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

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?

• 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 $B(0,T)=\mathbb{E}^\mathbb{Q}\left[\exp\left(-\int_0^T r_s\mathrm{d}s\right)\right]$. Apr 2 '20 at 17:38
• @KeSchn Thanks for the comment. I agree with what you said, and that is my understanding as well. Apr 2 '20 at 19:12
• Good to hear (: everything clear now or are there any open questions left? Apr 2 '20 at 20:05