Lognormal Distribution and Monte Carlo valuation

Respond to the following questions.

1. Suppose you observe the following month-end stock prices for stocks Y and Z:

                                                                 Day
   
                                           0               1            2            3            4
   
   Stock Y                         100           105        102         97        100
   
   Stock Z                         100           105        150         97        100

Compute the following for each stock.

a. The mean monthly continuously compounded return

b. The annual return

c. The mean monthly standard deviation

d. The mean annual standard deviation

1. Use Monte Carlo to compute prices for claims that pay the following if S₀= 100, r= 0.06, σ_S= 0.4 and δ= 0:

   a. S²1

   b. √S₁

   c. S₁^-2

2. Suppose S(0)= 100, r= 0.06, σS= 0.4 and δ= 0. Use the equation below to compute prices for claims that pay the following:

   F^p_0.1(S^a) = S^a₀exp((a-1)r+1/1a(a-1)σ²)

   a. S²

   b. √S

   c. S^-2

   Compare your answers in question 3 to the ones you obtained in question 2.