Binomial tree: random interest rates I
Consider the two-step binomial tree in Chapter 8 Question 1. However, now suppose that if the stock is at 120, then the annually compounded interest rate from time T = 1 to T = 2 is 5%, not 10%.
(a) Write down the value of the money market account Mm at all states of the tree. Does the joint tree for (Sm, Mm) recombine?
(b) By using the martingale condition for Sm/Mm, find the risk-neutral probabilities with respect to the money market numeraire Mm at each node of the tree.
(c) Hence by using the martingale condition for Z(m, 2)/Mm show that
(d) Use (c) and an appropriate martingale condition to prove that the risk-neutral probability, with respect to the numeraire Z(m, 2), of the stock having value 120 at T = 1 is 44/65. Hence show that the risk-neutral probabilities of this state, with respect to the money market account and the ZCB with maturity T = 2, differ by 2/195. Do you want to revisit your comments in Question 1(b)?
Question 1
Binomial tree: European and American puts Consider a two-step binomial tree, where a stock that pays no dividends has current price 100, and at each time step can increase by 20% or decrease by 10%. The possible values at time T = 2 are thus 144, 108 and 81. The annually compounded interest rate is 10%.
(a) Calculate the price of a two-year 106-strike European put using (i) a replication argument and (ii) risk-neutral expectation.
(b) Calculate the price of a two-year 106-strike American put using replication, and hence verify that the American put has price strictly greater than the European.
(c) Calculate the prices of a two-year 86-strike European put and American put. What is different to (b)?