A four-step binomial tree for the price of a stock St is to be calculated using the up- and downticks given as follows:
u = 1.15
d = 1/u
These up and down movements apply to one-month periods denoted by ∆ = 1. We have the following dynamics for St,
Sut+∆ = Stu
Sdt+∆ = Std
where up and down describe the two states of the world at each node. Assume that time is measured in months and that t = 4 is the expiration date for a European call option Ct written on St. The stock does not pay any dividends and its price is expected (by "market participants") to grow at an annual rate of 15%. The risk-free interest rate r is known to be constant at 5%.
(a) According to the data given above, what is the (approximate) annual volatility of St if this process is known to have a lognormal distribution?
(b) Calculate the four-step binomial trees for the St and the Ct.
(c) Calculate the arbitrage-free price C0 of the option at time t = 0.