Hi, i was wondering why we can't just substitute the values for mu, sigma & t=1/2 into the lognormal model to calculate the probability. Why doesn't that work here?
The (continuous-time) lognormal model has different parameters to the lognormal distribution that is obtained as a solution to the geometric Brownian Motion SDE, so you need to be careful about which you are given in the question. In this question, you are given the geometric Brownian Motion SDE, which has solution as shown in (ii), so you use that. This is equivalent to the distribution: log(St/S0) ~ N((mu-0.5*sigma^2)*t, sigma^2*t) The form of the (continuous-time) lognormal model is given in Section 1.1 of Chapter 10, and is differently parameterised. Here, the equivalent distribution would be: log(St/S0) ~ N(mu*t, sigma^2*t)
