Call price as numeraire A stock that pays no dividends has current price S0. In one year's time the stock price is 111 with probability 0.75 (state A), and 75 with probability 0.25 (state B). Annually compounded interest rates are 10%. The one-year 70-strike European call has price 10 at T = 0.
(a) Write down the payout of the 70-strike call at states A and B.
(b) By using a martingale condition for C70(m, 1)/Mm, find the risk-neutral probability of state A with respect to the money market numeraire.
(c) Hence, or otherwise, prove that S0 = 810/11 (= 73.64).
(d) Find the price of a one-year 80-strike American put. For K > 0, write down a simple formula for the price of the one-year K-strike American put.
(e) By using an appropriate martingale condition, show that q∗, the risk-neutral probability of state A with respect to the 70-strike call numeraire, satisfies
(f) Which of the following assets cannot be used as a numeraire in this binomial tree?
I. The stock.
II. The one-year 100-strike European call.
III. A forward contract with delivery price 90 and maturity one year.
IV. The ZCB with maturity T = 1.
V. The one-year 115-strike European put.