I have been going through chapter 10 and 11 of the Acted material and I have been confused regarding the normal distribution that we consider for the Log normal model. For example, in Question 10, September 2014: we consider: ln(Su) ~ N[ ln(St) + mu(u-t), sigma^2 (u-t)] But in Quesion 5, October 2015 : we consider : ln(Su) ~ N[ ln(St) + (mu - 0.5 * sigma^2)(u-t), sigma^2 (u-t)] What is the reason for this difference?
The first expression you have written is the log Normal model whose SDE looks like : dS_t = S_t*((mu+0.5*sigma^2)*dt + sigma*dW_t) Hence ln(Su) ~ N[ ln(St) + mu(u-t), sigma^2 (u-t)] But for second expression , it is Geometric Brownian Motion whose SDE looks like : dS_t = S_t*(mu*dt + sigma*dW_t) Hence ln(Su) ~ N[ ln(St) + (mu - 0.5 * sigma^2)(u-t), sigma^2 (u-t)] We can equate both these Brownian motion models with use of parameterization as : Let eta = u+0.5*sigma^2 So , the SDE of lognormal model looks like : dS_t = S_t*(eta*dt + sigma*dW_t) which is same as the SDE of Geometric Distribution
