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stochastic interest rate model

A

asmkdas

Member
please explain me the Q.15.3 in CT1-15:Stochastic Interest Rate Model

What is the probability that Sn will take the value 1.02*1.04^(n-1)?
Ans: There are n different years in which the 2% could fall and so:
Probability=n*(1/3)*(1/3)^(n-1)=n/3^n
 
There are n places the interest rate 0.02 could be appear in the list of interest rates each year:

- the 0.02 could be in the first year, then all the others are 0.04
- the first year could be 0.04, then the second year is 0.02, then all the others are 0.04
etc..etc
- the first n-1 years could be 0.04, then the last year is 0.02.

This gives n ways of getting 0.02 in one year and 0.04 in the other n-1 years.

The probability of choosing 0.02 in a given year is 1/3.
The probability of choosing 0.04 in n-1 years is (1/3)^(n-1).

Putting it all together the overall probability is n*(1/3)^n.
 
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