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Setup

I read that when simulating forward curves $(r_t(s_i))_i$ at some future time $t>0$, one is supposed to center them not around $F(0;t,t+s_i)$, but around $$\mathbb E^{\mathbb Q_{t}}[r_t(s_i)],$$ which is not equal to $F(0;t,t+s_i)$.

Question

Why is that the case? What makes the numeraire $B(\cdot,t)$ so special? Is there simple intuitive reason why we choose $B(\cdot,t)$ over $B(\cdot,t+s_i)$ for example?


Notation (most probably unnecessary, but I'll still state it just in case I used non-standard notation)

Given some financial market $\mathcal S=(S_0,S_1,\dots,S_n)$ of tradeable assets $S_i$ we define the T-forward measure $\mathbb Q_T$ in such a way that $\frac{S_i(\cdot)}{B(\cdot,T)}$ is a $\mathbb Q_T$-martingale, where $B(t,T)$ is the prize at time $t$ of a T-zero coupon bond.

Now let $r_t(s):=\frac{1}{s}\int_t^{t+s}r(u)du$ be the average interest rate over $[t,t+s]$, then we assume that $\mathbb E^{\mathbb Q_{t+s}}[r_t(s)] = F(0;t,t+s)$, where $F(u;t,t+s) = \frac{r_u(t+s)(t+s)-r_u(t)t}{s}$ is the continuously compounded (from $t$ to $t+s$, as seen from time $u\leq t$) forward rate.

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  • $\begingroup$ Question is a bit unclear, you can remove some stuff and get the point directly I think. $\endgroup$ – byouness May 16 '18 at 20:21
  • $\begingroup$ I'm new to the field and not sure which parts of the notation are well established and which are not. What would you remove? $\endgroup$ – Peter May 16 '18 at 21:35

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