St+1=St exp((r-1/2²)t + t)

Where

St is the stock price at t ( 10USD)

St+1 is the stock price at t+1

r is the expected annual stock return (say 0.15)

is the annualized volatility of the underlying stock.(0.2)

T is the time in year(1year)

n is the number of step involved in the calculation (12 step)

n is the number of steps involved

t is the size of the unit step size =T/n

is the distribution term with zero mean(a random value from a normal sample) =0.15

Consider the following value ;

St = 10$

R= 0.15 (15% expected return per year)

= 0.02 (20% annual volatility in prices)

T= 1 year

N =100

is 0.15

question::

(1) starting with the initail stock price and considering 100 step calculate the expected value of the stock price at the end of every successive interval of time.

(2)Plot the entire movement of price over the T period under obervation

3) instead of considering a fixed e in the previous step randomly assign value to from a stardard normal distribution

4) perform 5 trial of 100 steps to each plot the probable movement of the stock price over a 1 year period.plot each trajectory of price as a seperate line.