Problem 1. The price of a non-dividend-paying stock is currently $50. It is known that in six months the stock price will be either $45 or $55. The continuously compounded risk-free interest rate is 10% per year. What is the risk-neutral probability that the stock price reaches $45 in six months?

A.0.244

B.0.373

0.5

D.0.627

E.0.756

Problem 2. Consider the same stock as in Problem 1 and a six-month European put option on the stock with a strike price of $47. A trader sells one put option. To hedge, he should long ______ shares.

-0.4

-0.3

-0.2

0.2

0.3

Problem 3. Consider the same stock and European put option as in Problem 2. What is the current value of the option?

A.$0.46

B.$0.71

C.$1.16

D.$1.90

E.$2.16

Problem 4. A stock price is currently $80. It is known that in 4 months it will be either $75 or $85. The risk-free interest rate is 5% per year with continuous compounding. Consider a 4-month European call option with a strike price of $80. (1) Draw the binomial tree. (2) Compute of the option. (3) Compute the risk-neutral probability. (4) Find the current value of the call option.

Problem 5. A stock price is currently $40. Over every 3-month period the stock prices goes either up by 10% or down by 10%. The continuously compounded risk-free interest rate is 12% per year. Use a 2-step binomial tree to find the value of a 6-month American put option with a strike price of $45.