Binomial tree: random interest rates II
A stock that pays no dividends has current price 100. In one year's time the stock price is 120 with probability 0.7, and 90 with probability 0.3. If the stock price at T = 1 is 120, then the price at T = 2 is 150 with probability 0.5, and 110 with probability 0.5. If the stock price at T = 1 is 90, then the price at T = 2 is 110 with probability 0.6, and 80 with probability 0.4. Annually compounded interest rates are 10%, except if the stock has price 120 at T = 1, in which case the interest rate from T = 1 to T = 2 is 20%.
(a) Draw the two-step binomial tree and give the value of the money market account at each state.
(b) Prove that the price Z(0, 2) at T = 0 of the ZCB with maturity T = 2 is
(c) Find the price of a two-year 120-strike European call option.
(d) What is the risk-neutral distribution of the random variable Z (1, 2) with respect to the numeraire that is the ZCB with maturity T = 2?