# Price a forward contract on a zero-coupon bond

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 $t_1$) ?

• Yes. Think about how you would replicate the forward contract by trading in the two zero bonds. Feb 7, 2017 at 15:02

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
----------------------------------------------------------------------------
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
-----------------------------------------------------------------------------
long ZCB t2   | -P(t, t2)          | +P(t1, t2)              | 0
short ZCB t1  | +P(t, t2)          | -P(t, t2) / P(t, t1)    |
-----------------------------------------------------------------------------
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. $K = P(t, t_2) / P(t, t_1)$.

• Shouldn't it rather be $1$ at $t_2$ for the forward as well as for the long ZCB $t_2$ ? Nov 5, 2018 at 16:27
• 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 $t_2$. Nov 8, 2018 at 7:38
• 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. Nov 11, 2018 at 19:15
• 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. Nov 11, 2018 at 21:37

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 $t$.

• 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. Feb 21, 2017 at 5:10
• The price is given by the answer. Feb 21, 2017 at 12:18
• @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? Aug 15, 2017 at 4:17
• @A.Oreo: I will say yes to both. Aug 15, 2017 at 12:57

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 $$y$$.