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Is anybody able to help me understanding why does $P_t(S)$ appear in the solution to the following problem; deriving the price of bond forward contracts?
Thank you
Given:

  • $r_t$, the instantaneous rate process
  • $P_t(T)$, the price of a zero-coupon bond at $t$ and expiring at $T$
  • $F_t(S,T)$, the price at $t$ of a forward expiring at $S\leq T$
  • $B_S(T)$, the price at $S$ of a coupon-bearing bond expiring at $T>t$

The payoff for going long the forward is $B_S(T)-F_t(S,T)$, and the position is costless at inception such that, in the absence of arbitrage, the forward satisfies
$F_t(S,T)=\frac{1}{P_t(S)}\mathbb{E}_t\left(e^{-\int_t^Sr_\tau d\tau} B_S(T)\right)$
where $\mathbb{E}_t$ denotes the expectation

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  • $\begingroup$ What is $F_t(S,T)$? You say is the forward expiring... you mean the forward rate? or is $F_t(S,T)$ the forward price of the coupon-bearing bond $B_S(T)$? $\endgroup$
    – Sebastian
    Nov 25 '21 at 11:54
  • $\begingroup$ $F_t(S, T)$ is the price at $t$ of the forward (or futures) contract $\endgroup$
    – bl00mb3r8
    Nov 25 '21 at 13:47
  • $\begingroup$ What contract? The underlying asset is the coupon-bearing bond, right? $\endgroup$
    – Sebastian
    Nov 25 '21 at 13:52
  • $\begingroup$ Yes, the contract $F_t(S, T)$ is a forward contract in which the underlying asset $B_S(T)$ is a coupon-bearing bond. $\endgroup$
    – bl00mb3r8
    Nov 25 '21 at 14:23
  • $\begingroup$ Is the conditional expectation taken under the risk-neutral measure or under the forward measure? When is the inception, $t=0$ or just $t$? $\endgroup$
    – Sebastian
    Nov 25 '21 at 16:58
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I assume the forward contract matures at time $S$ and $0 \leq t < S < T.$ Let $F_t(S,T)$ be the price that makes the forward contract be worth zero at time $t$. Then by the risk-neutral valuation formula we obtain $$0= \mathbb E^{\mathbb Q} \left[(B_S(T) - F_t(S,T))e^{-\int_t^S r(u)du} | \mathcal F_t \right].$$ Since $ F_t(S,T))e^{-\int_t^S r(u)du}$ is $\mathcal F_t$-measurable, $$F_t(S,T)=e^{\int_t^S r(u)du} \mathbb E^{\mathbb Q} \left[B_S(T)e^{-\int_t^S r(u)du} | \mathcal F_t \right].$$ Under the risk-neutral probability measure $\mathbb Q$ the discounted bearing-bond is a martingale, so $$F_t(S,T)=e^{\int_t^S r(u)du} B_t(T)=\frac{1}{P_t(S)}B_t(T).$$

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  • $\begingroup$ I guess there's a small typo in the first equation. It should $F_t(S,T)$ instead of $F_S(S,T)$ $\endgroup$
    – bl00mb3r8
    Nov 26 '21 at 7:36
  • $\begingroup$ Yes, it's a typo. $\endgroup$
    – Sebastian
    Nov 26 '21 at 10:43

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