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I'm trying to understand relationship between short rates and forward rates

Let $f(t,T)$ is forward rates compounding at $T$ as seen from $t$, and $r(t)$ just a short rate

For a Zero Coupon Bond paying \$1 at maturity $T$ following relations holds:

$$P(t,T) = e^{-\int_t^Tf(t,s)ds}$$ $$P(t,T) = E[e^{-\int_t^Tr(s)ds}|\mathcal{F}_t]$$

Can we say that forward rates is conditional expectations of short rates? $$f(t,T) = E[r(T)|\mathcal{F}_t]$$

Does it holds under special assumption if not?

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    $\begingroup$ Think of it. Is $e^{-x}$ a function that commutes with the expectation? $\endgroup$ Commented Dec 6, 2024 at 12:44
  • $\begingroup$ @kurtg already ... even tried to expand ln(P(t,T)) as Taylor series, but expectations of powers of integrals is too much for me :( could you recommend some articles to clarify relation between forward and short rates? specifically if i have forward rates what can i say about implied/expected short rate dynamics? $\endgroup$ Commented Dec 11, 2024 at 23:40
  • $\begingroup$ To get $f(t,T)$ from $P(t,T)$ you differentiate $-\ln P(t,T)\,.$ To get $r(t)$ from $f(t,T)$ you set $r(t)=f(t,t)\,.$ More explicit relationships can be obtained when the model for $r(t)$ allows for a closed form solution of $P(t,T,r(t))$ (Markovian short rate models). The prime examples are Vasicek, generalized Hull White, CIR. See the book of Musiela and Rutkowski. $\endgroup$ Commented Dec 12, 2024 at 9:19

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