Consider $r(t)$ the spot rate of default-free interest where $B(t,T)$ represents the $T$-maturity zero-coupon bond price at time $t$. Also assume, we are taking the expectation under the risk-netural probability measure $P$. Assume $0\leq j< k.$

$E_P(e^{-\int_j^k r(t)dt})=B(0,j)$

(1) Is this expression correct?

(2) If so, why is it so?

  • $\begingroup$ If $r_t$ is the short rate, then the time $t$ price of a default-free zero-coupon bond maturing at time $T$ is given by $B(t,T)=\mathbb{E}^\mathbb{Q}\left[\exp\left(-\int_t^T r_s\mathrm{d}s\right)\mid\mathcal{F}_t\right]$. This follows directly from no-arbitrage, i.e. price = expected discounted payoff with respect to risk-neutral probability measure. If $t=0$, you get $B(0,T)=\mathbb{E}^\mathbb{Q}\left[\exp\left(-\int_0^T r_s\mathrm{d}s\right)\right]$. $\endgroup$
    – Kevin
    Apr 2 '20 at 17:38
  • $\begingroup$ @KeSchn Thanks for the comment. I agree with what you said, and that is my understanding as well. $\endgroup$ Apr 2 '20 at 19:12
  • 1
    $\begingroup$ Good to hear (: everything clear now or are there any open questions left? $\endgroup$
    – Kevin
    Apr 2 '20 at 20:05

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