I'm trying to calculate the price of a forward contract on a zero-coupon bond (ZCB). The forward contract matures at $t_1$ and the ZCB matures at $t_2$. So is the price of the forwards contract just the ratio of (price ZCB that matures at $t_2$) / (price ZCB that matures at $t_1$) ?
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2$\begingroup$ Yes. Think about how you would replicate the forward contract by trading in the two zero bonds. $\endgroup$– LocalVolatilityFeb 7, 2017 at 15:02
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3 Answers
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Another way to obtain this result is, as I mentioned in the comment, to think about how you would replicate the forward contract. It has the following cash-flow structure:
type | t | t1 | t2
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forward | 0 | +P(t1, t2) - K | 0
Here, I also use $P(t_1, t_2)$ to denote the time $t_1$ price of the zero coupon bond with maturity in $t_2$. $K$ is the fair delivery price of the forward contract.
You replicate this contract by taking a long position in the zero-coupon with maturity $t_2$ and financing the purchase by selling the zero coupon bond with maturity in $t_1$ for a notional that yields a current cash-inflow of $P(t, t_2)$. You get
type | t | t1 | t2
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long ZCB t2 | -P(t, t2) | +P(t1, t2) | 0
short ZCB t1 | +P(t, t2) | -P(t, t2) / P(t, t1) |
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total | 0 | +P(t1, t2) | 0
| | -P(t, t2) / P(t, t1) |
The portfolio has the same cash-flows as the forward in both $t$ and $t_2$. It has the same random cash-flow in $t_1$ ($+P(t_1, t_2)$) and thus the non-random cash-flows at this time also have to agree, i.e. $K = P(t, t_2) / P(t, t_1)$.
answered Feb 8, 2017 at 9:11
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$\begingroup$ Shouldn't it rather be $1$ at $t_2$ for the forward as well as for the long ZCB $t_2$ ? $\endgroup$– OlórinNov 5, 2018 at 16:27
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$\begingroup$ No - that's all fine. Don't understand your question regarding the forward but the position in the zero coupon bond is unwound upon the maturity of the forward contract - so there are no more cash-flows in $t_2$. $\endgroup$ Nov 8, 2018 at 7:38
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$\begingroup$ I am saying that long ZCB t2 gives you +1 at t2, that you don't take in account. And that should modify what is happeing at t2 for the forward as well, to be still able to apply your non arbitrage argument. $\endgroup$– OlórinNov 11, 2018 at 19:15
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$\begingroup$ As I wrote before, the position in the zero bond is unwound in $t_1$. Thus, even though the zero bond has a cash-flow in $t_2$ of one, it is not relevant to this argument. Everything is correct here. $\endgroup$ Nov 11, 2018 at 21:37
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$\begingroup$ (Sorry for the delay, I just realized while browsing through old stuff that I forgot to answer.) At t for the long-short position it's not -P(t,t2)+P(t,t2) but -P(t,t2)+P(t,t1) and this is not 0 in general and hence as not the same cashflows at t=0 than the foward zero-coupon position. The correct idea is to be at t long a zc expiring at t2 and short a quantity P(t,t2)/P(t,t1) of zc's expiring at t1. This indeed cost 0 at t etc etc. $\endgroup$– OlórinJul 21 at 19:56
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Let $E^{t_1}$ be the expectation operator under the $t_1$-forwad probability measure $Q^{t_1}$, which takes the bond price process $\{P(t, t_1), \, 0\le t \le t_1\}$ as the numeraire. Then, the price of the forward contract, at time $t$, where $0\le t \le t_1$, is given by \begin{align*} E^{t_1}\big(P(t_1, t_2)\mid \mathcal{F}_t\big) &= E^{t_1}\left(\frac{P(t_1, t_2)}{P(t_1, t_1)}\mid \mathcal{F}_t\right)\\ &=\frac{P(t, t_2)}{P(t, t_1)}, \end{align*} as $\left\{\frac{P(t, t_2)}{P(t, t_1)}, 0\le t \le t_1\right\}$ is a martingale under the $t_1$-forwad probability measure $Q^{t_1}$. Here, $\mathcal{F}_t$ is the information set at time $t$.
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$\begingroup$ Thank you for your replies. I'm not sure if I was calculating the (price ZCB that matures at t2t2) correctly. When the I calculated (price ZCB that matures at t2t2) / (price ZCB that matures at t1t1) I got a ratio of like 1.5. What seemed to work was multiplying the strike times the forwards contract divided by the price that matures at t1. So like K*F(t2)/P(t1) I'm sorry if my notation is terrible. $\endgroup$ Feb 21, 2017 at 5:10
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$\begingroup$ @Gordon I have question that, if it is a ZCB, so the yield of forward bond price FBP is exactly the same as forward rate? If it's not a ZCB, they are not always same? Suppose we have the relation $$B(t_1,t_2) = G(y_{t_1}),$$ $y_{t_1}$ is the yield of $B(t_1,t_2)$ at $t_1,$ then we have $\textrm{FBP} _0 = G(y_0).$ Is it right? $\endgroup$– A.OreoAug 15, 2017 at 4:17
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Elaborating on @Gordon's answer. Denoting with $P(t,T,S)$ the $T$-forward price of a zero-coupon bond maturing in $S$ ($S \geq T$), we have the relation:
$$ P(t,T) P(t,T,S) = P(t,S) $$
which, in terms of spot ($L$) and forward ($F$) simply-compounded yields, which are related by the relation:
$$ (1+\tau(t,T) L(t,T))(1+\tau(T,S)F(t,T,S))=(1+\tau(t,S)L(t,S)) $$
from which the standard forward yield, defined as the (see e.g., Brigo-Mercurio (2001) chap 1
$$ F(t,T,S) = \frac{1}{\tau(T,S)} \left(\frac{P(t,T)}{P(t,S)} - 1\right) $$
can be interpreted as the appropriate yield matching the forward price of a zcb
$$ P(t,T,S) = \frac{1}{1+ \tau(T,S) F(t,T,S)} $$
where $\tau(x,y)$ is the time measure between times $x$ and $y$.
