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Math Help - Probability of interest rates falling below strike, over a period

  1. #1
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    Unhappy Probability of interest rates falling below strike, over a period

    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?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by sam13 View Post
    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?
    There is no way of doing it without knowing something like the autocorrelation function of interest rates, since each days interest rate
    is not independent of its imeadiate predecessors.

    RonL
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  3. #3
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    Lognormal Distribution

    Yes, but doesnt it help to know that the distribution is lognormal? Can a conditional probability be calculated from that?
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  4. #4
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    Autocorrelation

    In any case, I can calculate the autocorrelation etc. if needed since I have a historical time series.
    If I had the autocorrelation, how would I compute this probability?
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