You are given following information regarding 3-step binomial tree:

1. Each step is 3-month

2. The current stock price S(0) = 60

3. Time to expiry is 9 months

4. The strike price K = 65

5. The continuously compounded risk free rate is 6%

6. the stock pays no dividend

7. volatility = 17%

8. Use log-normal expressions (Jarrow-Hud form) to calculate how much stock price goes up or down in each time step

9. The option is a put option with following specific characteristics: Put option can be exercised only at the end of 6 month or at expiry.

(a) Calculate u

(b) Calculate d

(c) Calculate risk neutral probability

(d) Calculate put option value at time zero

(e) Calculate replicating portfolio at time zero

(f) Under general binomial pricing model, we require u > e(r)h > d where r is continuously compounded risk free rate, is continuously compounded dividend rate, h is size of time step. Prove that if this condition is violated then arbitrage opportunity exist.