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March 25th, 2013, 09:00 AM   #1
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Formula for bond price at time t

Hi everybody, I am reading a book called "Mathematical Methods for Foreign Exchange" and I am not quite sure I understand the very first (quite unexplained) equation there. It's supposed to express a price of a zero coupon bond at time t as:



where the refers to the price at time t = 0 f the obligation to pay $1 dollar at time T in the future. In my opinion, the (constant) could be something like $0.613913 for a 10-years bond at yield of 5%. I am not sure what would be the graph of . I tend to think that it models prices for ever more distant maturity dates, so if the t is in the (0, 10) years interval, the value at t = 0 should be $1 and the value at t = 10 should be those $0.613913, i.e. equal to . For those two values, the should be 0.613913 (for t = 0) and 1 (for t = 10) respectively. That makes sense. But I have a problem with the values in between. I am trying to picture the graph of the intermediate values of like this (modeling using t in (0, 1) interval):

(see http://forums.babypips.com/attachment.p ... 1364195247)

If the graph of above is correct, then the graph of looks like this:

(see http://forums.babypips.com/attachment.p ... 1364196164)

...which seems weird, because I believe it shouldn't be convex just like the graph of wasn't convex. The reasoning I have for that is that I believe the first derivative (slope) of the function as it approaches the value 1 (or $1; from right in the first graph and from left in the second graph) should be the same, because that simply expresses that the outlook an investor has - say - over the next year is a lot more "certain" that the outlook she has over a longer period of time.

Can anyone shed some light on this for me please?
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March 26th, 2013, 07:30 PM   #2
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Re: Formula for bond price at time t

To partially answer my own question, the book further models the bond prices as:



, which makes the formula above look like this:



So the fraction holds. But still, the restriction on the the bond price given by the first formula seems a little far fetched, considering how many different ways there are for bond pricing... I guess it is based off of the "continuous compound interest formula" as in here: http://cs.selu.edu/~rbyrd/math/continuous/
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