 days to go before expiration. Put strike at 90.

Q1a. What is the prob of option expring ITM?

Q1b. What is the avg underlying stock price when put expires ITM?

Q1c. What is the conditional avg put pmt at expiration?

Q1d. How much should the put be priced today

(for additional exercise, redo the question above for 110 put; 90 call; 110 call)

Q2a. What is the delta from shorting 10 puts from Q1? How do you hedge that position with stocks? Long or short, how many shares?

Q2b. What is the gamma of the 10 short put position, hedged by shares to be delta neutral? If underlying moves down by $5, what is PnL from delta and gamma?

Q2c. What is the one day theta value?

Q2d. If stock moves up by $5 in one day, what is the total pnl? How do you decompose total pnl to delta/gamma/ theta?

Q3. What is the vega of the hedged position above? If IV goes up by 3 points in one day, when underlying stock falls by $5 (stock goes from 100 to 95 and IV goes from 20 to 23%), what is your vega pnl? What is your total pnl from delta/gamma/theta/vega?