Arithmetic Brownian motion

Question I:
Let {X(t),t ≥ 0} be an arithmetic Brownian motion with a drift factor of 0.35 and a volatility of 0.43. Given that X4 = 2.
(a) Find the distribution of X(13).
(b) What is the probability that X(13) > 9.
(c) Find the confidence interval of X(13) at level 90%.
Question II:
(a) Show that the probability that a European call option will be exercised in a risk-neutral world, with the notation introduced in the BSM chapter, is equal to N(d2).
b) At time T, find an expression for the value of a derivative that pays off $100 if ST > K? (c) What is the value of this security at time zero using risk-neutral valuation.
Question III:
We assume that the stock price at time t, St, has a LogNormal model with S0 = 100, µ = 0.08 and σ = 0.3. It is assumed that the stock pays no dividend.
(a) Find P (S1 ≥ 105).
(b) Find P (S1 < 98).
(c) Let Kt = S0ert, compute P (St ≥ Kt).
Question IV:
Find the price of a 3-month European put option on a non-dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% and the volatility is 30% per annum.

Sample Solution