A four-step binomial tree for the price of a stock St is to be calculated using the up and sown ticks 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 S_v

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 date given above, what is the (approximate) annual volatility of St if this process is known to have a log-normal distribution?

(b) Calculate the four-step binomial trees for the St and the C_t.

(c) Calculate the arbitrage-free price C₀ of the option at time t = 0,

sdoun - dS = uS, +4