Consider the Black-Scholes model with stock price process

S(t) = S(0) exp(( ^2/2)t + B(t));

where t 0 is the time (in years), B(t) is a standard Brownian motion, is the drift, and > 0 is the volatility of the stock. If = 1.1157, = 2.0402 and the stock price on January 1 is S(0) = 13/10,

(a) Determine the probability that the stock price is below 0.66 on September 1 of that year.

(b) How does the result in a) change if you know that the stock price is equal to 0.66 on December 1 of that year?

(c) How does the result in a) change if you know that the stock price is equal to 0.66 on March 1 of that year?

(d) Determine the probability that the stock price is below 0.55 on September 1 of that year and larger than 0.99 on January 1 of next year.