Hi, I am looking at interest rates that are assumed to follow a lognormal distribution over a period.
I want to calculate the probability of the interest rate falling below a specified strike price, on any date on or before Dec 1st.
Now, I know how to calculate the prob that the interest rate on Dec 1st will be at or below the strike price: integrate over the distribution using the following formula:
= normdist ( (ln(S/X) - sigma x sigma x 0.5 x t )/ (sigma x sqrt(t) ) using the excel options mean=0, stdev = 1, cdf=true.
where S = current spot interest rate
X = strike
t = time from today to Dec 1st.
however this gives me the probability of it hitting on Dec 1st, and I want the probability of it hitting on or before Dec 1st.
Is there a more sophisticated way of doing it other than calculating the probability on each day and taking the max?