Constant continuous forward rate interpolation

Question

Assume that the continuously compounded forward rate is constant between two node points. What is the interpolated discount factor between these two points?

So you have the two discount factors $D_{10}$ and $D_{12}$. What is $D_{11}$?

Answer 1

Assume the (annualised, continuously compounded) forward rate between two nodes, say $t_{10}$ and $t_{12}$, is constant, say $ f_{10,12}$, then the discount factors of the two consecutive knots will be linked as follows:

$D_{12}=D_{10}e^{-f_{10,12} \left(t_{12}-t_{10}\right)}=D_{10}e^{-2f_{10,12}}$

From which is then easy to infer the formula for $t_{11}$,

$D_{11}=D_{10}e^{-f_{10,12} \left(t_{11}-t_{10}\right)}=D_{10}e^{-f_{10,12}}$

or alternatively, you can use

$D_{11}=D_{12}e^{f_{10,12}}$

Re-first comment, we can rearrange the first equation to get f in terms of D's:

$D_{12}= D_{10}e^{-f_{10,12} \left(t_{12}-t_{10}\right)}$

$\frac{D_{12}}{ D_{10}}=e^{-f_{10,12} \left(t_{12}-t_{10}\right)}$

$\ln \frac{D_{12}}{ D_{10}}=-f_{10,12} \left(t_{12}-t_{10}\right)$

$f_{10,12} =-\frac{1}{t_{12}-t_{10}}\ln \frac{D_{12}}{ D_{10}}=\frac{1}{t_{12}-t_{10}}\ln \frac{D_{10}}{ D_{12}}$

Re-second comment, assume $s<t<T$, I assume your $\hat t=s$ in this sense. So, as per above, the $D_t$ and $D_s$ will be linked as follows:

$D_t=D_{ s}e^{-f(t-s)}$

Now I think you are assuming that f is constant across tenors:

$f=-\frac{1}{T-s}\ln \frac{D_T}{D_s}$

Substitute this f into the previous equation, and then rearrange the term in the exponent so that we can cancel e and ln:

$D_t=D_{s}e^{\frac{t-s}{T-s}\ln \frac{D_T}{D_s}}=D_{ s}e^{ln \left(\frac{D_T}{D_s}\right)^\frac{t-s}{T-s}}$

Thus,

$D_t=D_{ s} \left(\frac{D_T}{D_s}\right)^\frac{t-s}{T-s}=D_{ s} D_s^{-\frac{t-s}{T-s}}D_T^{\frac{t-s}{T-s}}=D_s^{\frac{T-t}{T-s}}D_T^{\frac{t-s}{T-s}}$