Assume that the price of a share in the XYZ stock satises the following SDE:

dSt = Stdt + StdWt

where

The annual drift is 15%

the annual volatility is 30%

The share price now, S0 is $90.

An investor has big condence in the XYZ stock and decides to invest in the stock in order to build up a retirement fund of $ 1,000,000 in 20 years time. The continuously compounded annual interest rate over the 20 year period is assumed to be 5%.

(a) Determine a formula for the distribution of St.

(b) In how many shares does the investor need to invest now, in order to give a 50:50 probability of building up the planned $ 1,000,000 retirement benedit in 20 years time? What would the current cost of the total investment be?

(c) In how many shares does the investor need to invest now, in order to give a 90 percentile probability of building up the planned $ 1,000,000 retirement benet in 20 years time. What would the current cost of the total investment be?

(d) Suppose the investor decides to rather invest in bonds to secure the $1,000,000 retirement benet in 20 years time. What would the current cost of this investment be.