Q1. Discrete option pricing

Stock takes 6 scenarios at option expiration:

scenario outcome prob

1 1 0.05

2 2 0.15

3 3 0.30

4 4 0.30

5 5 0.15

6 6 0.05

Consider a call option with strike X=4. Assume rf=0

a. what is the probability that call with expire ITM?

b. what is the conditional probability for the two scenarios that call will be in the money (outcome 5 and 6)

c. what is the conditional value of the underlying stocks when call expires ITM?

d. what is the conditional call payoff when it expires ITM?

e. how much should the option be valued today?

for additional exercise: redo the question with a put option.

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Q2. Underlying stock follows uniform distribution with mean at 100 and MAD=10. Stock price today is priced at the mean value.

Consider a call option with strike at 110.

a. what is the prob that option will expire ITM?

b. what is the avg underlying stock price when call expires ITM?

c. what is the avg call payoff when it expires ITM?

d. how much should the call be priced now?

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Q3. Consider a call with x=110. Underlying stock follows uniform distribution, and right now is at 100 and MAD=10.

a. what is the option delta and gamma?

b. if you want to long 1 call contract, how do you hedge your delta with underlying shares?

c. what it the delta and gamma of the hedged position above (your hedged position here is long call and short shares).

d. if underlying stock moves up to 104, what is the the total pnl for the hedged portfolio in part b? how much of the total pnl is due to stock and how much is due to option price movement?

e. if underlying stock moves dn to 96, what is the the total pnl for the hedged portfolio in part b? how much of the total pnl is due to stock and how much is due to option price movement?

additional exercise: redo the question with a short put position with strike at 90.

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Q4. You observe in the market that a call option with strike at 110 is priced at $1.25. What is the implied MAD?