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Thread: Proof of short rate equivalent exponential equation

  1. #1
    Junior Member mkmath's Avatar
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    Proof of short rate equivalent exponential equation

    Hi,

    Is there a proof demonstrating the equivalency of the following equations?

    I see the following equation for the gain on a continuously compounded investment using the short rate model A LOT in finance text books.

    Proof of short rate equivalent exponential equation-screen-shot-2018-04-16-8.33.55-pm.png

    The first equation makes complete sense and is trivial to understand:

    The value of the investment at some time t+dt = the investment at t multiplied by 1+interest rate over the period dt.

    However, I cannot internalize the second equation with the integral in the exponent.

    Sorry if the question is too vague but I really don't know where to start. I tried writing out the first equation as a discrete series of compounding steps (2 steps of size dt here):

    B(2dt)=B(0)(1+r(t)dt)(1+r(t+dt)dt)

    and then trying to find a rule for exponential forms of series when the number of terms goes to infinity and dt goes to zero, but I haven't been able to work it out.

    This makes me think I'm missing something fundamental in my attempt to conceptualize exactly what is happening here.

    Any assistance is proving the equivalence would be greatly appreciated.

    M
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  2. #2
    MHF Contributor

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    Re: Proof of short rate equivalent exponential equation

    The equation B(t+ dt)= B(t)(1+ r(t)dt) is equivalent (taking the limit as dt goes to 0) to $\displaystyle \frac{dB}{B}= r(t)dt$. Integrating both sides, $\displaystyle ln(B)= \int r(t) dt+ C$, where C is an arbitrary constant, and, taking the exponential of both sides, $\displaystyle B(t)= e^{\int r(t)dt+ C}= C'e^{\int r(t)dt}$, where $\displaystyle C'= e^C$. Taking t= 0, $\displaystyle B(0)= C'e^0= C'$ so C'= B(0). In the case that B(0)= 1, C'= 1 and $\displaystyle B(t)= e^{\int r(t)dt}$. (Your textbook uses "s" for the dummy variable inside the integration where I have retained "t". It makes no difference.)
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  3. #3
    Junior Member mkmath's Avatar
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    Re: Proof of short rate equivalent exponential equation

    Great! I see that the key was to recognize the first line of your reply...and r(0)=0.

    Thank you!
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