Stochastic Interest Rate Models - Q15.11

Hi

I have been doing some questions on the stochastic interest rate models chapter and have a question regarding question 15.11.

I had expected the parameters for the log normal distribution of Sn to be n(mu) and n(sigma^2). I find in the answers that the second parameter is (n^2)(sigma^2).

Is somebody able to explain why this is not the case?

Thanks

AJ

 

Ok, so I've also been doing exam question 2 on page 21 of chapter 15 and I see the same parameters used in this question.

Has this got something to do with the interest rate being constant opposed to each annual rate of interest, i, being independent? Is there some explanation/proof of where these parameters come from?

 

I was rather hoping for an explanation/reasoning for using these particular parameters. The thread you've linked only confirms that the answer given is correct.

The parameters used make sense as the question, exam style Q2 (page 21), asks for the probability of 14k accumulating to 20k in 4 years which is equivalent to 14k accumulating to 15305.71 in 1 year. (as the interest rate is constant but unknown - so as you say the fixed interest rate model)

In the latter the parameters mu and sigma can be used to give the same result. If you are able to explain how the time period changes the parameters that would be great.

Thanks

 

The solution to 15.11 may make more sense after you have done CT3.
Basically, the core of the proof is:
if ln(1+i) ~ N(u,s^2) => n ln(1+i) ~ N(nu,n^2s^2)

 

Thank you for the replies. That makes more sense now.