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Distribution of Log normal model for securities

G

George Philip

Active Member
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
 
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