My Math Forum Formula for bond price at time t

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 March 25th, 2013, 09:00 AM #1 Newbie   Joined: Mar 2013 Posts: 5 Thanks: 0 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: $B_{t,T}= \frac{B_{0,T}}{B_{0,t}}$ where the ${B_{0,T}$ 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) $B_{0,T}$ 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 ${B_{t,T}$. 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 ${B_{0,T}$. For those two values, the $B_{t,T}$ 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 ${B_{0,t}$ like this (modeling using t in (0, 1) interval): (see http://forums.babypips.com/attachment.p ... 1364195247) If the graph of ${B_{0,t}$ above is correct, then the graph of ${B_{t,T}$ 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 ${B_{0,t}$ 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?
 March 26th, 2013, 07:30 PM #2 Newbie   Joined: Mar 2013 Posts: 5 Thanks: 0 Re: Formula for bond price at time t To partially answer my own question, the book further models the bond prices as: $B_{t,T}=e^{-r(T-t)}$ , which makes the formula above look like this: $B_{t,T}=\frac{B_{0,T}}{B_{0,t}}=\frac{e^{-r(T-0)}}{e^{-r(t-0)}}=\frac{e^{-rT}}{e^{-rt}}=e^{rt-rT}=e^{r(t-T)}=e^{-r(T-t)}$ 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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